CT2010 Abstracts of talksThe unification of Mathematics via Topos Theory

Olivia CaramelloI will propose a new view of Grothendieck toposes as unifying spaces in Mathematics beingable to serve as 'bridges' for transferring information between distinct mathematical theories.This approach, first introduced in my Ph.D. dissertation, has already generated ramificationsinto distinct mathematical fields and points towards a realization of Topos Theory as a unifyingtheory for Mathematics.In the lecture, I will explain the fundamental principles that characterize my view of toposes asunifying spaces, and demonstrate the technical usefulness of these methodologies by providingapplications in several distinct areas.

Advances in semi-abelian categorical algebraGeorge Janelidze

"Semi-abelian categorical algebra" attempts to extend constructions and results known forgroups, rings, associative and Lie algebras, and other similar algebraic structures to the contextof abstract semi-abelian categories. Unlike the abelian case of modules over rings versusabelian categories, such an extension requires introducing additional elementary categoricalaxioms in the form of exactness properties and even introducing new universal constructions -since:(a) every variety of groups with multiple operators is semi-abelian, and one certainly does notexpect every such variety to be as "nice" as, say, the category of groups;(b) although all the above-mentioned classical algebraic structures form semi-abeliancategories, clearly there are many specific first order categorical properties that distinguishthem from each other.The aim of this talk is to list those possible additional axioms and constructions, and to explainhow some of them imply/allow others. This involves beautiful work of several young and lessyoung mathematicians present here at CT2010, but the list seems to be still far from beingcomplete... Very briefly - at the level of key words - its main items are:1. All extensions with abelian kernels must be torsors.2. The Huq commutator must coincide with the Smith-Pedicchio commutator.3. Every internal star-multiplicative graph must be an internal groupoid.4. Strong protomodularity in the sense of D. Bourn.5. Internal object actions must be representable; this is not true e.g. for rings and suggests aweaker condition of action accessibility.6. New properties of pullback functors between "points".7. There is a monoidal category structure that measures non-abelianness.8. Semisimple classes in the sense of radical theory must be hereditary.More properties should be suggested by most recent work of D. Bourn, but I am not yet readyto discuss them.

The Newton product of chain complexesAndré Joyal

The normalised chain complex functor in the Dold-Kan correspondance is bilax monoidal by atheorem of Aguiar and Mahajan. By transporting the tensor product of simplicial abeliangroups along the Dold-Kan correspondance, Birgit Richter has recently introduced a newtensor product of chain complexes, called the large tensor product.

We shall give an explicit description of the large product and explains how it is related toNewton's finite difference calculus.

The Hopf algebra of Möbius intervalsMatias Menni

We recall the definition of Möbius category and Leroux's characterization in terms of incidencealgebras [1]. But the main purpose of the talk is to give an objective proof of a 1980's resultdue to Lawvere, which was unpublished until very recently [2]. This result states that thereexists a Hopf algebra H, such that for every incidence algebra X there exists a canonical mapfrom X to the dual of H.The key ideas underlying Lawvere's result throw new light on the concept of Möbius cartegorythat, in particular, leads to a new characterization.

References[1] P. Leroux. Les categories de Möbius. Cahiers de Top. et Geom. Diff. Cat., 16, 1975[2] F. W. Lawvere and M. Menni. The Hopf algebra of Möbius intervals, TAC, 2010

Lax monoidal categories and higher operadMark Weber

We begin with a survey of the theory of lax monoidal categories and the operads they give riseto. This process enjoys various functorialities, which together with the Formal Theory ofMonads, explains why monad distributuve laws are so ubiquitous in the operadic approaches tohigher category theory. One of the desired outcomes of this work is to describe the higherdimensional analogues of the Gray tensor product of 2-categories. In the second part of the talk,the general analogues of the funny tensor product of categories, which is what one has whenone ignores the highest dimensional data of the Gray tensor product, will be presented.The general theory being discussed involves *lax* monoidal structures on a given category, soapriori it isn't obvious whether the sought-after generalised Gray tensor products are "genuine"tensor products, lax, or "genuine up to homotopy". To allow all possibilities to be considered,it is important to organise closed lax monoidal structures into nice categories, and tocharacterise the genuine closed monoidal structures abstractly among them. The third part ofthe talk will discuss some work in progress intended to address this issue.

From analysis to living systems through category landAndrée Ehresmann

Initially an analyst (under the direction of Choquet), I came to category theory through theproof-reading with Charles Ehresmann of his important paper "Gattungen von lokalenStrukturen" (1957), and I then tried to adapt its constructions in analysis, which requiredintroducing notions such as (in more modern terms): - (internal) 'partial' actions of a categoryor of a presheaf of categories, and their 'completion' into global ones, for constructing'distructures', an infinite dimensional generalization of Schwartz vector distributions, withapplications to control and optimization problems (1960-65); - closure on a category (1962-1964) for constructing a 'convenient' category of infinite dimensional differentiable manifolds,and different concrete internal categories and groupoids for extending fibre bundle theory inthis infinite dimensional frame. These notions paved the way of my future research withCharles, of which I'll recall some results: - completion theorems (up to isomorphism) leading tothe explicit construction, by induction, of the prototype and of the type associated to a sketch(1972), - introduction and construction of monoidal closed structures on categories of internalcategories and, more generally, on categories of sketchable categories (1969-1972), - theory ofmultiple categories, with construction of several monoidal closed structures on the category of

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n-fold categories (1974-79). For the last 25 years I have been developing (with Jean-PaulVanbremeersch) the theory of Memory Evolutive Systems, a categorical model for complexnatural self-organised systems, e.g. living systems (summed up in our 2007 book). I shall showhow my former results are at the basis of this work and in particular lead to criteria for theemergence of structures of strictly increasing complexity, with an application to the formationof higher cognitive processes in our model MENS for a neural and cognitive system (2009-2010). The itinerary would not be complete without mention of my 50 years old implication inthe development and publication of the "Cahiers"; and of their Supplements (1980-83) devotedto Charles' complete works to which I added about 450 pages of comments to translate them ina more modern language, to replace them in the historical context, and to extend some results.

The number of countable models of a sentence of L1, in a Grothendieck topos

Nate AckermanAt the end of his paper on Denumerable models of complete theories, Vaught asked thequestion, “Can it be proved, without the use of the continuum hypothesis, that there exists acomplete theory having exactly Aleph-1 non-isomorphic denumerable models?” This questionhas stood to this day as one of the oldest open problems in model theory. The statement that ithas a negative answer, i.e. that no countable first order theory has exactly Aleph-1 manycountable models (assuming CH), has become known as “Vaught’s Conjecture”.Over the years much work has been done on Vaught’s conjecture and one of the mostsignificant advances towards a solution was made in Morley’s paper on The number ofcountable models. In this paper he proved that any sentence of L

1, which does not have a

perfect set of countable models must have either countably many or Aleph-1 many countablemodels.Among the many reasons why Morley’s paper represented significant progress towards aresolution of Vaught’s conjecture was that it extended the scope of the conjecture from firstorder theories to sentences of L

1, . By extending the scope of Vaught’s conjecture, Morley

brought the conjecture into the realm of infinitary logic as well as descriptive set theory andthereby opened up the use of techniques from these areas to its study.In my talk I will discuss an extension of Morley’s result to countable models of L

1, in an

arbitrary Grothendieck topos. When we move from models in the category of sets to anarbitrary Grothendieck topos, the notion of a “countable model” bifurcates into four distinctnotions which we call, purely countable, countably generated, monic countable and epi

countable. I will discuss a proof that, under 13-determinacy and a mild countability

assumption on a site (C,J), Morley’s theorem extends to the category of sheaves on (C,J), i.e.for any of the four notions of countable, any sentence of L

1, which does not have a perfect

kernel of countable models in the category of sheaves on (C,J) has at most Aleph-1 manycountable models.

Relatively Terminal CoalgebrasJiri Adamek

Terminal coalgebras of endofunctors F play an important role in the theory of systemsexpressed by F-coalgebras. Jan Rutten demonstrated in [R] that the terminal coalgebra is thecoalgebra of behaviours of states in such systems. The classical construction (dualizing that of

initial algebras in [A]) is to form the limit of the op-chain 1 F1 FF1 ... which iterates theapplication of F to the unique map F1 1 to the (trivial) terminal algebra 1.Another source of interest in terminal coalgebras stems from the first model of untyped -

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calculus presented by Dana Scott. However, Scott did not use a terminal coalgebra: rather, he

used, for a "suitable" algebra :FA A, the limit of the analogous op-chainA FA FFA ... The properties of the endofunctor F he used made it clear that F preserves this limit. Wheneverthis happens, we are going to prove that the limit carries the structure of a coalgebra for whichthe first projection (into A) is a universal coalgebra-to-algebra morphism. But in general, this

limit F A = limi< FiA carries itself an obvious structure of an algebra :F(F A) F A. We

prove that this algebra has always the same relatively terminal coalgebra as the originalalgebra.For finitary set functors relatively terminal coalgebras are always obtained in + steps: we

first form the algebra F A in steps, and then we perform the same construction on it-in thenext steps we get a limit preserved by F, thus, yielding a relatively terminal coalgebra for

F A, consequently, also for A. This generalizes the result of James Worell [W] that terminalcoalgebras of finitary functors take + steps. All accessible (=bounded) set functors F have relatively terminal coalgebras: if F preserves -filtered colimits, we need + steps of iteration. More generally, every mono-preserving,accessible endufunctor of a locally presentable category has relatively terminal coalgebrasobtained by the iterative construction. This is a new result even for (absolutely) terminalcoalgebras: the proof that a terminal coalgebra exists, presented by Michael Barr [B], was notconstructive.

References [A] J. Adámek, Free algebras and automata realizations in the language of categories,Comment. Math. Univ. Carolinae 14 (1974), 589-602 [B] M. Barr, Terminal coalgebras in well-founded set theory, Theoret.Comput.Sci. 114 (1993),299-315[R] J. Rutten, Universal coalgebra: a theory of systems, Theoret.Comput.Sci 249 (2000), 3-80 [W] J. Worrell, On the final sequence of a finitary set functor, Theoret.Comput.Sci. 338(2005), 184-199

Categorical methods for Hopf algebra theoryAna Agore

We prove that there exists a cofree Hopf algebra on every bialgebra (resp. algebra): that is theforgetful functor from the category of Hopf algebras to the category of bialgebras (resp.algebras) has a right adjoint. This is an affirmative answer to a forty years old problem posedby Sweedler. On the route the arbitrary coproducts in the category of Hopf algebras areexplicitly constructed. On the other hand the construction of the product of an arbitrary familyof coalgebras, bialgebras and Hopf algebras is also given: it turns out that the product of anarbitrary family of coalgebras (resp. bialgebras, Hopf algebras) is the sum of a family ofcoalgebras (resp. bialgebras, Hopf algebras). As a consequence the categories of coalgebras,bialgebras and Hopf algebras are shown to be complete and a complete description for limitsin the above categories is given.Moreover, necessary and sufficient conditions for a Hopf algebra map to be a monomorphism,respectively an epimorphism are given.

Composing Lawvere theoriesAndrei Akhvlediani

It has been known since [4] that strong monads provide a uniform categorical account of

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computational effects such as exceptions and interactive input/output. In [2] enriched Lawveretheories rather than monads are taken as the primitive structure for capturing those effects. Oneadvantage of the latter is that the category of Lawvere theories comes equipped with naturalsum and tensor operations, which act as a mechanism for combining the associated effects,yielding a natural method for modelling modularity of computational effects.In this talk we propose another method for combining Lawvere theories. Following the ideas of[3], we show that each Lawvere theory can be seen as a monad in Prof(Mon) - the bicategoryof profunctors internal in the category of monoids. This point of view allows us to composethem using distributive laws. We present several examples of this composition and discuss itsimportance with regards to computational effects.We also briefly touch on the applications of those ideas to the categorical approach to QuantumInformation and Computation [1].

References[1] S. Abramsky, B. Coecke, A categorical semantics of quantum protocols, in Proceedings ofthe 19th IEEE conference on Logic in Computer Science (LiCS’04). IEEE Computer SciencePress (2004)[2] M. Hyland, G. Plotkin, J. Power. Combining Effects: Sum and Tensor TheoreticalComputer Science, 357, 2006, 70-99[3] S. Lack, Composing PROPs, Theory and Applications of Categories, Vol. 13, (2004)[4] E. Moggi, Notions of computation and monads. Information And Computation, 93(1), 1991

Duality between data and process: an enriched categorical point of viewLuca Albergante

Enriched categories have been shown to be an important tool in understanding the nature ofdistributed computation ([KL10]). Of particular interest is the characterization of processes bymeans of experiments and observers ([KL91]).Database theory is an active field of research with many theoretical and technical problemsstill to be solved. As described in [JR07], category theory has been shown to be an effectiveway of dealing with various aspects of databases.This talk discusses how an enriched categorical framework can be applied to databases. To thisend, we will describe three approaches enriched over the same bicategory:– Approach 1) considers a fixed database state with views as experiments in the sense of[KL91]– Approach 2) considers variable database states and uses a view ([JR07]) as the enrichment– Approach 3) considers the sequences of SQL operations that led to a particular database stateenriched as in [KL99]Approaches 2) and 3) exhibit the process-data duality, and we will show that they arecomponents of a single mathematical object: they take part in a two-sided enrichment in thesense of [KLSS02].

References[JR07] M. Johnson, R. Rosebrugh. Fibrations and universal view updatability.Theor.Comput.Sci., 388(1-3):109-129, 2007[KL91] S. Kasangian, A. Labella. On continuous time agents. In S.D. Brookes, M.G. Main, A.Melton, M.W. Mislove, D.A. Schmidt, eds., MFPS, volume 598 of Lecture Notes in ComputerScience, pages 403-425. Springer, 1991[KL99] S. Kasangian, A. Labella. Observational trees as models for concurrency. MathematicalStructures in Comp.Sci., 9(6):687-718, 1999

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[KL10] S. Kasangian, A. Labella. Conduch property and Tree-based categories. Journal of Pureand Applied Algebra, 214(3):221-235, 2010[KLSS02] M. Kelly, A. Labella, V. Schmitt, R. Street. Categories enriched on two sides.Journal of Pure and Applied Algebra, 168(1):53–98, 2002

Type theory and homotopy: Some recent developmentsSteve Awodey

This talk surveys recent research on a newly-discovered connection between logic andtopology in the form of an interpretation of constructive Martin-Löf type theory into abstracthomotopy theory, resulting in new examples of higher-dimensional categories.Martin-Löf type theory is a framework for constructive mathematics at least as strong assecond-order logic, which has been used to formalize large parts of constructive mathematicsand for the development of high-level programming languages. It is prized for its combinationof expressive strength and proof-theoretic tractability. One aspect of the type theory that has ledto difficulties in providing semantics, however, is the intensionality in the treatment of equalitybetween terms. The research surveyed here uses homotopy to interpret intensionality byinterpreting the type theory into Quillen model categories and related structures, constructing abridge between type theory and axiomatic homotopy theory. With respect to this newhomotopical semantics, the type theory is sound and essentially complete.The (weak) higher-dimensional groupoids arising as fundamental groupoids of (abstract) spacesalso appear in type theory, providing a new way of constructing examples of such structureswith specific properties. The homotopical interpretation thus presents both new semanticaltools for the study of type theory as well as a new field of possible applications of type theoryin the study of homotopy and higher-dimensional algebra. Existing machine implementationsof type theory also suggest the possibility of computational applications in homotopy theory.

The Diaconescu theorem for 2-topoiIgor Bakovic

The investigation of sheaf-theoretical and cohomological structures associated to highercategories led Street to define the notion of a Grothendieck 2-topos in the case of strict 2-categories, which he later generalized to the case of bicategories. Street defined a bisite as abicategory supplied with a suitable notion of a Grothendieck topology and he definedGrothendieck 2-topoi as those bicategories which are biequivalent to bicategories of stacksover the bisite. The main result of his work was a bicategorical version of Giraud’s theoremproviding a characterization of Grothendieck 2-topoi in terms of bilimits, bicolimits, exactnessand size conditions.We discuss different (non)equivalent notions of regularity and exactness for 2-categories andbicategories by reviewing basic examples: the 2-category at of small categories, then the 2-category of B-indexed categories for a small bicategory B, and the 2-category of stacks over atopological space X.This is joint work with Branislav Jurco.

Higher braided operads and stabilisation hypothesisMichael Batanin

Breen and Baez-Dolan suggested the following important stabilisation hypothesis in highercategory theoryHypothesis 1. The category of n-tuply monoidal k-categories is equivalent to the category of(n+1)-tuply monoidal k-categories provided n k+2.Let V be a symmetric monoidal category. Classically one defines the category ofnonsymmetric, braided and symmetric operads in V.

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If V is also symmetric monoidal model category we introduce the category of n-braidedoperads for any 1 n . Under some more technical assumption on V we also construct a

model category of n-braided operads Olocn(V) as an appropriate localization of the category of

n-operads. Then we prove

Theorem 1 (Operadic stabilization). The model categories Olocn(V) and Oloc

n+1(V) are

Quillen equivalent provided V is a k-truncated symmetric monoidal model category andn k+2.The Breen-Baez-Dolan stabilisation hypothesis for any model of the theory of weak k-categories which has sufficiently "nice" homotopy theory is an immediate corollary from thistheorem. There are several such models in the literature (for example the category of Rezkweak n-categories) and other models are expected to be equivalent to one of them.

The nerve theorem and Grothendieck's hypothesis on homotopy typesClemens Berger

The nerve theorem gives sufficient conditions for a monad so that the category of its algebrasembeds fully faithfully into a presheaf topos. According to Grothendieck the homotopy type ofan n-truncated space should be representable by a suitably defined fundamental n-groupoid. Inthis expositary talk, I will first give some details entering into the nerve theorem, then explainin which way the nerve theorem offers a strategy to prove Grothendieck's hypothesis, andfinally discuss the present state-of-art of the subject.

On higher homotopy operations and higher categoriesDavid Blanc

Higher homotopy and cohomology operations play an important role in classical algebraictopology, but their precise definition has never been quite clear. In this talk we will describeseveral (equivalent) approaches to giving a general definition in the context of certain"categories enriched in n-stems", and explain their manifestations as differentials in spectralsequences and cohomology classes. We will also show why a good model for (1,n)-categoriescould lead to a complete set of homotopy invariants for reasonable model categories.

Torsors, herds and flocksTom Booker

We define what it means for a comonoid A in a braided monoidal category V to be a Herd; thenotion of a higher dimensional version of a herd, called a flock, is introduced.Write Cmrf(A) for the V-category of right A-comodules whose underlying objects in V have

duals, and Vf for the full subcategory of V consisting of objects with duals.

Once the obvious flock structure on Vf is understood we show how this lifts to give a V-flock

structure on Cmrf(A).

Finally we give a Tannaka duality for flocks and herds.This material is joint work with Ross Street to appear.

Centralizer and faithful groupoidDominique Bourn

Recent works [1,2] paid a special attention to the universal property of objects as the groupAutG in the category Gp of groups, or as the Lie-algebra DerA in the category K-Lie of Lie-algebras: namely, the split extension

determines, via the pulling back, a bijection between the set Gp(K,AutG) of group

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homomorphisms and the set of isomorphic classes of split extensions between G and K. Butwhat was rather awkward and uncomfortable was that this universal property apparently didnot give rise to any larger functorial process. Moreover, parallel kinds of considerations wereextended, on one side, to the non-pointed context [3] and, on the other side, to pointedcategories as those of (non unitary) rings or of associative algebras with the notion of actionaccessible category [4]. So that there was the growing feeling that there should be a generalscheme hidden behind those scattered situations.Eventually the heart of the question is concentrated in the notion of -faithful object in acategory E, where is a class of maps stable under composition. With this notion, thefunctorial process is restored, but only with respect to the class . On the other hand all theprevious scattered situations can be understood as contexts where there are enough -faithfulobjects. More precisely, these are categories D, now called groupoid accessible categories, suchthat the category GrdD of internal groupoids in D has enough -faithful groupoids, where isthe class of discrete fibrations.

References[1] F. Borceux, D. Bourn, Split extension classifier and centrality, in Categories in Algebra,Geometry and Math. Physics, Contemporary Math., 431, 2007, 85-14[2] F. Borceux, G. Janelidze, G.M. Kelly, Internal object actions, CommentationesMathematicae Universitatis Carolinae, 46, 2005, 235-255[3] D. Bourn, Action groupoids in protomodular categories, Th. and Applications ofCategories, 16, 2006, 46-58[4] D. Bourn, G. Janelidze, Commutator theory and action accessible categories, Cahiers deTop. et Gom. Diff., 50, 2009, 211-232

Some strict higher homotopy groupoids: intuitions, examples, applications, prospectsRonnie Brown

The aim is to show how the idea of `algebraic inverse to subdivision' led to a family of stricthigher homotopy groupoids more intuitive and powerful than the earlier relative and n-adichomotopy groups, through having structure in a full range of dimensions. The advantage ofstrict structures is that they help not only to understand traditional structures of such homotopygroups, such as actions and Whitehead products, but allow specific calculations of some suchgroups through calculation of richer structures.

Revisiting Differential Calculus in the light of an Uncanny FunctionElisabeth Burroni

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This is joint work with Jacques Penon.

The universal loop space operad and generalisationsEugenia Cheng

Grothendieck hypothesised that " -groupoids" could be used to model homotopy types, andmany different ways of approaching this have been proposed in recent years. I will discuss thetheory proposed by Batanin/Leinster using a generalised form of operad called "globularoperads". There are various analogies with the use of operads to study loop spaces; I will recallthe loop space operad E called “universal” by Salvatore [4], and analyse an analogous globularoperad G which can be used to recognise the fundamental -groupoid of a space. This operadwas introduced by Batanin [1] and given a different abstract treatment by Leinster [3]. I willgive a categorical proof of the universal property of E, generalise it to prove a universalproperty of G and give a general framework for universal operads acting on structures.One key to the success of operads in the study of loop spaces is the repertoire ofcombinatorially convenient operads that have been constructed for use in different situations,as the universal ones are too large to be useful for calculations. This sort of repertoire has untilnow been lacking for the globular operads used to study higher-dimensional categories.Identifying the universal property of the operad G helps us to find other suitable operads forrecognising -groupoids, as we have a simple criterion we can check. I will end my talk bydiscussing how this, combined with my previous work on comparing operadic theories of n-categories [2] enables us to use several well-known non-universal classical operads to buildanalogous non-universal globular operads. The hope is that these smaller operads will enable aproof that globular operadic -groupoids model homotopy types.

References[1] M. A. Batanin. Monoidal globular categories as a natural environment for the theory ofweak n-categories. Adv. Math., 136(1):39–103, 1998[2] Eugenia Cheng. Comparing operadic theories of n-category, 2008. Accepted in Homology,Homotopy and Applications. Also E-print 0809.2070[3] Tom Leinster. Higher operads, higher categories. Number 298 in London MathematicalSociety Lecture Note Series. Cambridge University Press, 2004[4] Paolo Salvatore. The universal operad of iterated loop spaces. Talk at Workshop onOperads, Osnabrück, 2000

Action accessibility and centralizersAlan Stefano Cigoli

Action accessible categories were introduced by D. Bourn and G. Janelidze in "Centralizers inaction accessible categories" [Cah.Topol.Geom.Differ.Categ. 50 (3) (2009) 211–232] as anatural context in which centralizers of subobjects and equivalence relations exist and arecharacterized as kernels of suitable maps. Moreover, in the same paper, the authors proved thatin actions accessible categories the following important condition holds:(*) two equivalence relations commute if and only if the associated normal subobjectscommute.The aim of this talk is to show that the existence of (well-behaved) centralizers, together withadditional conditions, characterizes those pointed protomodular categories which are actionaccessible.More precisely, we shall prove that a homological category C is action accessible if and onlyif the two conditions below hold:1. for every split extension in C:

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there exists an object Z(X,B), called the centralizer of X in B, which is the greatest subobjectof B commuting with X in A and it is a normal subobject of A;2. a split extension in C is faithful if and only if its centralizer is the zero object.We shall also introduce a new notion of centralizer, called ns-centralizer for equivalencerelation, that in pointed exact protomodular categories, together with condition (*), guaranteesthat the two conditions above are fulfilled.

On generalized multicategoriesMaria-Manuel Clementino

Barr's description of topological spaces as lax algebras for the ultrafilter monad [Ba70],together with Lawvere's study of metric spaces as enriched categories [La73], led to theintroduction of the notion of (T,V)-category (see [CH03], [CT03]). When V=Set they coincidewith T-categories of Burroni [Bu71], and so in particular multicategories are (T,Set)-categories, where T is the free-word monad. Our main goal then was to use (T,V)-categories to investigate topological descent theory andexponentiability in topological settings, based on known characterizations of effective descentand exponentiable maps using convergence. However, the study of (T,V)-categories went beyond this first goal. As generalized enrichedcategories, the study of Lawvere's notion of (Cauchy) complete enriched category could becarried on, encompassing the study of (T,V)-bimodules, of a dual (T,V)-category and a YonedaLemma. This is accomplished only when the symmetric monoidal-closed category V is apreordered category (that is, a quantale) in [CH09]. In this talk we present some steps towards the study of general (T,V)-categories, when themonad T is not assumed a priori to be cartesian, proposing a notion of (T,V)-bimodule and ofdual of a (T,V)-category. This can be applied to the case of V=Set and T=free-word monad,leading to a Yoneda structure for multicategories.This is joint work with Dirk Hofmann.

References[Ba70] M. Barr, Relational algebras, in: Reports of the Midwest Category Seminar, IV,Springer Lecture Notes in Mathematics, Vol. 137, 1970, pp. 39-55[Bu71] A. Burroni, T-categories, Cahiers Topologie Géom. Différentielle 12 (1971), 215-321[CH03] M.M. Clementino, D. Hofmann, Topological features of lax algebras, Appl. Categ.Structures 11 (2003), 267-286[CH09] M.M. Clementino, D. Hofmann, Lawvere completeness in topology, Appl. Categ.Structures 17 (2009), 175-210[CT03] M.M. Clementino, W. Tholen, Metric, topology and multicategory -- a commonapproach, J. Pure Appl. Algebra 179 (2003), 13-47[La73] F.W. Lawvere, Metric spaces, generalized logic, and closed categories, Rend. Sem.Mat. Fis. Milano 43 (1973), 135-166

A Brief Introduction to Turing CategoriesRobin Cockett

Turing categories [1] provide a broad basis for investigating computability categorically. ATuring category is a cartesian restriction category [2] with a Turing object-that is a universalweak exponential object. There is (depending on one’s choice of morphisms) an "initial"Turing category T0 which is in some sense the mother of all notions of computation. This

category is a "unitary" restriction category and this observation allows the image of its global

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sections functor :T0 Par to be completely described.

Also of interest are Turing categories with special properties. In particular, Turing categorieswith meets and joins are closely related to Di Poala and Heller's original setting [3] and allowone to reconstruct the basic results of recursion theory in a more general form.This is joint work with Pieter Hofstra.

References [1] R. Cockett, P. Hofstra. Introduction to Turing categories. Annals of Pure and Applied Logic156 (2008) 183-209[2] R. Cockett, S. Lack. Restriction Categories I: Categories of partial maps. TheoreticalComputer Science, Vol.270 (2002) 223-259Y [3] R. Di Paola, A. Heller. Dominical categories:recursion theory without elements. Journal of Symbolic Logic, Vol. 53(3) (1987) 594-6635

Combinatorial Game CategoriesGeoff Cruttwell

In 1977, Joyal showed that the set of Conway combinatorial games could be given acategorical structure. In this talk, I'll discuss some additional structure that the category ofcombinatorial games has, as well as other interesting models of the resulting "combinatorialgame categories". This is joint work with Robin Cockett and Kevin Saff.

Toposes in Semi-Algebraic GeometryNicholas Duncan

One of the key motivations for topos theory was solving problems in Algebraic Geometry, andto this end a number of toposes were constructed. Semi-Algebraic geometry is similar toalgebraic geometry in which it studies systems of polynomials, but it is different in that itconsiders inequalities as well as polynomial equations. Instead of working over an algebraicallyclosed field like the complex numbers we use real closed fields. In algebraic geometry rings serve as the basic geometric objects, and the opposite of ringhomomorphisms form the geometric maps. In semi-algebraic geometry rings play a part, but apartial order relation is also required. The different axioms for rings and for the partial ordergive rise to a number of different categories which can be used as basic geometric buildingblocks. In this talk I will give examples of toposes derived from different categories used in semi-algebraic geometry, and also various geometric morphisms between these toposes.

Coherent families of Hermitian adjunctionsJeff Egger

I will give an equivalent definition of dagger pivotal (resp., tortile, compact closed) categorywhich makes it clear that every such category is, in some sense, a category of inner productspaces.

Euler Characteristics of CategoriesThomas M. Fiore

The Euler characteristic is among the earliest and most elementary homotopy invariants. For afinite simplicial complex, it is the alternating sum of the numbers of simplices in eachdimension. This combinatorially defined invariant has remarkable connections to geometricnotions, such as genus, curvature, and area.Euler characteristics are not only defined for simplicial complexes or manifolds, but for many

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other objects as well, such as posets and, more generally, categories. We propose in this talk atopological approach to Euler characteristics of categories. The idea, phrased in homologicalalgebra, is the following. Given a category and a ring R, we take a finite projective R -module resolution P* of the constant module R (assuming such a resolution exists). The

alternating sum of the modules Pi is the finiteness obstruction o( ,R). It is a class in the

projective class group K(R ), which is the free abelian group on isomorphism classes offinitely generated projective R -modules modulo short exact sequences. From the finiteness

obstruction we obtain the Euler characteristic respectively L2-Euler characteristic, by adding the

entries of the R -rank respectively the L2-rank of the finiteness obstruction. This topologicalapproach has many advantages, several of which now follow. First of all, this approach iscompatible with almost anything one would want, for example products, coproducts, coveringmaps, isofibrations, and homotopy colimits. It works equally well for infinite categores andfinite categories. There are many examples. Classical constructions are special cases, for

example, under appropriate hypotheses the functorial L2-Euler characteristic of the proper orbitcategory for a group G is the equivariant Euler characteristic of the classifying space for properG-actions. The K-theoretic Möbius inversion has Möbius-Rota inversion and Leinster’s Möbiusinversion as special cases. We also obtain the classical Burnside ring congruences. In certain

cases, the L2-Euler characteristic agrees with the groupoid cardinality of Baez-Dolan and theEuler characteristic of Leinster, and comparisons will be made.This is joint work with Wolfgang Lück and Roman Sauer. Our preprint is available onlinehttp://arxiv.org/abs/0908.3417.

A double categorical analysis of the tripos-to-topos constructionJonas Frey

A tripos is a kind of fibration which has enough structure to allow to interpret intuitionistichigher order logic. The concept of tripos and the tripos-to-topos construction were introducedin 1980 by Hyland, Johnstone and Pitts [4], originally to give a conceptual framework for theconstruction of the effective topos [3]. Nowadays, it remains a central tool in the field ofcategorical realizability.The motivation of the presented work is to get a better understanding and in particular auniversal characterization of the tripos-to-topos construction itself. Such a characterization canbe useful for example to reduce questions about geometric morphisms between realizabilitytoposes to questions about certain cartesian functors between the associated triposes (which areusually much easier to study).Triposes and toposes form 2-categories in a natural way, so the obvious question is whetherthe tripos-to-topos construction is left biadjoint to a forgetful functor from toposes to triposes.If we choose adequate 2-categories (regular functors and cartesian functors that commute withexistential quantification as 1-cells between toposes and triposes, respectively), then the answerto this question is positive. A similar statement appears in the abstract of Pitts’ article [6], andrecent unpublished work by Rosolini and Maietti [7] gives a characterization in a generalframework which contains the tripos-to-topos construction as a special case.However, this result is not entirely satisfactory for our purposes, because the 2-categories aretoo restrictive. In order to talk about geometric morphisms and Lawvere-Tierney topologies,we need to consider 1-cells between triposes and toposes that only commute with finite limitstructure. Indeed (a bit surprisingly), the construction still works in this broader context as wasalready described in [4]. In this more general case, however, the search for a characterization iscomplicated by the fact that the construction is not 2-functorial anymore, but only oplaxfunctorial, and (op)lax concepts are generally very badly behaved (for example the horizontal

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composition of a lax functor with a lax transformation may not even be well defined).To cope with these complications, we suggest here to apply double categorical techniques.Concretely, we start from the idea that a 2-category with a designated class of 1-cells can beviewed as a double category, and this leads to natural and well behaved concepts of (op)laxfunctors and transformations. Using these oplax functors and transformations, we can define aconcept of biadjunction, which we identify as an adequate setting for the desired universalcharacterization. This approach originates from different places, variants can be found forexample in [5, 1, 2, 8].As a second contribution (besides the characterization), we give a decomposition of the tripos-to-topos construction in two steps which is motivated by the usage of 'weakly completeobjects' in [4]. The intermediate step is given by a weakened version of quasi-toposes whichwe call q-toposes. We argue that q-toposes are interesting themselves, since although they haveless structure than quasitoposes they have a powerful internal language.

References[1] A. Carboni, G.M. Kelly, D. Verity, R.J. Wood. A 2-categorical approach to change of baseand geometric morphisms II. Theory and Applications of Categories, 4(5):82–136, 1998[2] B. Day, P. McCrudden, R. Street. Dualizations and antipodes. Appl. Categ. Structures,11(3):229–260, 2003[3] J.M.E. Hyland. The effective topos. In The L.E.J. Brouwer Centenary Symposium(Noordwijkerhout, 1981), volume 110 of Stud. Logic Foundations Math., pages 165–216.North-Holland, Amsterdam, 1982[4] J.M.E. Hyland, P.T. Johnstone, and A.M. Pitts. Tripos theory. Math. Proc. CambridgePhilos. Soc., 88(2):205–231, 1980[5] P.T. Johnstone. Fibrations and partial products in a 2-category. Appl. Categ. Structures,1(2):141–179, 1993[6] A.M. Pitts. Tripos theory in retrospect. Math. Structures Comput. Sci., 12(3):265–279,2002[7] G. Rosolini and M. Maietti. Structuring models of quotients. Conference talk at "Advancesin Formal Topology and Logical Foundations" in Padova.[8] M. Shulman. Framed bicategories and monoidal fibrations. Theory and Applications ofCategories, 20(18):650–738, 2008

Monoidal quantaloids as a framework for a (bi)categorical Quantum MechanicsEmmanuel Galatoulas

From many aspects, the theory of categories enriched over a quantaloid is viewed as aparticularly straightforward generalisation of enriched category theory ([2]) when theunderlying basis of enrichement is not a (symmetric) monoidal category V but rather abicategory (e.g. [1]). Most of the enriched category theory and its calculus carries over in thecase of a quantaloid, ie. a small locally complete and cocomplete bicategory whose hom-categories are suplattices.For instance, Q-enriched categories and distributors among them (also known as profunctors ormodules) form a quantaloid denoted as Dist(Q) (if we consider functors among the Q-enrichedcategories, we get a locally (pre)ordered 2-category, denoted as Cat(Q)) which are the analogueof the categories V-Mod and V-Cat of V-enriched categories and modules (or functors)amongst them.Our interest in quantaloidal enrichmemt is not just purely categorical though. We argue thatquantaloids may provide a fruitful basis for a (bi)categorical approach to Quantum Mechanics(QM) both at the conceptual and the formal level. Such an approach has to address, amongst

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others, the issue of composite systems and this leads naturally to the question of endowing aquantaloid with a monoidal structure.Monoidal bicategories is not a novel notion (although its presentation remains a bitcomplicated). It dates back to work like that of Street and Day (e.g. [3]), where in fact themonoidal bicategory in question is the (bi)category V-Mod itself. Given that a quantaloid is asimplified bicategory as well as that Q-enrichmemt is a generalisation of V-enrichmemt, it isreasonable to deal with the monoidal structure of a quantaloid as a specialisation of the genericbicategorical notion.This is exactly the topic of the present work. We give a consistent definition of a monoidalquantaloid and we prove that the monoidal structure is naturally inherited by the quantaloidDist( Q) as well (and that even Cat( Q) becomes a monoidal 2-category). We also point tosome of the implications of these formalities for our ambient goal of elaborating a(bi)categorical QM.

References[1] I. Stubbe, Categories enriched over quantaloids: categories, distributors and functors,Theory and Applications of Categories 14, 1-45[2] G.M. Kelly: Basic concepts of enriched category theory, London Mathematical SocietyLecture Note Series, Cambridge University Press[3] B. Day, R. Street: Monoidal Bicategories and Hopf Algebroids, Advances in Mathematics129, pp. 99-157 (1997)

Lax aspects of abstract homotopy theoryRichard Garner

The tools of abstract homotopy theory-Quillen model structures, weak factorisations systems,and so on-allow us to delineate higher-dimensional structure borne by a category. Typically,one thinks these tools adequate only to the description of (weakly) invertible higher structure.The aim of this talk is to describe some situations which go beyond this. In particular, we showthat this machinery yields a satisfying account of the lax morphism classifiers of 2-dimensionalmonad theory, and of the homotopy-coherent nerve of simplicial category theory.

Higher extensions and the relative Kan propertyJulia Goedecke

We explore the concept of higher extensions in an axiomatic way, treating it from a relativehomological viewpoint, and draw comparisons between higher extensions and simplicialresolutions. We start with very few axioms on the class of extensions, which nevertheless givefairly strong results such as "being a higher extension is symmetric" and "an augmentedsimplicial object is a resolution if and only if all its truncations are higher extensions". Thislast result can be viewed as saying "resolutions are infinite-dimensional extensions" or "higherextensions are finite-dimensional resolutions", which is indeed how they are used inpractice.Adding a fairly week cancellation axiom and a relative version of image factorisationintroduced by Tamar Janelidze, we prove a relative version of the fact that a regular category isMal'tsev if and only if every simplicial object is Kan. The different steps in this proof need avaryingly strong set of axioms, and when using a stronger cancellation axiom which impliesthat all split epis are extensions, a stronger result is obtained even without the relative imagefactorisation, using only axioms which go up to higher dimensions.This is joint work with Tomas Everaert and Tim Van der Linden

Weak Crossed BiproductsRamon Gonzalez Rodriguez

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In this talk we present the general theory and universal properties of weak crossed biproducts.We prove that every weak projection of weak bialgebras induces one of these weak crossedstructures. Finally, we compute explicitly the weak crossed biproduct associated to a groupoidthat admits an exact factorization.

Pullback functors between points, and internal homology and cohomologyJames Gray

A point over an object B in an abstract category is a split epimorphism into B with a specifiedsplitting. As initiated by D. Bourn, the pullback functors between the categories of points canbe used to define various exactness properties, including Bourn protomodularity, permutabilityof internal equivalence relations, additivity, and others. We introduce a new kind of internalhomology and cohomology of objects in a finitely complete pointed category, and describetheir relationship with the left the right adjoints respectively of those pullback functors. Wealso consider the problem of existence: while the left adjoints are well known to exist wheneverthe ground category has pushouts, the problem of existence of the right adjoints is highly non-trivial, and so far we are only able to give some simple reformulations, examples and counter-examples. Finally, we briefly discuss the relationship between what we call internal homologyand cohomology and some well known constructions of homological algebra.

Structures as N-regimes in algebraic universesRene Guitart

Iterating the icon constructionNick Gurski

Pseudonatural transformations are the 2-cells in the traditional 3-dimensional structure whoseobjects are bicategories. Since these compose in an associative and unital manner only up toisomorphism, they are not suitable top-dimensional cells in a 2-dimensional structure whoseobjects are bicategories. The appropriate 2-cells for this job are the icons of Steve Lack. I willexplain how icons can be constructed in a very general setting as the appropriate notion of 2-cell in one approach to 2-dimensional enriched categories. This construction provides a goodframework for viewing the Stabilization Hypothesis iteratively, as well as encompassing somewell-known examples.This is joint work with Eugenia Cheng.

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Split structures in topologyDirk Hofmann

In this talk we will report on further progress on our work on topological objects as categories.A great source of inspiration for us is the work of R. Rosebrugh and R.J. Wood [1,2] onconstructive complete distributive lattices where the authors employ elegantly the concepts ofadjunction and module in their study of ordered sets. Both notions (suitably adapted) areavailable in topology too, and consequently we study various kinds spaces in a similar spirit:* Relative to a choice of modules, we consider spaces which admit all colimits with weightin . Here we follow closely the classical case of enriched categories, but also indicate crucialdifferences.* In continuation of the point above, we introduce -distributive and -algebraic spaces.Analogously to [1,2], we show that the category of -distributive spaces and -cocontinuousmaps is dually equivalent to the idempotent splitting completion of a suitable category ofconvergence relations. We discuss examples and explain how these results entail known andnot-so-known(?) duality theorems in topology.* Finally, we study properties and structures of the resulting categories, in particular monoidal(closed) structures.

References[1] R. Rosebrugh, R.J. Wood, Constructive complete distributivity IV, Appl. Categ. Structures2 (1994), 119–144[2] R. Rosebrugh, R.J. Wood, Split structures, Theory Appl. Categ. 13 (2004), 172–183

Relative modularity and semi-left-exact reflectionsTamar Janelidze

It is known that: (a) the reflection of a variety C of universal algebras into a subvariety X in Cis not semi-left-exact in general; (b) if C is congruence modular, then the semi-left-exactnesscondition restricted to pullbacks of regular epimorphisms always holds - and the same is truewhen C is an abstract Barr exact category and X a (suitably defined) Birkhoff subcategory inC [1]. We extend the result (b), some related results, and the concepts involved, to a contextwhere the class of all regular epimorphisms is replaced with an arbitrary class E of regularepimorphisms satisfying certain conditions. In particular, (C,E) can be any relative semi-abelian category in the sense of [2].

References[1] G. Janelidze and G. M. Kelly, Galois theory and general notion of central extension,Journal of Pure and Applied Algebra, 97 (1994) 135-161[2] T. Janelidze, Relative semi-abelian categories, Applied Categorical Structures, 1(1) (2009)373-386

Characterization of reflective subcategories of presheaves whose reflection preservesfinite products

Panagis KarazerisWe study conditions that imply the preservation of finite products by left Kan extensions offunctors into cocomplete categories, along the Yoneda embedding (the guiding example beingthe categorical realization of a simplicial set). As a corollary we show that the left Kanextension of a functor from a small category into a Grothendieck topos preserves finiteproducts iff the functor is sifted-flat in the internal logic of the topos. As a further corollary wecharacterize reflective subcategories of presheaves whose reflection preserves finite products.

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Generalised Span ConstructionsToby Kenney

In the category of spans on a category with pullbacks, composition of morphisms is bypullback. We look at the properties of pullback that make this into an appropriate way tocompose spans, and consider other possible ways to compose spans. We use our generalisedspan construction to iterate the span construction, and obtain comparisons between the iteratedspans.This is joint with D. Pronk.

Categorification of Hopf algebras of rooted treesJoachim Kock

I will exhibit a monoidal structure on the category of finite sets indexed by P-trees, for afinitary polynomial endofunctor P. This structure categorifies the monoid scheme (heremeaning `functor from semirings to monoids') represented by (a P-version of) the Connes-Kreimer bialgebra from renormalisation theory. (The antipode arises only after base changefrom N to Z.) The multiplication law is itself a polynomial functor, represented by three easilydescribed set maps, occurring also in the polynomial representation of the free monad on P.(Various related Hopf algebras that have appeared in the literature result from varying P.)The construction itself is not difficult, so most of the talk will be spent introducing theinvolved notions: after some comments about combinatorial Hopf algebras and the Connes-Kreimer Hopf algebra in particular, I will recall some notions from the theory of polynomialfunctors. These enter in two different ways: one is the `operad' sort of way (where the naturaltransformations are cartesian), employed for talking about trees. The other is the`categorification-of-polynomial-algebra' aspect (where natural transformations are not requiredto be cartesian): the distributive category of polynomial functors (in tree-many variables) is the`coordinate ring' of the `affine space' given by tree-indexed finite sets.

Hopf monadsSteve Lack

For any braided monoidal category C and any monoid M in C , there is a monad on C givenby tensoring with M , whose algebras are objects of C equipped with an action of M . Write

CM for the category of such algebras, and U:CM C for the forgetful functor.

There is a bijection between monoidal structures on CM for which U is strict monoidal, andbialgebra structures on M (Pareigis). As observed by Moerdijk, there is a further equivalentcharacterization of this situation: the endofunctor part T of the monad is opmonoidal, in thesense that there are natural coherent morphisms X,Y:T(X Y) TX TY which are

compatible with the unit and multiplication of the monad. A monad T equipped with such has variously been called a Hopf monad, a bimonad, an opmonoidal monad, and a comonoidalmonad. Such monads can be considered on any monoidal category, not necessarily braided.I will describe various extensions of these results, allowing a treatment not just of bialgebrasbut of Hopf algebras, weak bialgebras, and weak Hopf algebras.This is based on joint work with Alain Bruguières and Alexis Virelizier, with DimitriChikhladze and Ross Street, and with Gabriella Böhm and Ross Street.

Axioms and Models for Concrete HomotopyFrancois Lamarche

By concrete homotopy we mean homotopy based on a path functor, as in the original modelfound in the category of k-spaces. The point of the axiomatics is not that there are manymodels (right now we know of only three, which we will call here a, b and c ) but that the

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construction of the higher homotopy structure of an object in such a model is extremely simpleand that models b and c have remarkable properties.So to be a model a finitely complete category C is required to have– A class of maps F, the fibrations which obey standard properties: closure under pullback andcomposition, with projections and isos being fibrations.– A path endofunctor P with the usual two natural projections to the identity endofunctor andunit path inclusion from the identity, wich the requirement that PX X X is a fibration. Theessential point is that this functor is fibered w.r.t. F and that the above map is a fibration.These suffice to define a homotopy between maps, and we need additional axioms to say thatPX is a groupoid modulo homotopy and that any fibration is equipped with an action of it,with the usual equations holding modulo homotopy. Thus in model a fibrations are justHurewicz fibrations and P is a suitably fibered version of the usual path functor. In general P isnot required to have a left ("cylinder") adjoint, and it is not the case for models b, c. The factthat it is fibered allows an easy construction of the groupoid 1 for any fibration and since this

groupoid is internal inside the world of fibrations, the 1 construction can be just iterated to

give the higher homotopy structure. Thus the homotopy higher groupoid of X is not a set-groupoid, but something living in a suitable category of "covers" of X .Models c, d are based on small categories. First a path functor is defined in Cat, in whichobjects are zig-zags of maps, and morphisms suitably defined "bimodules". This constructionis similar but distinct from the “Hammock” construction given by Dwyer-Kan. Its naturalnotion of concatenation of paths is unfortunately not functorial. We know of two ways toremedy this:– In b concatenation is tweaked, which makes the unit valid only up to homotopy. Fibrationsthere are a kind "generalized (cleft) Grothendieck bifibrations".– In c the natural—in an intuitive, not technical sense—order structure on P is quotiented out,which makes the fibrations ordinary (cleft) Grothendieck fibrations.Thus constructions b, c give a new approach to homotopy in Cat, much more concrete (and lesstechnical) than the Quillen model structure defined by Thomason in 1980. In addition fibrationsin model c are exponentiable, which allows the interpretation of dependent types and thelambda-calculus, with P obeying the necessary axioms to interpret the Martin-Löf identitypredicate. This is being worked out by my student Robert Hein.Moreover this model has the counterintuitive property that path composition is strictlyassociative, with a strict unit. The first offshoot of this surprising result is a conjecture thatgeneralizes the well-known theorem that for a category X the groupoid 1X is its universal

associated groupoid: we formulate a conjecture that holds for any n.

Coalgebras, Monads and Dynamic ModalitiesR.A. Leal

Coalgebras for an endofunctor B:Set Set have proved to be a useful model of observablebehaviour for many types of state-based systems (see [4] for an overview). Coalgebraic modallogic (see e.g. [3]) can be described as the generalisation of modal logic over Kripke frames tomodal logic over B-coalgebras. Here, a modality for B-coalgebras is a natural transformation

:Set[-,2] Set[B(-),2]. Dynamic modal logics are modal languages in which a collection oflabelled modalities , ,... are used to reason about state change resulting from certain"actions" , ,... L; intuitively reads: "there is an execution of after which, holds".Moreover, the actions are usually assumed to carry an algebraic structure. A well knownexample is given by Propositional Dynamic Logic (PDL) [1], where modalities are labelled byprograms and the algebraic structure comes from the program operations: sequentialcomposition ( ; :after executing , execute ), choice ( : execute either or ), iteration

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( *: execute some finite number of times). In categorical terms, a PDL structure is a

coalgebra X (PX)L, where P is the covariant powerset functor, PDL modalities are labelled(Kripke) P-modalities, and the program operations form an algebra on the set of relations on X.Our aim is to place dynamic modal logics in a categorical setting. In particular, we wish tounderstand the interplay of algebraic and coalgebraic structure and its role in axiomatisations.In our view, dynamic modal logics are modal logics where actions are interpreted as B-

coalgebras. Dynamic modalities , L, are modalities for BL-coalgebras. These modalitiesarise from a uniform construction of labelling B-modalities. An algebraic structure onSet[X,BX] induces the axioms. For example, when B is a monad then Kleisli compositiongives rise to a generalised form of sequential composition (as observed in [2]). Our mainobservation is that the PDL axiom for sequential composition ; holds moregenerally when the adjoint :B Set[Set[-,2],2] of the underlying B-modality is a monadmorphism. Going further we can show that axioms like can be obtainedwhen B= UB' for some monadic functor U. In general, dynamic coalgebraic modalities andtheir axiomatisations seem related to the enrichment of the category of Bcoalgebras and to thelifting of a suitable fibration to the Kleisli category of B. This talk presents work in progress.

References[1] R. Goldblatt. Logics of Time and Computation. Number 7 in CSLI Lecture Notes. Centerfor the Study of Language and Information, 2nd edition, 1992[2] E. Moggi, Notions of Computation and Monads. Information and Computation Vol. 93,(1991), pp. 55–92[3] D. Pattinson. Coalgebraic modal logic: soundness, completeness and decidability of localconsequence. Theoretical Computer Science 309:177-193, 2003[4] J.J.M.M. Rutten, Universal coalgebra: a theory of systems, Theoretical Computer Science249(1), 2000, pp. 3-80

Magnitude and diversity: how an invariant from category theory solves a problem inmathematical ecology

Tom LeinsterIn mathematics, many objects come with a canonical notion of size: sets have cardinality,vector spaces have dimension, topological spaces have Euler characteristic, and so on. Acategorical approach illuminates the connections between these invariants and throws up atleast two more, the Euler characteristic of a category and the magnitude of a metric space.In ecology, on the other hand, there are some long-standing questions about biodiversity. Howshould one quantify it? And, if we are given a list of species from which to build acommunity, in what proportions should the species be represented in order to maximize thecommunity's diversity?As we shall see, the answer to the maximum diversity question is surprisingly categorical - inboth senses of the word.

Hopf algebra generalisationsIgnacio Lopez Franco

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Higher Categorical Structures from Intensional Type TheoryPeter LeFanu Lumsdaine

Category theorists know that identity is not a simple question. Two objects may be isomorphic,but it is not enough just to know this; one must keep track of the isomorphisms involved. On aset, equality is just a relation; but in a category, the isomorphisms form a groupoid, andgenerally a (weak) n-category has a (weakly) groupoidal sub-n-category of equivalences,which may contain non-trivial higher dimensional structure.In traditional foundations, strict (relational) equality has long the primitive concept, withweaker/higher equivalences as derived notions. Martin-Löf's Intensional Type Theory, underthe "propositions as types" paradigm, expresses the identity relation on a type A as a dependenttype IdA(a,b), governed by a version of the Lebniz equality rule. Unexpectedly, this brought

the possibility of seeing types not just as "sets", but as something more like weak highergroupoids, with Id-types as their hom-higher-groupoids; the Leibniz rule ensures that all logicalconstructions then respect the appropriate "equivalence" of each type.Much recent work has gone into making this idea precise. On the one hand, simplicial sets,strict higher groupoids, and various related categories have been shown to give "homotopy-theoretic" models on ITT; dually, various higher-categorical structures have been constructedfrom the theory's syntax. In this talk, I will discuss recent developments in the latter direction.

Interior Operators and Topological CategoriesJoaquín Luna-Torres

The categorical notion of closure operators has unified various important notions and has led tointeresting examples and applications in diverse areas of mathematics (see for example, thework of Dikranjan and Tholen). For a topological space it is well-known that the associatedclosure and interior operators provide equivalent descriptions of the topology, but this is nottrue in general. Consequently, it makes sense to define and study the notion of interioroperators I in the context of categories.In this talk we introduce interior operators in a category C with a fixed class M ofmonomorphisms in C closed under composition in such a way that C is M-complete and theinverse images of morphisms have both left and right adjoint. Then we construct a concretecategory CI over C which is a topological category and discuss some of their prop- erties.

Furthermore, we provide some examples, such as: Kuratowski interior operator, Grothendieckinterior operator, interior operators on Grothendieck topos and interior operators on thecategory of fuzzy topological spaces.

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Pseudomonads: no iteration versionFrancisco Marmolejo

In his book Algebraic Theories, E. Manes has the following exercise:

providing thus a version of a monad in which there is no need to iterate the functor S. In arecent article, "Monads as Extension Systems—No Iteration is Necessary", such a descriptionis extended to the corresponding algebras, leading furthermore to a description of distributivelaws and wreaths in which there is no iteration of the functors involved.In an attempt to give a similar no-iteration version for pseudo-distributive laws betweenpseudomonads, we show how the description for monads and their algebras extends to asimilar description for pseudomonads and their algebras. In the case where the pseudomonad islax idempotent, this description is given in terms of Kan Extensions.This is joint work with R.J. Wood.

On naturally Mal'tsev, Mal'tsev and weakly Mal'tsev categoriesNelson Martins-Ferreira

In the context of finite limits the notion of naturally Mal'tsev category, introduced by P.T.Johnstone in [1], characterizes those categories for which every internal reflexive graph isnaturally equipped with a canonical groupoid structure. Also a well known property of Mal'tsevcategories [2], sometimes even used as a definition, is that every internal reflexive relation isalready an equivalence relation. The notion of weakly Mal'tsev category was introduced in [3]with the purpose of having a context where every internal reflexive graph admits at most onemultiplicative structure. This new notion is weaker than the notion of a Maltsev category: in aweakly Mal'tsev category only those internal reflexive relations r:R X X, where r is a strongmonomorphism, need to be equivalence relations, and, unlike in a Mal'tsev category, not everyinternal category is an internal groupoid. This raised the following problem: characterize thoseweakly Mal'tsev categories for which an internal category is already and internal groupoid. Weshow that in the context of a weakly Mal'tsev category with kernel pairs and equalizers, everyinternal category is an internal groupoid if and only if every internal preorder is anequivalence relation. In the context of a variety of universal algebras, this latter condition-thatreflexivity and transitivity together imply symmetry-is proved to be equivalent to thequasivariety being n-permutable, for some n. References[2] A. Carboni, M. C. Pedicchio, and N. Pirovano, Internal graphs and internal groupoids inMal'cev categories, Proceedings of Conf. Category Theory 1991, Montreal, Am. Math. Soc. for

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the Canad. Math. Soc., Providence, 1992, pp. 97-109[3] N. Martins-Ferreira, Weakly Mal'cev categories, Theory Appl. Categ. 21 (2008), no. 6, 91-117

Enrichment inherits cyclicity, but does not inherit symmetryMicah Blake McCurdy

Kimmo Rosenthal [1] showed, for V a symmetric *-autonomous category and X a V-category,the category V-Prof(X,X) of V-endoprofunctors is *-autonomous. He shows that, while V-Prof(X,X) is not symmetric, it is cyclic, which can be thought of as a faint sort of symmetry.We describe a generalization of this result, in the following ways:1) We replace *-autonomous categories with linearly distributive categories; which wereintroduced by Cockett and Seely [2] and shown to be the suitable notion of "*-autonomouscategory without negation". Following Blute, Ruet, and Lamarche [3] and Maltsiniotis [4], weintroduce a suitable notion of "cyclicity" for linearly distributive categories.2) We remove the restriction to endoprofunctors; producing instead a linear bicategory (seeCockett, Koslowski, and Seely [5]) of V-profunctors; we adapt the notion of cyclicity from 1)to this new setting.3) We replace the symmetric monoidal category V with a cyclic linear bicategory W, to obtainin the end:If W is a cyclic linear bicategory, then W-prof is a cyclic linear bicategory.For enrichment, therefore, we see that the suitable level of commutativity is neither symmetry,nor braidedness, but cyclicity.

References[1] K. Rosenthal. *-Autonomous Categories of Bimodules, JPAA 97-2, 1994[2] R. Cockett, R. Seely. Weakly Distributive Categories, JPAA 114, 1997[3] R. Blute, P. Ruet, F. Lamarche. Entropic Hopf Algebras and Models of Non-commutativeLogic, TAC 10, 2002[4] G. Maltsiniotis. Traces dans les Categories Monoidales, Dualite, et Categories MonoidalesFibrees, Cah. de Top. 36-3, 1995[5] R. Cockett, J. Koslowski, R. Seely. Introduction to Linear Bicategories, Math. Struct. inC.S., 10-2, 2000

Four problems regarding representable functorsGigel Militaru

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The Notion of Topological toposJose Reinaldo Montanez Puentes

The present paper suggests a topos construction method which extends categories oftopological spaces. The analysis of the relation between the topos and the category oftopological spaces which it extends promotes the notion of Topological Topos, a notion thathas been inexistent in today’s literature. It is worth pointing out that the theory is developedtaking as starting point the Category of topological spaces and it is generalized to topologicalconstructs.The categories of topological spaces which we consider to be appropriate to be extended bytopos are those generated by a special kind of endofunctors defined in Top, which we havecalled Elevators. By means of final topologies, each topological space W determines anelevator which is idempotent and whose image is both a topological subcategory of Top and asubcategory co-reflexive of Top, which we name Cw. The endomorphism monoid of W, which

we name Mw, generates a topos, Mw-sets which we will call Ew. Cw naturally submerges into

Ew and determines a funtor, which is faithfull and full and has a left adjoint. This fact

motivates the notion of Topological topos, which is presented below.Definition: Let C be a topological subcategory of Top and E a topos. We may say that E is aC-Topological Topos if E contains an isomorphic reflexive subcategory to C. In such case, thepair (E,C) will be called a Topological Tandem. When imposing a Grothendieck topology onthe monoid Mw, which we have called Extensive Topology, a sub-topos of Ew, Shw is

determined, which we have called Extensive topos. Shw is a Cw topological topos and the

couple (Cw,Shw) is a Tandem. We have shown that Jonhstone’s topos [1] is an Extensive

topos and the Bornologic topos suggested by Lawvere in [2] is also an Extensive topos.

References[1] P.T Johnstone, On a topological Topos, Proc. London Math. Soc (3) 38 (1979), 237-271

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[2] F. Lawvere, The Bornologic Topos. Unpublished lecture, Bogotá, Colombia, 1983

A Schreier-Mac Lane extension theorem in action accessible categoriesAndrea Montoli

The aim of this talk is to give an intrinsic version of the Schreier-Mac Lane extension theorem,classically known for groups ([4] IV, Theorems 8.7 and 8.8). Any extension of groups

determines, via conjugation in the group X , a group homomorphism :Y Aut(K)/Int(K),called the abstract kernel of the extension. On the set Ext (Y,K) of isomorphism classes of

extensions with abstract kernel there is a simply transitive action of the abelian groupExt '(Y,ZK), where ZK denotes the center of K and ' is given by restricting the

automorphism (y) to ZK.In [2], Bourn showed that this result fully holds in any semi-abelian category C with splitextension classifier [1]. This setting, however, excludes many interesting algebraic structures,like the category of rings. In the present talk we show that the same Schreier-Mac Laneextension theorem holds in a much wider class of categories, called action accessiblecategories [3], provided they are exact. This family of categories includes, beyond groups andLie algebras, the categories of rings, associative algebras, Poisson algebras, Leibniz algebras,associative dialgebras (in the sense of J.L. Loday) and any variety of groups.This is joint work with Dominique Bourn.

References[1] F. Borceux, D. Bourn, Split extension classifier and centrality, Contemporary Mathematics,vol. 431 (2007), 85-104[2] D. Bourn, Commutator theory, action groupoids, and an intrinsic Schreier-Mac Laneextension theorem, Advances in Mathematics 217 (2008), 2700-2735[3] D. Bourn, G. Janelidze, Centralizers in action accessible categories,Cah.Topol.Geom.Diff.Cat. 50 (2009), no. 3, 211-232[4] S. Mac Lane, Homology, Springer-Verlag, 1963

Uniform fields vs. fibrewise uniform spaces. An adjoint situationClara Marina Neira Uribe

Inspired by the concept that the objective of General Topology is the study of continuousfunctions, a branch of General Topology known as Fibrewise Topology was originated. Sincethe 50’s, this approach has lead to the idea of considering properties of functions instead ofproperties of topological spaces with the aim of finding more general results.From the categorical point of view, this means replacing the study of the properties of thecategory of topological spaces and continuous functions by the study of the properties of thecategory whose objects are all continuous functions from a topological space to the space Tand whose morphisms from the object p:E T to the object q:F T are all the continuousfunctions :E F such that q =p.The uniform structure on a fibrewise uniform space has been defined in terms of fibrewiseentourages and the topology on the fiber space is described by fibrewise uniformneighbourhoods of each point.On the other hand, the theory of uniform fields (uniform bundles) introduced by J. Dauns andK. H. Hofmann generalizes three classic theories: the theory of sheaves, where fibers vary overa base space but are discrete, the theory of spaces of functions, where the fibers are constantbut not discrete and the theory of fiber bundles, where the fibers are locally constant. In a

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uniform bundle (E,p,T) a uniformity for p is considered and a base for the topology on E isgiven by tubes around local sections.The purpose of this work is to describe an adjoint functor between the category of fibrewiseuniform spaces and the category of uniform fields.

The Glueing Construction and Double CategoriesSusan Niefield

Classical Logic and Frobenius AlgebrasNovak Novakovic

The once-intractable problem "is there such a thing as a Boolean category" - a category whichis to a Boolean algebra what a cartesian-closed category with coproducts is to a Heytingalgebra - has been given several solutions during the last six years or so. But, we still cannotsay we have a really satisfying solution.The present work started as a denotational semantics for classical logic in the category ofposets and bimodules, furthering the work in [Lam07]. After noticing that the objectsrepresenting Boolean propositions were equipped with a Frobenius algebra structure, it wasalso found that one could use it to get a faithful representation of the "Free Frobeniuscategory", i.e. the free symmetric monoidal category generated by one object equipped with aFrobenius algebra structure [Dij89]. Actually one can also get a faithful representation of the

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"Free Frobenius compact-closed category", which happens to be a minor variation on it. Inaddition, with a little bit more work, we managed to construct subcategories of Posets andbimodules which are equivalent to these two free monoidal categories, thus giving a purelysemantical construction for these syntactical/geometric objects.The cleanest way to represent a category of logical formulas and proofs is by the means of so-called proof nets, where the objects are ordinary formulas but the maps belong to a category Cwhere some logically distinct objects have been identified (e.g., conjunction and disjunction).The original structure of the logical formulas introduces constraints on these maps, so there is afaithful functor from the proof net category to C, but one which is neither full nor injective onobjects.The work above allows us to construct a new category of proof nets for classical logic wherethe category C is a free Frobenius category, and give the required "full completeness theorem"i.e., show that every map in that category comes from a proof in the sequent calculus. This useof Frobenius algebras in classical logic is quite different from the one proposed by Hyland[Hyl04] and furthered by Garner [Gar05].It more resembles the work of Lamarche-Strassburger [LS05], where the category C is builtfrom sets and relations using an "interaction category" construction [Hyl04, Section 3], wherecomposition is obtained by the means of a trace operator. Our new category has the desirableproperty of being resource-sensitive, i.e. counting how many times contractions have been usedto superpose bits and pieces of a proof.

References[Dij89] R.H. Dijkgraaf. A geometric approach to two dimensional conformal field theory.Technical report, Universiteit Utrecht, 1989[Gar05] R. Garner. Three investigations into linear logic. Technical report, University ofCambridge, 2005[Hyl04] J.M.E. Hyland. Abstract interpretation of proofs: Classical propositional calculus. In J.Marcinkowski and A. Tarlecki, eds., Computer Science Logic, CSL 2004, volume 3210 ofLNCS, pages 6–21. Springer-Verlag, 2004[Lam07] F. Lamarche. Exploring the gap between linear and classical logic. Theory andApplications of Categories, 18(17):473–535, 2007[LS05] F. Lamarche and L. Strassburger. Naming proofs in classical logic. In P. Urzyczyn, ed.,TLCA Proceedings, volume 3461 of LNCS, pages 246–261. Springer, 2005

Connective completeness in topoiArnold Oostra

In classical logic the basic connectives , , , are functionally complete, but inintuitionistic logic there are new connectives that are no combination of the usual ones. Thissituation was studied by Caicedo [1], who found that in the topos of sheaves over anytopological space it suffices to add the constant connectives to obtain a functionally completeset.Caicedo’s results readily generalize to localic topoi [4]. For the topos of directed graphs, whichis not localic, a similar result holds [3] and in this talk we extend the techniques used to awider family of topoi. We show also remarkable connections with recent work in GeneralAlgebra ("Universal Algebra") [2,5].

References[1] X. Caicedo, Conectivos intuicionistas sobre espacios topológicos. In: Revista de laAcademia Colombiana de Ciencias Exactas, Físicas y Nat- urales XXI 81 (1997) 521–534

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[2] X. Caicedo, R. Cignoli, An algebraic approach to intuitionistic connectives”. In: Journal ofSymbolic Logic 66 (2001) 1620–1636[3] A. Oostra, Conectivos en el topos de grafos dirigidos. In: Boletín de Matemáticas 3 (1996)55–62[4] A. Oostra, Conectivos en Topos. Master’s degree dissertation. Bogotá: UniversidadNacional de Colombia, 1997[5] A. Oostra, Operaciones Implícitas en Variedades Ecuacionales. Ph.D. Dissertation. Bogotá:Universidad Nacional de Colombia, 2006

On the duality between trees and disksDavid Oury

A combinatorial category Disks was introduced by André Joyal to play a role in his definitionof weak -category. He defined the category to be dual to Disks. In the ensuing literature, amore concrete description of was provided. In this talk I will describe the constructions usedand sketch another proof of the dual equivalence. In so doing I'll introduce various categoriesequivalent to Disks or , each providing a helpful viewpoint.

Algebraic Theories over Nominal SetsDaniela Petrisan

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This is joint work with Alexander Kurz and Jirí Velebil.

On some laws relating left and right actionsClaudio Pisani

ReferencesC. Pisani (2010), A Logic for Categories, preprint available on arXiv.

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Applications of the weighted-limit-theorem to Hopf algebra theoryHans-E. Porst

The category of Hopf algebras over a field k can be shown to shown to be locally finitelypresentable, using classical results of Hopf algebra theory. However, the arguments to be usedcannot be generalized to the general case where k is just a commutative unital ring, notnecessarily a field. Using the Weighted-Limit-theorem of Makkai and Paré we will show thatfor any such ring R the categories CoalgR, BialgR and HopfR of R-coalgebras, bialgebras and

Hopf algebras respectively are locally presentable. As a consequence a number of existencetheorems for adjunctions in the area of Hopf algebras (free and cofree Hopf algebras, (co)freeadjunction of antipode) will follow easily.

Representation theory of MV-algebrasYuri A. Poveda

We develop a general representation theory for MV-algebras. Our guide line is the theory ofclassifying topoi of coherent extensions of universal algebra theories. Our main resultcorresponds, in the case of MV-algebras and MV-chains, to the representation of commutativerings with unit as rings of global sections of sheaves of local rings. We prove that any MV-algebra is isomorphic to the MV-algebra of all global sections of a sheaf of MV-chains on acompact topological space. This result is closely related to McNaughton’s theorem.If we have a universal algebra theory T and a geometric extension of it G, the classifying topoihave the necessary constructions for developing a representation theory of T-algebras in G-algebras (models of G).Given a T-algebra A in Set, we obtain a topos :EA Set, and a model of G, SA in EA with a

morphism *(A) SA. Such SA is the free G-algebra over A. The morphism *(A) SA

corresponds by adjunction to morphism A *(SA) with * the functor of global sections. We

show in our case that this morphism is in fact an isomorphism.This is a join work with Eduardo Dubuc.

Equivariant Homotopy Theory for Representable OrbifoldsDorette Pronk

Orbifolds are spaces that can locally be described as the quotient of an open subset ofEuclidean space by the action of a finite group. To obtain a notion of morphism betweenorbifolds that is appropriate to study orbifold homotopy theory, orbifolds can be representedby proper foliation groupoids (i.e., Lie groupoids witha proper diagonal and discrete isotropygroups). Two such groupoids represent the same orbifold precisely when they are Moritaequivalent. More specifically, the category of orbifolds can be viewed as the bicategory offractions of the 2-category of Lie groupoids with respect to the essential equivalences.An orbifold is representable when it can be obtained as the quotient of a manifold by the actionof a compact Lie group. We show that the subcategory of representable orbifolds is equivalentto the bicategory of fractions of the 2-category of translation groupoids with equivariant mapswith respect to equivariant essential equivalences.This prepares the way to generalize invariants from equivariant Bredon cohomology toorbifolds. We construct an equivariant fundamental groupoid and describe Bredon cohomologywith twisted coefficients for orbifolds.

Algebraic model structuresEmily Riehl

Natural weak factorization systems (nwfs) are algebraizations of weak factorization systems(wfs), which are ubiquitous in homotopy theory. As a practical consequence of Richard

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Garner's modified small object argument, many wfs that arise in nature can be algebraicized togive nwfs. This technique applies immediately to anything that is cofibrantly generated, butalso can be used in some cases which are not, at least classically, such as the wfs from theHurewicz model structure on chain complexes or Steve Lack's trivial model structure on certaindiagram 2-categories.This talk explores the consequences of these facts in the theory of model categories. We definean "algebraic model structure," which is a ordinary model category with the additionalstructure that can be expected when wfs are replaced with nwfs. In particular, cofibrations andfibrations are equipped respectively with coalgebra and algebra structures and there is a naturalcomparison map between the two functorial factorizations.Of pure categorical interest is a new notion of adjunction of nwfs, generalizing ordinarymorphisms of nwfs. We prove a change of base result, extending the "semantics" functor andits reflection, whose construction is the content of Garner's small object argument. This leadsto a proof that "algebraic Quillen adjunctions" exists in many familiar situations. In place of aclassical result that says that these left adjoints preserve cofibrations and right adjoints preservefibrations in such situations, we identify five adjunctions of nwfs, which give algebraiccharacterizations of the relationship between every reasonable pair of nwfs involved.

Goursat categories and the 3 3 LemmaDiana Rodelo

We present a new characterisation of Goursat categories in terms of special kind of pushouts,that we call Goursat pushouts. This allows one to prove that, for a regular category, theGoursat property is actually equivalent to the validity of the denormalised 3-by-3 Lemma.Goursat pushouts are also useful to clarify, from a categorical perspective, the existence of thequaternary operations characterising 3-permutable varieties.

Enriched weaknessJiri Rosicky

The basic notions of category theory, such as limit, adjunction, and factorization system,involve assertions of the existence of a unique morphism with certain properties. For example,an object X is orthogonal to a morphism f:A B if the induced mapping hom(f,X):hom(B,X)

hom(A,X) is a bijection. This formulation makes enriched version possible - one replacesthe bijection by the isomorphism of hom-objects in the base category V. Weak notions arisewhen one asks just for the existence of a morphism, e.g., X is injective to f if hom(f,X) is asurjection. In order to obtain an enriched version of weak notions, one has to choose a class Eof morphisms in V playing the role of surjections. If we take E to be the isomorphisms, weobtain the "non-weak notion of weakness".Assuming that V is locally finitely presentable as a closed category, we show that, taking E tobe pure epimorphisms (= filtered colimit of split epimorphisms), we obtain a satisfactory theoryof enriched injectivity. For example, accessible V-categories with E-weak colimits coincidewith small E-injectivity classes in locally presentable V-categories, with accessible V-categories with products and finite cotensors, and with models of (limit,E)-sketches.The case V=Cat with E the equivalences leads to interesting questions about 2-categories. Forinstance, A is a small-injectivity class in a locally finitely presentable 2-category K iff it isaccessible, accessibly embedded, closed under flexible limits and 2-replete. We hope to furtherdevelop our approach in order to get the homotopy theory for enriched categories. For instance,in the case V=SSet and E the weak equivalences, our injectivity is related to homotopyorthogonality for simplicial model categories K. Given a set S of morphisms, an object X iscalled S-local if it is fibrant and E-injective to S and a morphism h is called an S-localequivalence if each S-local object is E-injective to the cofibrant replacement of h. These

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concepts are crucial for left Bousfield localizations of K.This is joint work with S. Lack.

The Unique Solution PropertyPeter Schuster

Every complete metric spaces possesses the unique solution property: if a uniformlycontinuous real-valued function has approximate roots and in a uniform manner at most oneroot, then it actually has a root. The unique solution moreover becomes a uniformly continuousfunction in the parameters once one imposes a natural generalisation of the uniquenesshypothesis upon the given equation. This property of complete metric spaces seems to haveoccurred first in the early 1970s within the Russian school of recursive mathematics; has sinceproved productive in constructive and computable analysis; and stood right at the beginningsof proof mining. Its technical usefulness aside, what is the inner reason for the uniqueexistence property of complete metric spaces? What is the "metatheorem" logicians were askedfor by Beeson to explain the phenomenon that uniqueness helps to find the solution above any"pure existence proof" e.g. by means of the Bolzano-Weierstrass principle?Continuity is an issue; another one is (avoiding countable) choice. Neither gives a definitiveanswer, but the latter has at least shown the right direction. When working with completionsdefined without sequences we have observed that completeness is even equivalent to theunique solution property. This equivalence is easily seen from the choice-free constructiveproof [3] of the unique solution property for metric spaces completed by what Richman [2]called locations, which are specific examples of the functions to which the unique solutionproperty applies.To get a more satisfying explanation we now show how the equivalence can be related to acategorical characterisation of completeness given by Lawvere in [1]: a (generalised quasi-)metric space is complete if and only if every adjoint pair of R-bimodules is induced by an R-functor. In brief, an adjoint pair of R-bimodules corresponds to a location a' la Richmanenriched with parameters, whereas an R-functor corresponds to the unique solution as afunction in the parameters. As a by-product we get a choice-free approach to Lawvere'scharacterisation.The author is indebted to Peter Hancock and Martin Hyland for brief but substantial hints.

References[1] F. W. Lawvere, Metric spaces, generalized logic and closed categories. Originallypublished as: Rendiconti del Seminario Matematico e Fisico di Milano, XLIII (1973), 135-166.Republished in: Reprints in Theory and Applications of Categories, No. 1 (2002), 1-37[2] F. Richman, The Fundamental Theorem of Algebra: a. Constructive Development withoutChoice. Pacific Journal of Mathematics, 196 (2000), 213-230[3] P. Schuster, Problems, Solutions, and Completions. Journal of Logic and AlgebraicProgramming, 79, Issue 1 (2010), 84-91

Strong functors and monadsKruna Segrt

A strong functor T:A B between categories A and B, tensored over a closed symmetricmonoidal category E, is a functor equipped with a tensorial strength X,A:X TA T(X A)

satisfying some natural unit and associativity axioms. If A and B are enriched over E, there isa correspondence between giving a strength on T, and giving an enrichment of T over E. Thiscorrespondence extends to a 2-isomorphism between the 2-category of tensored E-categories,E-functors and E-natural transformations, and the 2-category of tensored E-categories, strong

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functors and strong natural transformations.In particular, we recover Anders Kock’s correspondence between enriched monads and strongmonads. In this context, categories equivalent to a module category for a monoid in E can bedescribed intrinsically. If time permits, we formulate a homotopy-theoretical generalizationusing Quillen model categories: we indicate sufficient conditions for a strong monad T on amonoidal model category E in order that the homotopy category of T-algebras be equivalent tothe homotopy category of modules over a monoid in E.A result of this type has previously been obtained by Stefan Schwede for strong monads on thecategory of simplicial sets.

Autonomous categories in which A is isomorphic to A*Peter Selinger

There are several examples of autonomous categories (in particular compact closed ones) inwhich the objects are self-dual, in the sense that A is isomorphic to A*, for all objects A.Sometimes, A=A* even holds strictly. Such a requirement is less benign than one initiallyassumes, and the examples range from natural to not very natural. In this talk, I will investigatewhich coherence conditions should be required of such a category. I will also discuss whatgraphical calculus could be used to reason about such a category.

The algebra of complicial setsRichard Steiner

Complicial sets are combinatorial objects equivalent to strict omega-categories. By definition,they are simplicial sets with additional structure subject to certain conditions. We willcharacterise them very explicitly as functors of a specific kind on a simple algebraic category.We will also give a more conventional algebraic description in terms of operations andidentities. This leads to a strictly algebraic simplicial description of strict omega-categories,similar to the previously known cubical description.

Symmetry and Cauchy-completion for quantaloid-enriched categoriesIsar Stubbe

For a category A enriched in an involutive quantaloid Q, we can compute its Cauchy-completion Acc, but also its symmetrisation As. For generalised metric spaces (B. Lawvere),

i.e. categories enriched in the quantale of positive real numbers, the Cauchy-completion of asymmetric generalised metric space is again symmetric; but it is not true in general (as pointedout by R. Betti and B. Walters). In this talk I will formulate an elementary condition on Qassuring the existence of a distributive law of the Cauchy-completion monad over thesymmetrisation comonad on Cat(Q), so that the Cauchy-complete and symmetric Q-categoriescan be computed as the Cauchy-completions of the symmetric Q-categories. This condition issatisfied by the quantale of positive real numbers, and also by every quantaloid associated witha Grothendieck topology (B. Walters).This is joint work with H. Heymans.

The Web Monoid and Opetopic SetsStanislaw Szawiel

In the original works of [HDA3] and [HMP] the categories of opetopic and multitopic setswere constructed using two mysterious monoids. In the approach of Baez and Dolan this rolewas played by the "sliced operad for operads", whose algebras were required to be typed,symmetric operads over a given operad. Hermida, Makkai and Power used the "multicategoryof function replacement" to construct the category of multitopic sets. Its principal feature was

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the fact that composition in this multicategory repaced function symbols by formal compositesof such. We propose what we believe to be a significantly simpler structure – the web monoid,which combines the essential features of both approaches. Its multiplication is replacement offunction symbols by formal composites, and its algebras are multicategories with nonstandardamalgamation over a given one. The web monoid is constructed using the three tensorstheorem, which provides us with an abstract construction that works in any cocompletecategory with two suitable monoidal structures. With the web monoid in hand a simpleconstruction of the category of opetopic sets can be given.

References[HDA3] J.C. Baez, J. Dolan. Higher-Dimensional Algebra III: n-Categories and the Algebra ofOpetopes, Adv. Math. 135 (1998), 145-206[HMP] C. Hermida, M. Makkai, J. Power. On weak higher dimensional categories I, part 1: J.Pure Appl. Alg. 154 (2000), 221-246, part 2: J. Pure Appl. Alg. 157 (2000), 247-277, part 3: J.Pure Appl. Alg. 166 (2002), 83-104

Equideductive categories and their logicPaul Taylor

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Relative commutator theory and the associator of loopsTim Van der Linden

We fill in the question mark in the diagram

which relates several non-equivalent concepts of commuting normal subobjects, here namedafter the authors who introduced them. This diagram is meant to be read in the followingmanner.The bottom triangle restricts itself to theories which make sense for varieties of -groups,while the top triangle extends those theories to a categorical context. In the left hand sidecolumn we have theories which are one-dimensional and relative ; the theories in the righthand side column, however, are two-dimensional and absolute, while the ones in the middlecolumn are two-dimensional and relative. So we are looking for a categorical commutatortheory which is both relative and two-dimensional.We explain how the concept of double central extension from categorical Galois theory may beused to obtain precisely such a commutator theory. Then we show how the abstract definitionapplies to the case of loops, where the associator is an example of a relative commutator.This is joint work with Tomas Everaert.

On the 2-Category Theory of QuasicategoriesDominic Verity

Early on in his development of the theory of quasicategories, André Joyal observed that therewas much to be gained by regarding those structures as being the 0-cells of a certain 2-categoryof simplicial sets. So, for example, adjunctions of quasicategories may be described simply as

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adjunctions within this 2-category and can fruitfully be studied as such. Of course thisapproach doesn’t immediately tell us how to handle the homotopy coherence properties ofthese adjunctions, but it does allow us to give a perfectly satisfactory account of the universalproperties that they enjoy.So it would seem reasonable to ask ourselves if there exist other aspects of the category theoryof quasicategories which may be obtained by purely 2-categorical means. However, in order tocarry out such a 2-categorical development of abstract category theory we would usually startby insisting upon the existence of certain well behaved 2-categorical limits and colimits in ourambient 2-categories. So it is disappointing to discover that while Joyal’s 2-category certainlyadmits constructions which look as if they ought to play the role of these things, those fail topossess the required 2-categorical (or even bicategorical) universal properties.This kind of situation is already quite familiar to us from the theory of homotopy limits andcolimits. These are constructions in the homotopy category of spaces which look very muchlike limits and colimits, but which fail to be characterised by the usual kinds of universalproperty. Following Heller [1] and Grothendieck [2], however, we might observe that we canconstruct an indexed category, defined upon the 2-category of all small categories and called adérivateur, wherein we can actually describe these constructions as genuine adjoints to re-indexing. In this talk we define, and begin the process of studying, a variant of the dérivateurconcept which is designed to describe the limit and colimit-like constructions that exist inJoyal’s 2-category. Our thesis is that such higher dérivateurs provide a convenient domainwithin which to further pursue a 2-categorical account of the category theory ofquasicategories.

References[1] A. Heller. Homotopy theories. Memoirs of the American Mathematical Society, 71(383),1988[2] A. Grothendieck. Les dérivateurs. available athttp://people.math.jussieu.fr/~maltsin/groth/Derivateurs.html, 1991

An induction principle for implications in arithmetic universesSteven Vickers

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This is joint work with Maria Emilia Maietti.

Weighted automataRobert F.C. Walters

At CT2009 we introduced a notion of Markov automaton, together with parallel operationswhich permit the compositional description of Markov processes. We illustrated by showing

how to describe a system of n dining philosophers (with 12n states), and we observed thatPerron-Frobenius theory yields a proof that the probability of reaching deadlock tends to one asthe number of steps goes to infinity. In this lecture we add sequential operations to the algebra(and the necessary structure to support them). The extra operations permit the description ofhierarchical systems, and ones with evolving geometry. We illustrate our algebra by describinga system related to the dining philosopher system called Sofia’s Birthday Party. The algebrahas close connections with the pioneering work of S. Eilenberg on automata (1974).This is joint work with Luisa de Francesco and Nicoletta Sabadini.

Approximation in quantale-enriched categoriesPawel Waszkiewicz

I will tell three short stories about the interaction of topology and order that can be observed incontinuous quantale-enriched categories, which are distinguished among all quantale-enrichedcats exactly in the same way as continuous lattices are distinguished among all completelattices. In my talk I will focus on continuous [0, ]-categories, although the results areapplicable much more widely.(1) Firstly, I will show that as soon as the natural topology on a continuous [0, ]-category isT1, then it is homeomorphic to the subspace Scott topology on the maximal elements of a

continuous dcpo.(2) Secondly, I will sketch a duality theorem that is a generalization of the Lawson duality forcontinuous dcpos.(3) Thirdly, I will use D. Pataraia's construction of fixed points for monotone maps on dcpos to

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obtain some constructive fixed point theorems for arbitrary endomaps in complete metricspaces.What are continuous quantale-enriched categories? Categorically speaking, a partial order P isa continuous dcpo, or simply a domain, if and only if the supremum map S:Idl(P) P thatmaps any ideal of P to its supremum S exists and has both right (the Yoneda embedding)and left adjoint. The left adjoint :P Idl(P) defines a notion of approximation in P: we havey x if and only if y occurs in every ideal that has its supremum above x. Intuitively, in thiscase y is essentially smaller than x. However S ( x)=x for all x P.By analogy, a Q-category X is continuous if it is relatively cocomplete with respect to acertain saturated class of weights Idl (this gives the supremum Q-functor S:Idl(X) X and theadjunction S |Y) such that S has a left adjoint .

Two 2-tracesSimon Willerton

Over recent years, in several areas of mathematics the notion of 'categorified trace' or '2-trace'has arisen. For instance, in higher representation theory where groups act on linear categoriesthere is the notion of a '2-character'; in Khovanov knot homology the Hochschild homology isviewed as a categorical trace. It transpires that there are actually two orthogonal, andsometimes dual, notions of 2-trace in common usage and I will explain how they arise andgive various examples.The general setting is in a monoidal bicategory. One can try to take the trace of an endo-1-morphism, f:X X. The two common ways to do this are by using the bicategorical structure,or by using the monoidal structure. In the setting of a bicategory of enriched profuctors onetrace is expressible as an end, the other as a co-end. In the setting of the bicategory ofcorrespondences between algebraic varieties one gives Hochschild homology and the otherHochschild cohomology.Taking the ordinary trace of the identity map on a vector space gives the dimension of thevector space. Analogously there are two notions of '2-dimension' for an object given by takingthe different '2-traces' of its identity 1-morphism. One of these dimensions is a monoid and theother is a module over that monoid; for instance, in the algebraic varieties setting this recoversthe standard structure on cohomology and homology.This talk will be illustrated throughout by examples from various areas of mathematics.

Completely and Totally Distributive CategoriesRichard J. Wood

A locally small category is said to be totally cocomplete if its Yoneda functor Y has a leftadjoint X. In a paper characterizing the category of sets among totally cocomplete categories,Rosebrugh and Wood introduced the terminology totally distributive for those total categoriesfor which X has a left adjoint W. They showed that small powers of the category of sets aretotally distributive. (In fact they characterized these as the totally distributive categories forwhich the inverter of the canonical W Y is dense and Kan.) It is also known that the categoryof sheaves on a CCD lattice is totally distributive.A locally small category K is cocomplete if it is a P-algebra, where P is the KZ-monad on Catwith PK the small colimit completion of K. In "Limits of small functors", Day and Lackshowed that the monad P lifts to the category of L-algebras, where L is the coKZ completionmonad on Cat. (Part of this result is contained in Freyd’s "Several new concepts".) It followsthat there is a distributive law LP PL and we say that a category K is completely distributiveif K is a PL-algebra. This means that K is small cocomplete, small complete, and PK Kpreserves small limits. It also follows from Day and Lack that every totally distributivecategory is completely distributive.

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In this talk, time permitting, we will investigate further the relationship between complete andtotal distibutivity. But we will also show that there is a further supply of totally distributivecategories provided by categories of (interpolative) bimodules between small taxons in thesense of Koslowski’s "Monads and interpolads in bicategories". A taxon is defined like acategory except that the existence of identities is dropped in favour of requiring merely thatcomposition be a certain colimit. There is a 2-category of taxons and a 2-functor i:Cat Taxthat interprets a category as a taxon. Our result above about bimodules is equivalent to sayingthat for a small taxon T, Tax(Top,i(set)) is a totally distributive category.This is joint with Francisco Marmolejo and Bob Rosebrugh.

Admissibility, stable units and connected componentsJoao Xarez

Consider any full reflection H| I:C M, with unit :1C HI, such that C has finite limits.

Suppose there is also a functor U:C Set, into the category of sets, which preserves finitelimits, reflects isomorphisms and U( c) is a surjection for every object c in C. If, in addition,

all the maps homM(t,m) homSet(1,U(m)) induced by the functor U are surjections, where t

and 1 are respectively terminal objects in C and Set, for every object m in the subcategory M,then it is true that* the reflection H| I is semi-left-exact (admissible in the sense of Galois categorical theory) ifand only if its connected components are "connected";* it has stable units if and only if any finite product of connected components is connected.The meaning of "connected" is the usual in Galois categorical theory, and the very simpledefinition of connected component with respect to the ground structure above will be given.For instance, both reflections of compact Hausdorff spaces into Stone spaces, and ofsemigroups into semilattices are admissible, furthermore, they have stable units, since theyobey the conditions above.

Categorical Models for Intuitionistic Existential GraphsFernando Zalamea

(A) Peirce’s Existential Graphs Systems Alpha and Beta (1896-1910) provide the only knownaxiomatization of classical propositional logic and first-order (purely relational) classical logicwith the same rules of inference for both the propositional and the first-order segments. Behindthe language and the rules of the Existential Graphs, a topological logic is in action [Burch1991, Zalamea 2003], and one may expect that a natural intuitionistic logic should emerge.This has been the case with the recent construction of intuitionistic Existential Graphs systems[Oostra 2010].(B) A semantics for Existential Graphs had been provided (via equivalence with the usualpropositional and first-order systems) by Boolean algebras and first-order models.Nevertheless, a natural interpretation of Peirce's rules was due (particularly theiteration/deiteration rule with which one can define a general notion of intuitionistic connective[Caicedo, Cignoli 2001]). [Brady, Trimble 2000a, 2000b] filled the gap and presentedcategorical models for Alpha and Beta, where Peirce's rules correspond to adequate naturaltransformations. Brady and Trimble used *-autonomous categories to model Alpha and aJoyal-Street string calculus to model Beta.In our talk, we will

present Oostra's new intuitionistic Existential Graphs (A)discuss some modifications needed in Brady and Trimble's approach (B) in order toobtain intuitionistic categorical models for Graphs:

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reducing the language to negation and conjunction: shift of *-autonomouscategories to symmetric monoidal categories with non-involutive negationextending the semantics to independent intuitionistic connectives: shift ofmonoidal categories to a new sort of "trimoidal" categories (coherencediagrams open)shift of categorical first-order (classical) theories to functors into the categoryof Heyting algebras and shift of Joyal-Street's inversion deformation rule to anon-involutive one (adjunction diagrams open).

ReferencesG. Brady, T. Trimble (2000a). A Categorical Interpretation of C. S. Peirce's PropositionalLogic Alpha. Journal of Pure and Applied Algebra 149 (2000): 213-239G. Brady, T. Trimble (2000b). A String Diagram Calculus for Predicate Logic and C. S.Peirce's System Beta. PreprintR. Burch (1991). A Peircean Reduction Thesis. The Foundations of Topological Logic.Lubbock: Texas Tech University Press, 1991X. Caicedo, R. Cignoli (2001). An Algebraic Approach to Intuitionistic Connectives. Journalof Symbolic Logic 66 (2001): 1620-1636A. Oostra (2010). A Lattice of Intuitionistic Existential Graphs Systems. PreprintF. Zalamea (2003). Peirce's Logic of Continuity: Existential Graphs and Non-cantorianContinuum. The Review of Modern Logic 9 (2003): 115-162

Representing multicategories as monadsMarek Zawadowski

If a functor p:E B is a bifibration then it is an exponentiable object in Cat/B and the

exponential fibration pp is again a bifibration. Moreover, pp is a lax monoidal fibration, i.e. ithas a lax (in fact strict) monoidal structure in fibers given by compostion of functors and thereindexing functors are lax monoidal. When C has pullbacks then the exponential (bi)fibration

of the basic (bi)fibration C C, denoted End(C) C, has as objects functors on slices of C,and a morphism in End(C) over u:c c' is a natural transformation :Fu* u*F' where F andF' are endofunctors on C/c and C/c', respectively.Using such fibrations one can explain the connections between various notions ofmulticategories and various notions of monads on slices of the category Set (and any otherlocally cartesian closed category). From the action of the lax monoidal fibration ofamalgamated signatures Siga Set on the basic fibration over Set

we get, by the exponential adjunction, a representation

of the fibration of amalgamated signatures in the exponential bifibration. This representation isa strong monoidal morphism of fibrations, faithful, full on isomorphism, and its essential

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image consists of the (finitary) polynomial endofunctors and cartesian natural transformations.It induces an equivalence of categories (fibrations, in fact) of multicategories with non-standardamalgamations considered by HermidaMakkai-Power and the fibration of (finitary) polynomialmonads. In a similar way we get an equivalence between the fibration of the symmetricmulticategories considered by Baez-Dolan and the fibration of the analytic monads, a versionof the notion considered by A. Joyal.The generalizations of the structures mentioned above, can be naturally constructed from thesymmetrization monad, the fibered monoidal monad on the Burroni (lax monoidal) fibrationdefined from any polynomial monad on S et. The 'classical' notions are obtained by taking thefree monoid monad on the category Set.

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