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Theory and Applications of Categories, Vol. 33, No. 4, 2018, pp. 67–94.

DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS

CLARK BARWICK, SAUL GLASMAN AND DENIS NARDIN

Abstract. In this technical note, we proffer a very explicit construction of the dualcocartesian fibration of a cartesian fibration, and we show they are classified by the samefunctor to the ∞-category of ∞-categories.

Anyone who has worked seriously with quasicategories has had to spend some qualitytime with cartesian and cocartesian fibrations. (For a crash course in the basic definitionsand constructions, see Appendix A; for an in-depth study, see [HTT, §2.4.2].) The pur-pose of (co)cartesian fibrations is to finesse the various homotopy coherence issues thatnaturally arise when one wishes to speak of functors valued in the quasicategory Cat∞of quasicategories. A cartesian fibration p : X S is “essentially the same thing” as afunctor X : Sop Cat∞, and a cocartesian fibration q : Y T is “essentially the samething” as a functor Y : T Cat∞. We say that the (co)cartesian fibration p or q isclassified by X or Y (A).

It has therefore been a continual source of irritation to many of us who work withquasicategories that, given a cartesian fibration p : X S, it seems difficult to constructan explicit cocartesian fibration p∨ : X∨ Sop that is classified by the same functorSop Cat∞. Many constructions require as input exactly one of these two, and if onehas become sidled with the wrong one, then one is left with two options:

(1) One may extrude the desired fibration through tortuous expressions such as “thecocartesian fibration p∨ classified by the functor by which the cartesian fibration pis classified.” We know of course that such a thing exists, but we have little hope ofusing it if we don’t have access to a model that lets us precisely identify an n-simplexof X∨ in terms of p.

(2) Alternately, one may use Lurie’s construction Dl of [Lurie, 2011, Cnstr. 3.4.6].1 Thisis at least more precise: an n-simplex of his simplicial set Dl(pop) is an n-simplex σ ofSop along with a functor ∆n ×Sop Xop Top such that each functor Xop

σ(k) Topis representable. It is stated — but not shown — that Dl(pop) Sop is a cocartesianfibration classified by the same functor [Lurie, 2011, Rk. 3.4.9]. Nevertheless, Dl(pop)isn’t particularly explicit: representability is of course a difficult matter in general, soit’s not easy to say in elementary terms what an object of Dl(pop) actually is. Worse,this model is unhelpfully large relative to X. (In fact, as written, it only works when

Received by the editors 2014-11-20 and, in final form, 2017-12-23.Transmitted by Kathryn Hess. Published on 2018-01-08.2010 Mathematics Subject Classification: 18D30.Key words and phrases: cocartesian fibrations, cartesian fibrations, quasicategories.c© Clark Barwick, Saul Glasman and Denis Nardin, 2018. Permission to copy for private use granted.

1We thank a referee for pointing us to this construction, of which we were previously unaware.

67

68 CLARK BARWICK, SAUL GLASMAN AND DENIS NARDIN

p has small fibers.) When S = ∆0, for example, this construction replaces X with itsessential image under the Yoneda embedding, which, while equivalent, is obviouslymuch larger.

In this paper, we give a concrete construction of the dual cocartesian fibration p∨ ofa cartesian fibration p (which works in general), and we show they are classified by thesame functor to Cat∞. In particular, the objects of X∨ are precisely the objects of X,and when S = ∆0, the dual X∨ is isomorphic to X. As evidence for the robustness ofthis construction, we will construct a relative twisted arrow ∞-category for a cocartesianfibration and its dual. One of us (S.G.) uses this in his construction of the Day convolutionfor ∞-categories [Glasman, 2013]. Our description of the dual will be used heavily in theforthcoming paper [BDGNS, 2014].

Amusingly, the construction of the dual itself is quite simple; however, proving that itworks as advertised (and for that matter, even proving that p∨ is a cocartesian fibration) isa nontrivial matter. The main technical tool we use is the technology of effective Burnside∞-categories and the unfurling construction of the first author [Barwick, 2014].

In the first section, we will give an informal but very concrete description of the dual,and we will state the main theorem, Th. 1.4. In §2, we briefly recall the definition of thetwisted arrow category, which plays a significant role in the construction. In §3, we give aprecise definition of the dual of a cartesian fibration, and we prove that it is a cocartesianfibration. In particular, we can say exactly what the n-simplices of X∨ are (3). In §4, weprove Pr. 4.1, which asserts that the double dual is homotopic to the identity, and we usethis to prove the main theorem, Th. 1.4. Finally, in §5, we construct a relative versionof the twisted arrow ∞-category for a cocartesian fibration and its dual, which providesanother way to witness the equivalence between the functor classifying p and the functorclassifying p∨.

1. OverviewBefore we describe the construction, let us pause to note that simply taking opposites willnot address the issue of the day: if p : X S is a cartesian fibration, then it is true thatpop : Xop Sop is a cocartesian fibration, but the functor Sop Cat∞ that classifiespop is the composite of the functor X : Sop Cat∞ that classifies p with the involution

op : Cat∞ Cat∞

that carries a quasicategory to its opposite.This discussion does, however, permit us to rephrase the problem in an enlightening

way: the morphism (p∨)op : (X∨)op S must be another cartesian fibration that isclassified by the composite of the functor that classifies p with the involution op. Thedual cocartesian fibration to (p∨)op should be equivalent to pop, so that we have a dualityformula

((p∨)op)∨ ' pop.

DUALIZING CARTESIAN AND COCARTESIAN FIBRATIONS 69

In particular, it will be sensible to define the dual q∨ of a cocartesian fibration q : Y Tas ((qop)∨)op, so that p∨∨ ' p. We thus summarize:

The cartesian fibration and the cocartesian fibration are each classified byp : X S p∨ : X∨ Sop X : Sop Cat∞;

(p∨)op : (X∨)op S pop : Xop Sop op ◦X : Sop Cat∞;q∨ : Y ∨ T op q : Y T Y : T Cat∞;qop : Y op T op (q∨)op : (Y ∨)op T op ◦Y : T Cat∞.

We can describe our construction very efficiently if we give ourselves the luxury oftemporarily skipping some details. For any quasicategory S and any cartesian fibrationp : X S, we will define X∨ as a quasicategory whose objects are those of X and whosemorphisms x y are diagrams

u

x y

f g (1)

of X in which f is a p-cartesian edge, and p(g) is a degenerate edge of S. Composition ofmorphisms in X∨ will be given by forming a pullback:

w

u v

x y z

The n-simplices for n ≥ 3 are described completely in 3. One now has to explainwhy this defines a quasicategory, but it does indeed (Df. 3.4), and it admits a naturalfunctor to Sop that carries an object x to p(x) and a morphism as in (1) to the edgep(f) : p(x) p(u) = p(y) in Sop. This is our functor p∨ : X∨ Sop, and we have goodnews.

1.1. Proposition. If p : X S is a cartesian fibration, then p∨ : X∨ Sop is acocartesian fibration, and a morphism as in (1) is p∨-cocartesian just in case g is anequivalence.This much will actually follow trivially from the fundamental unfurling lemmas of thefirst author [Barwick, 2014, Lm. 11.4 and Lm. 11.5], but the duality statement we’re afteris more than just the construction of this cocartesian fibration.

If one inspects the fiber of p∨ over a vertex s ∈ Sop, one finds that it is the quasicategorywhose objects are objects of Xs := p−1(s), and whose morphisms x y are diagrams (1)of Xs in which f is an equivalence. This is visibly equivalent to Xs itself. Furthermore,we will prove that this equivalence is functorial:


1.2. Proposition. The functor Sop Cat∞ that classifies a cartesian fibration p isequivalent to the functor Sop Cat∞ that classifies its dual p∨.Equivalently, we have the following.

1.3. Proposition. If X : Sop Cat∞ classifies p, then op◦X : Sop Cat∞ classifies(p∨)op.

We will define the dual of a cocartesian fibration q : Y T over a quasicategory Tas suggested above:

q∨ := ((qop)∨)op.

In other words, Y ∨ will be the quasicategory whose objects are those of Y and whosemorphisms x y are diagrams

u

x y

f g

of Y in which q(f) is a degenerate edge of T , and g is q-cocartesian. Composition ofmorphisms in Y ∨ will be given by forming a pushout:

w

u v

x y z

The three propositions above will immediately dualize.In summary, the objects of X∨ and (X∨)op = (Xop)∨ are simply the objects of X,

and the objects of Y ∨ and (Y ∨)op = (Y op)∨ are simply the objects of Y . A morphismη : x y in each of these ∞-categories is then as follows:


In η is a diagram of in which f and g

X∨u

x y

f g X is p-cartesian, lies over an identity;

(X∨)opu

x y

f g X lies over an identity, is p-cartesian;

Y ∨u

x y

f g Y lies over an identity, is q-cocartesian;

(Y ∨)opu

x y

f g Y is q-cocartesian, lies over an identity.

The propositions above are all subsumed in the following statement of our main the-orem, which employs some of the notation of A.

1.4. Theorem. The assignments p p∨ and q q∨ define homotopy inverse equiva-lences of ∞-categories

(−)∨ : Catcart∞/S

∼ Catcocart∞/Sop : (−)∨

of cartesian fibrations over S and cocartesian fibrations over Sop. These equivalences arecompatible with the straightening/unstraightening equivalences s in the sense that the di-agram of equivalences

Catcart∞/S Catcocart

∞/Sop

Fun(Sop,Cat∞)

Fun(Sop,Cat∞)

Catcocart∞/Sop Catcart

∞/S

(−)∨

s

op

s

opop ◦ −

(−)∨

s s

commutes up to a (canonical) homotopy.


2. Twisted arrow ∞-categories2.1. Definition. If X is an ∞-category (i.e., a quasicategory), then the twisted arrow∞-category O(X) is the simplicial set given by the formula

O(X)n = Mor(∆n,op ?∆n, X) ∼= X2n+1.

The two inclusions

∆n,op ∆n,op ?∆n and ∆n ∆n,op ?∆n

give rise to a map of simplicial sets

O(X) Xop ×X.

The vertices of O(X) are edges of X; an edge of O(X) from u v to x y can beviewed as a commutative diagram (up to chosen homotopy)

u x

v y

When X is the nerve of an ordinary category C, O(X) is isomorphic to the nerve of thetwisted arrow category of C in the sense of [DK, 1983]. When X is an ∞-category, ourterminology is justified by the following.

2.2. Proposition. [Lurie, [Lurie, 2011, Pr. 4.2.3]] If X is an ∞-category, then the func-tor O(X) Xop ×X is a left fibration; in particular, O(X) is an ∞-category.

2.3. Example. To illustrate, for any object p ∈ ∆, the ∞-category O(∆p) is the nerveof the category

00

01 10

. . . . . .. . . . . .

02 13 31 20

01 12 . . .. . . 21 10

00 11 22 22 11 00


(Here we write n for p− n.)In [Lurie, 2011, §4.2], Lurie goes a step further and gives a characterization the left

fibrations that (up to equivalence) are of the form O(X) Xop ×X. We’ll discuss (anduse!) this result in more detail in §5.

3. The definition of the dualWe now give a precise definition of the dual of a cartesian fibration and, conversely, thedual of a cocartesian fibration. The definitions themselves will not depend on previouswork, but the proofs that the constructions have the desired properties follow triviallyfrom general facts about the unfurling construction of the first author [Barwick, 2014,Lm. 11.4 and 11.5].

3.1. Notation. Throughout this section, suppose S and T two∞-categories, p : X Sa cartesian fibration, and q : Y T a cocartesian fibration.

As in Nt. A.4, denote by ιS ⊂ S the subcategory that contains all the objects andwhose morphisms are equivalences. Denote by ιSX ⊂ X the subcategory that containsall the objects, whose morphisms are p-cartesian edges.

Similarly, denote by ιT ⊂ T the subcategory that contains all the objects, whosemorphisms are equivalences. Denote by ιTY ⊂ Y the subcategory that contains all theobjects and whose morphisms are q-cocartesian edges.

It is easy to see that

(S, ιS, S) and (X,X ×S ιS, ιSX)

are adequate triples of ∞-categories in the sense of [Barwick, 2014, Df. 5.2]. Dually,

(T op, ιT op, T op) and (Y op, Y op ×T op ιT op, (ιTY )op)

are adequate triples of ∞-categories.Furthermore, the cartesian fibrations p : X S and q : Y op T op are adequate

inner fibrations over (S, ιS, S) and (T op, ιT op, T op) (respectively) in the sense of [Barwick,2014, Df. 10.3].

3.2. Definition. For any ∞-category C and any two subcategories C† ⊂ C and C† ⊂ Cthat each contain all the equivalences, we define Aeff (C,C†, C†) as the simplicial set whosen-simplices are those functors

x : O(∆n)op C

such that for any integers 0 ≤ i ≤ k ≤ ` ≤ j ≤ n, the square

xij xkj

xi` xk`


is a pullback in which the morphisms xij xkj and xi` xk` lie in C† and the morphismsxij xi` and xkj xk` lie in C†.

When Aeff (C,C†, C†) is an∞-category (which is the case, for example, when (C,C†, C†)is an adequate triple of ∞-categories in the sense of [Barwick, 2014, Df. 5.2]), we call itthe effective Burnside ∞-category of (C,C†, C†).

Note that the projections O(∆n)op ∆n and O(∆n)op (∆n)op induce inclusions

C† Aeff (C,C†, C†) and (C†)op Aeff (C,C†, C†).

Now it is easy to see that p and q induce morphisms of simplicial sets

p : Aeff (X,X ×S ιS, ιSX) Aeff (S, ιS, S)

andq : Aeff (Y op, Y op ×T op ιT op, (ιTY )op)op Aeff (T op, ιT op, T op)op,

respectively. We wish to see that p is a cocartesian fibration and that q is a cartesianfibration, but it’s not even clear that they are inner fibrations.

Luckily, the fundamental unfurling lemmas [Barwick, 2014, Lm. 11.4 and Lm. 11.5] ofthe first author address exactly this point. The basic observation is that the unfurling

Υ(X/(S, ιS, S)) (respectively, Υ(Y op/(T op, ιT op, T op)) )

of the adequate inner fibration p (resp., qop) [Barwick, 2014, Df. 11.3] is then the effectiveBurnside ∞-category

Aeff (X,X ×S ιS, ιSX) (resp., Aeff (Y op, Y op ×T op ιT op, (ιTY )op) ),

and the functor Υ(p) (resp., the functor Υ(qop)op) is the functor p (resp., the functor q)described above. The fundamental lemmas [Barwick, 2014, Lm. 11.4 and Lm. 11.5] nowimmediately imply the following.

3.3. Proposition. The simplicial set Aeff (S, ιS, S) is an ∞-category, and the functor pis a cocartesian fibration. Furthermore, a morphism of Aeff (X,X ×S ιS, ιSX) of the form

u

x y

f g

is p-cocartesian just in case g is an equivalence.Dually, the simplicial set Aeff (T, T, ιT ) is an ∞-category, and the functor q is a carte-

sian fibration. Furthermore, a morphism of Aeff (Y op, Y op×T op ιT op, (ιTY )op)op of the form

u

x y

f g

is q-cocartesian just in case f is an equivalence.


3.4. Definition. The dual of p is the projection

p∨ : X∨ := Aeff (X,X ×S ιS, ιSX)×Aeff (S,ιS,S) Sop Sop,

which is a cocartesian fibration. Dually, the dual of q is the projection

q∨ : Y ∨ := Aeff (Y op, Y op ×T op ιT op, (ιTY )op)op ×Aeff (T op,ιT op,T op)op T T,

which is a cartesian fibration.The formation of the dual and the formation of the opposite are by construction dual

operations with respect to each other; that is, one has by definition

(pop)∨ = (p∨)op and (qop)∨ = (q∨)op.

Observe that the inclusions

Sop Aeff (S, ιS, S) and T Aeff (T op, ιT op, T op)op

are each equivalences. Consequently, the projections

X∨ Aeff (X,X ×S ιS, ιSX) and Y ∨ Aeff (Y op, Y op ×T op ιT op, (ιTY )op)op

are equivalences as well.Note also that the description of X∨ and Y ∨ given in the introduction coincides with

the one given here: an n-simplex of X∨, for instance, is a diagram

x00

x01 x10

. . . . . .. . . . . .

x02 x13 x31 x20

x01 x12. . .. . . x21 x10

x00 x11 x22 x22 x11 x00

in which any j-simplex of the form x0j x1j · · · xjj covers a totally degeneratesimplex of S (i.e., a j-simplex in the image of S0 Sj), and all the morphisms xij xi`are p-cartesian.

In particular, note that the fibers (X∨)s are equivalent to the fibers Xs, and the fibers(Y ∨)t are equivalent to the fibers Yt.


4. The double dual4.1. Proposition. Suppose S and T two∞-categories, p : X S a cartesian fibration,and q : Y T a cocartesian fibration. There are equivalences

p ' p∨∨ and q ' q∨∨

of cartesian fibrations X S and cocartesian fibrations Y T , respectively. Theseequivalence are natural in that they give rise to natural isomorphisms

id ' (−)∨∨ : Catcart∞/S Catcart

∞/S, id ' (−)∨∨ : Catcocart∞/S Catcocart

∞/S

We postpone the proof (which is quite a chore) till the end of this section. In the meantime,let us reap the rewards of our deferred labor: in the notation of A, we obtain the following.

4.2. Corollary. The formation of the dual defines an equivalence of ∞-categories

(−)∨ : Catcart∞/S

∼ Catcocart∞/Sop : (−)∨

Proof. The only thing left to observe that (−)∨ is a functor from the ordinary categoryof cartesian (respectively, cocartesian) fibrations to the ordinary category of cocartesian(resp., cartesian) fibrations, and this functor preserves weak equivalences (since they aredefined fiberwise), whence it descends to a functor of ∞-categories Catcart

∞/S Catcocart∞/Sop

(resp., Catcocart∞/Sop Catcart

∞/S).

Let sSetf be the 1-category of quasicategories, and let RelCat be the 1-category ofrelative categories and relative functors [BK, 2012]. Let

U : RelCat sSetf

be the underlying∞-category functor, so that U(C) is a fibrant replacement of the markedsimplicial set N(C)\, whose marked edges are the weak equivalences, in the cartesian modelstructure on sSet+ [HTT, §3.1.3].

The following basic lemma will help prove the naturality of some of our constructions:

4.3. Lemma. Let A,B ∈ RelCat, and let F,G : A B be relative functors, and letλ : F G be a natural transformation. Then λ gives rise to a natural transformation

U(λ) : U(F ) ∼U(G).

If λ is a natural weak equivalence, then U(λ) is an equivalence of functors.Proof. A natural transformation between relative functors A B is the same data asa relative functor

k : A× [1][ B,


where [1][ is the relative category with two objects and a morphism between them whichis not a weak equivalence. But

N(A× [1][)\ ∼= N(A)\ × (∆1)[

as marked simplicial sets. Moreover, by [HTT, Proposition 3.1.4.2], the morphism

N(A)\ × (∆1)[ U(A)× (∆1)[

is a marked weak equivalence, and the target is fibrant since (∆1)[ is already fibrant. Sowe get a map

U(k) : U(A× [1][) ' U(A)× (∆1)[ U(B)which is exactly the data of a natural transformation from U(F ) to U(G).

In the case where λ is a natural equivalence, we use an almost identical argument.A natural equivalence between relative functors A B is the same data as a relativefunctor

l : A× [1]] B,

where [1]] is the relative category with two objects and a weak equivalence between them.Now

N(A× [1]])\ ∼= N(A)\ × (∆1)]

as marked simplicial sets, and by [HTT, Proposition 3.1.4.2], we have that

N(A)\ × (∆1)] U(A)× U([1]])

is a marked weak equivalence. But U([1]]) is a contractible Kan complex, and so theinduced map

U(l) : U(A× [1]]) ' U(A)× U((∆1)]) U(B)is a homotopy between U(F ) and U(G).

4.4. Notation. We recall the set-theoretic technicalities and notation used in [HTT,§1.2.15, Rk. 3.0.0.5]. Let us choose two strongly inaccessible uncountable cardinals κ < λ.Denote by Cat∞ (respectively, Top) the∞-category of κ-small∞-categories (resp., of κ-small Kan complexes). Similarly, denote by Cat∞ (resp., Top) the∞-category of λ-small∞-categories (resp., of λ-small Kan complexes).

Note that Cat∞ and Top are essentially λ-small and locally κ-small, whereas Cat∞and Top are only locally λ-small.

4.5. Lemma. The formation of the dual is natural with respect to pullback in the base;that is, it extends to a natural transformation of functors of∞-categories Catop

∞ Cat∞from

F : S 7→ Catcart∞/S

toG : S 7→ Catcocart

∞/Sop .


Proof. We describe a functor F : (sSetf )op RelCat as follows. Let F (S) be thecategory of diagrams

Φ : sSetf/S ×∆1 sSetf

satisfying the following conditions:

• for each object (T S) ∈ sSetf/S, Φ(T S, 1) = T ;

• for each object (T S) ∈ sSetf/S, the morphism

Φ(T S, 0) Φ(T S, 1)

is a cartesian fibration;

• for each morphism (T U S) in sSetf/S, the square

Φ(T S, 0) Φ(U S, 0)

Φ(T S, 1) Φ(U S, 1)

is a (strict) pullback square.

The morphisms in sSetf are the natural transformations, and the weak equivalences arethose which are objectwise categorical equivalences. If f : S0 S1 is a morphism insSetf , then we define F (f) : F (S1) F (S0) by

(F (f)(Φ))(T S0, i) = Φ(T S0fS1, i).

Evaluation at {idS} × ∆1 gives an equivalence of relative categories from F (S) to therelative category Cart/S of cartesian fibrations over S, but the assignment of Cart/S toS only admits the structure of a pseudofunctor [GHN, 2015, Definition A.2] to the (2, 1)-category of relative categories, while F is a functor on the nose. Replacing the coherencesnecessary when discussing pseudofunctors with a certain amount of flab makes it easierto discuss the naturality of dualization.

Thus constructed, F is a functor of relative categories: it takes categorical equivalencesto equivalences of relative categories: if f : S1 S0 is a map, the effect of F (f) onunderlying ∞-categories can be identified up to equivalence with the functor

(−)×S1 S0 : Catcart∞/S1 Catcart

∞/S0 ,

and if f is a categorical equivalence, then this functor is an equivalence of ∞-categories.Thus F descends to a functor

F : Cat∞ Cat∞.


Now consider the functor G : (sSetf )op RelCat which takes S to the category ofdiagrams

Ψ : sSetf/Sop ×∆1 sSetf

satisfying a set of conditions similar to those listed in the definition of F , except that themorphism

Ψ(T Sop, 0) Ψ(T Sop, 1)must be a cocartesian fibration. G is a point-set rectification of the functor

G : Cat∞ Cat∞

which takes S to the ∞-category coCart/Sop .We now define a natural transformation δ : F G as follows: for each S ∈ sSetf ,

δS : F (S) G(S) is given by

δS(Φ)(T S, i) =

Φ(T S, 0)∨T i = 0T op i = 1

where the subscript indicates that the dual is taken relative to T . The naturality of δis clear, but we must show that δS(Φ) satisfies the condition that for any commutativetriangle (T U Sop), the diagram

δS(Φ)(T Sop, 0) δS(Φ)(U Sop, 0)

δS(Φ)(T Sop, 1) δS(Φ)(U S, 1)

is a strict pullback square. Unwinding, what we really have to show is that given a strictpullback square

X Y

T U

in which the vertical maps are cartesian fibrations, the square

X∨T Y ∨U

T op Uop

is a strict pullback square. In this situation, the square

(X,X ×T ιT, ιTX) (Y, Y ×U ιU, ιUY )

(T, ιT, T ) (U, ιU, U)


is a pullback square of triples, and so induces a pullback square of effective Burnsidecategories. Now in the commutative cube

X∨T Y ∨U

Aeff (X,X ×T ιT, ιTX) Aeff (Y, Y ×U ιU, ιUY )

T op Sop

Aeff (T, ιT, T ) Aeff (U, ιU, U),

the front and side faces are pullback squares by definition, and so the back face is apullback square. The conclusion follows by Lemma 4.3.

Let’s now prove the main theorem, Th. 1.4.Proof of Th. 1.4. For any ∞-category S, consider the composite equivalence

Fun(Sop,Cat∞) ∼ Catcart∞/S

∼ Catcocart∞/Sop ∼ Fun(Sop,Cat∞),

where the first equivalence is given by unstraightening, the second is given by the forma-tion of the dual, and the last is given by straightening. By Lemma 4.5 and [GHN, 2015,Corollary A.31], all of these equivalences are natural in S. We thus obtain an autoequiv-alence η of the functor Fun((−)op,Cat∞) : Catop

∞ Cat∞, and thus of the functor

Map((−)op,Cat∞) : Catop∞ Top.

Now the left Kan extension of this functor along the inclusion Catop∞ Cat

op∞ is the

functor h : Catop∞ Top represented by Cat∞. The autoequivalence η therefore also

extends to an autoequivalence η of h.The Yoneda lemma now implies that η is induced by an autoequivalence of Cat∞

itself. By the Unicity Theorem of Toen [Toen, 2005], Lurie [Lurie, 2009, Th. 4.4.1], andthe first author and Chris Schommer-Pries [BSP, 2011], we deduce that η is canonicallyequivalent either to id or to op, and considering the case S = ∆0 shows that it’s theformer option.

This proves the commutativity of the triangle of equivalences

Catcart∞/S Catcocart

∞/Sop

Fun(Sop,Cat∞),

(−)∨

s s

and the commutativity of the remainder of the diagram in Th. 1.4 follows from duality.


We’ve delayed the inevitable long enough.Proof of Pr. 4.1. We prove the first assertion; the second is dual.

To begin, let us unwind the definitions of the duals to describe X∨∨ explicitly. First,for any ∞-category C, denote by O(2)(C) the simplicial set given by the formula

O(2)(C)k = Mor((∆k)op ?∆k ? (∆k)op ?∆k, C) ∼= C4k+3.

(This is a two-fold edgewise subdivision of C. It can equally well be described as a “twisted3-simplex ∞-category of C.”) Now the n simplices of X∨∨ are those functors

x : O(2)(∆n)op X

such that any r-simplex of the form

x(ab1c1d1) x(ab2c2d2) · · · x(abrcrdr)

covers a totally degenerate r-simplex of S, and, for any integers

0 ≤ a′ ≤ a ≤ b ≤ b′ ≤ c′ ≤ c ≤ d ≤ d′ ≤ n

(which together represent an edge abcd a′b′c′d′ of O(2)(C)) we have

(4.1.1) the morphism x(a′bcd) x(abcd) is p-cartesian;

(4.1.2) the morphism x(ab′cd) x(abcd) is an equivalence;

(4.1.3) the morphism x(abcd′) x(abcd) is an equivalence.

In other words, an object of X∨∨ is an object of X, and a morphism of X∨∨ is a diagram

u v

x y z

φ g ψ f

in X such that φ, g, and ψ all cover degenerate edges of S, and

(4.1.1-bis) the morphism f is p-cartesian;

(4.1.2-bis) the morphism ψ is an equivalence;

(4.1.3-bis) the morphism φ is an equivalence.

We will now construct a cartesian fibration p′ : X ′ S, a trivial fibration α : X ′ ∼ Xover S and a fiberwise equivalence β : X ′ ∼ X∨∨ over S. These equivalences will all bethe identity on objects. We will identify X ′ with the subcategory of X∨∨ whose morphisms


are as above with ψ and φ are degenerate; the inclusion will be the fiberwise equivalenceβ. The equivalence α : X ′ ∼ X will then in effect be obtained by composing g and f .

To construct p′, we write, for any ∞-category C,

O(C) := Fun(∆1, C).

Note that the functor s : O(C) C given by evaluation at 0 is a cartesian fibration (Ex.A.3). We now define X ′ as the simplicial set whose n-simplices are those commutativesquares

O(∆n) X

∆n S,

x

s p

σ

such that x carries s-cartesian edges to p-cartesian edges. We define p′ : X ′ S to be themap that carries an n-simplex as above to σ ∈ Sn. We remark thatX ′ X∨∨ is manifestlya fiberwise equivalence. In particular, this means that the assignment X 7→ X ′ preservesweak equivalence between cartesian fibrations, and thus descends to an ∞-functor

(−)′ : Catcart∞/S Catcart

∞/S.

We now construct the desired equivalences. The basic observation is that for anyinteger k ≥ 0, we have functors

∆k ∆k ×∆1 ∆k ?∆k ∆k ? (∆k)op ?∆k ? (∆k)op :

on the left we have the projection onto the first factor; in the middle we have thefunctor corresponding to the unique natural transformation between the two inclusions∆k ∆k ?∆k; on the right we have the obvious inclusion. These functors induce, forany n ≥ 0, functors

∆n O(∆n) O(2)(∆n)op.

These in turn induce a zigzag of functors

XαX ′

βX∨∨

over S, which are each the identity on objects. On morphisms, α carries a morphism givenby x y z to the composite x z, and β carries a morphism given by x y zto the morphism of X∨∨ given by the diagram

x y

x y z.

g f


We observe that since the construction of α and β is natural in X, we get a diagram of∞-functors

id α (−)′ β (−)∨∨ : Catcart∞/S Catcart

∞/S

by Lemma 4.3. We now have the following, whose proof we postpone for a moment.

4.6. Lemma. The morphism X ′ X constructed above is a trivial Kan fibration. Thusp′ is the composite of two cartesian fibrations, and therefore a cartesian fibration.

By Lemma 4.6, α is an equivalence of functors, and we know that β is an equivalenceof functors. Given the Lemma, we therefore obtain an equivalence of functors

id ' (−)∨∨ : Catcart∞/S Catcart

∞/S.

Let’s now set about proving that X ′ X is indeed a trivial fibration. For this, wewill need to make systematic use of the cartesian model categories of marked simplicialsets as presented in [HTT, §3.1].Proof of Lm. 4.6. We make O(∆n) into a marked simplicial set O(∆n)\ by markingthose edges that map to degenerate edges under the target map t : O(∆n) ∆n. Fur-thermore, for any simplicial subset K ⊂ O(∆n), let us write K\ for the marked simplicialset (K,E) in which E ⊂ K1 is the set of edges that are marked as edges of O(∆n)\.

Now write∂O(∆n) :=

n⋃i=0O(∆{0,...,i,...,n}) ⊂ O(∆n),

which is a proper simplicial subset of Fun(∆1, ∂∆n) when n > 2. Observe that ∂O(∆n)has the property that there is a bijection

Map(∂O(∆n), X) ∼= Map(∂∆n, X ′).

Recasting the statement the Lemma in terms of lifting properties, we see that it willfollow from the claim that for any n ≥ 0 and any morphism O(∆n)\ S] of markedsimplicial sets, the natural inclusion

ιn : ∂O(∆n)\ ∪(∂∆n)[ (∆n)[ O(∆n)\

is a trivial cofibration in the cartesian model structure for marked simplicial sets over S,where the ∂∆n in ∂O(∆n) is the boundary of the “long n-simplex” whose vertices are theidentity edges in ∆n.

In fact, we will prove slightly more. Let I denote the smallest class of monomorphismsof marked simplicial sets that contains the marked anodyne morphisms and satisfies thetwo-out-of-three axiom. We call these morphisms effectively anodyne maps of markedsimplicial sets. Clearly, for any morphism Y S], an effectively anodyne morphismX Y is a trivial cofibration in the cartesian model structure on marked simplicial setsover S.


It’s clear that ι1 is marked anodyne, because it’s isomorphic to the inclusion

(∆{0,2})[ (∆2)[ ∪(∆{1,2})[ (∆{1,2})].

Our claim for n > 1 will in turn follow from the following sublemma.

4.7. Lemma. The inclusion (∆n)[ O(∆n)\ of the “long n-simplex” is effectively ano-dyne.Let’s assume the veracity of this lemma for now, and let’s complete the proof of Lm. 4.6.It’s enough to show that the inclusion

(∆n)[ ∂O(∆n)\ ∪(∂∆n)[ (∆n)[

is effectively anodyne, for then ιn will be a effectively anodyne by the two-out-of threeproperty. We’ll deploy induction and assume that Lemma 4.6 has been proven for eachl < n. Now for each l, let

sklO(∆n)\ := colimI⊆n,|I|≤lO(∆I)\

so thatskn−1O(∆n)\ = ∂O(∆n)\.

By Lemma 4.6 for ιl, we have that

skl−1O(∆n)\ ∪(skl−1∆n)[ (∆n)[ sklO(∆n)\ ∪(skl∆n)[ (∆n)[

is a trivial cofibration, because it’s a composition of pushouts along maps isomorphic toιl. Since

sk0O(∆n)\ ∪(sk0∆n)[ (∆n)[ = (∆n)[,

iterating this up to l = n− 1 gives the result.Proof of Lm. 4.7. Write S for the set of nondegenerate (2n)-simplices

x = [00 = i0j0 i1j1 · · · i2nj2n = nn]

of O(∆n). For x ∈ S as above, define

A(x) = 12

(−n+

2n∑r=0

(jr − ir)).

Drawing O(∆n) as a staircase-like diagram and x as a path therein, it’s easily checkedthat A(x) is the number of squares enclosed between x and the “stairs” given by thesimplex

x0 = [00 01 11 12 · · · (n− 1)n nn] .


We’ll fill in the simplices of S by induction on A(x). For k ≥ 0, let

Sk = {x ∈ S |A(x) = k} and Tk = {x ∈ S |A(x) ≤ k}

andOk(∆n) :=

⋃x∈Tk

x ⊂ O(∆n).

We make the convention thatO−1(∆n) := ∆n.

We must now show that for all k with 0 ≤ k ≤ 12n(n− 1), the inclusion

Ok−1(∆n)\ Ok(∆n)\

is marked anodyne, and for each k it will be a matter of determining x ∩ Ok−1(∆n) foreach x ∈ Sk and showing that the inclusion

x\ ∩ Ok−1(∆n)\ x\

is effectively anodyne.The case k = 0 is exceptional, so let’s do it first. The set S0 has only one element, the

simplexx0 = [00 01 11 12 · · · (n− 1)n nn] .

We claim that the inclusion of O−1(∆n)\ x\0 is effectively anodyne. Sticking all themarked 2-simplices of the form

[ii i(i+ 1) (i+ 1)(i+ 1)]\

ontoO−1(∆n)\ is a marked anodyne operation, so let’s do that and call the result y. Clearlythe spine of x0 is inner anodyne in y, so the inclusion y x0 is a trivial cofibration. Thisproves the claim.

Now we suppose k > 0, and suppose

x = [00 = i0j0 i1j1 · · · i2nj2n = nn] ∈ Sk.

We call a vertex v = (irjr) of x a flipvertex if it satisfies the following conditions:

• 0 < r < 2n;

• jr > ir;

• ir−1 = ir (and hence jr−1 = jr − 1);

• jr+1 = jr (and hence ir+1 = ir + 1).


Observe that x must contain some flipvertices, and it is uniquely determined by them.Note also that if y is an arbitrary simplex of O(∆n) containing all the flipvertices of x,and if z ∈ S contains y as a subsimplex, then A(z) ≥ A(x), with equality if and only ifz = x.

We define the flip of x at v Φ(x, v) as the modification of x in which the sequence

· · · ir(jr − 1) irjr (ir + 1)jr · · ·

has been replaced by the sequence

· · · ir(jr − 1) (ir + 1)(jr − 1) (ir + 1)jr · · · .

Then Φ(x, v) ∈ Sk−1, so we have Φ(x, v) ⊂ Ok−1(∆n). We have therefore established thatx ∩ Ok−1(∆n) is the union of the faces

∂vx = x ∩ Φ(x, v)

as v ranges over flipvertices of x. Equivalently, if {v1, · · · , vm} is the set of flipvertices ofx, then x ∩ Ok−1(∆n) is the generalized horn

x ∩ Ok−1(∆n) ∼= Λ2n{0,··· ,2n}\{v1,··· ,vm} ⊂ ∆2n ∼= x

in the sense of [Barwick, 2014, Nt. 12.6].If m > 1, since flipvertices cannot be adjacent, it follows that the set

{0, · · · , 2n} \ {v1, · · · , vm}

satisfies the hypothesis of [Barwick, 2014, Lm. 12.13], and so the inclusion x ∩ Ok−1(∆n) xis inner anodyne, whence x\ ∩ Ok−1(∆n)\ x\ is effectively anodyne.

On the other hand, if m = 1, then x ∩ Ok−1(∆n) is a face:

x ∩ Ok−1(∆n) = ∂vx ∼= ∆{0,...,i+j,...,2n} ⊂ ∆2n ∼= x,

where v = (ij) is the unique flipvertex of x. We must show that the inclusion

x\ ∩ Ok−1(∆n)\ x\

is effectively anodyne. We denote by y the union of ∂vx with the 2-simplex

[i(j − 1) ij (i+ 1)j].

The inclusion ∂vx\ y\ is marked anodyne; we claim that the inclusion y x is inner

anodyne.Indeed, something more general is true: suppose s is an inner vertex of ∆m and F is a

subset of [m] which has s as an inner vertex and is contiguous, meaning that if t1, t2 ∈ Fand t1 < u < t2 then u ∈ F . Then the inclusion ∂s∆m ∪∆F ∆m is inner anodyne.


We prove this by induction on m − |F |. If |F | = m, then ∆F = ∆m and the claim isvacuous. Otherwise, let F ′ be a contiguous subset of [n] containing F with |F ′| = |F |+ 1.Then

∆F ′ ∩ (∆F ∪ ∂s∆m) = ∆F ∪ ∂s∆F ′ .

But ∆F ∪ ∂s∆F ′ is the generalized horn ΛF ′

F\{s}, and F \ {s} satisfies the hypothesis of[Barwick, 2014, Lm. 12.13] as a subset of F ′ since s was already an inner vertex of F .Thus ∂s∆n ∪∆F ∂s∆n ∪∆F ′ is inner anodyne, and by the induction hypothesis, weare done.

5. The duality pairingIn this section we give construct a pairing that concretely exhibits the equivalence betweenthe functor Y : T Cat∞ that classifies a cocartesian fibration q : Y T and theopposite of the functor that classifies the cocartesian fibration (q∨)op.

The way we’ll go about this is the following: we will construct a left fibration

M : O(Y/T ) (Y ∨)op ×T Y

such that for any object t ∈ T , the pulled back fibration

O(Y/T )t ((Y ∨)op)t × Yt ' Y opt × Yt

is a perfect pairing; i.e., it satisfies the conditions of the following result of Lurie.

5.1. Proposition. [[Lurie, 2011, Cor. 4.2.14]] Suppose σ : X A and τ : X Btwo functors that together define a left fibration λ : X A×B. Then λ is equivalent toa fibration of the form O(C) Cop × C (and in particular A ' Bop) just in case thefollowing conditions are satisfied.

(5.1.1) For any object a ∈ A, there exists an initial object in the ∞-category Xa :=σ−1({a}).

(5.1.2) For any object b ∈ B, there exists an initial object in the ∞-category Xb :=τ−1({b}).

(5.1.3) An object x ∈ X is initial in Xσ(x) just in case it is initial in Xτ(x).

In our case, the functor that classifies M will be the fiberwise mapping space functor

MapY/T : (Y ∨)op ×T Y Top.

This functor carries an object (x, y) ∈ (Y ∨)op ×T Y to the space MapY(t)(x, y), wheret = q(x) = q(y). If φ : s t is a morphism of S, then a morphism

(f, g) : (u, v) (x, y)


of (Y ∨)op ×T Y covering φ is given, in effect, by morphisms f : x Y(φ)(u) and g :Y(φ)(v) y of Y(s). The functor MapY/T will then carry (f, g) to the morphism

MapY(s)(u, v)Y(φ)

MapY(t)(Y(φ)(u),Y(φ)(v)) g ◦ − ◦ f MapY(t)(x, y).

Before we proceed headlong into the details of the construction, let us first give aninformal but very concrete description of both O(Y/T ) and M . The objects of O(Y/T )will be morphisms f : u v of Y such that q(f) is an identity morphism in T . Now amorphism f g from an arrow f : u v to an arrow g : x y is a commutativediagram

u xw

v y

ψ

g

φ

f

ξ

in which φ is q-cocartesian, q(ψ) is an identity morphism. Composition is performed byforming suitable pushouts on the source side and simple composition on the target side.We will establish below that there is indeed an ∞-category that admits this description.

The functor M will carry an object f ∈ O(Y/T ) as above to the pair of objects(u, v) ∈ (Y ∨)op×Y , and it will carry a morphism f g as above to the pair of morphisms w

u x

φ ψ , vξy

∈ (Y ∨)op × Y.

We call M the duality pairing for q. We will prove below that it is left fibration, whenceit follows readily from this description that the functor that classifies it is indeed be thefiberwise mapping space functor

MapY/T : (Y ∨)op ×T Y Top

defined above.

5.2. Proposition. Both an ∞-category O(Y/T ) and a left fibration M as describedabove exist.

We postpone the precise construction of O(Y/T ) and M till the end of this section(Constr. 5.4). Our concrete description suffices to deduce the main result of this section.

5.3. Theorem. For any object t ∈ T , the left fibration

O(Y/T )t ((Y ∨)op)t × Yt

pulled back from the duality pairing M is a perfect pairing; i.e., it satisfies the conditionsof Pr. 5.1.


Proof. Suppose x ∈ ((Y ∨)op)t and y ∈ Yt. Then it is easy to see that the identity mapidx is the initial object of the fiber O(Y/T )x: for any morphism g : x y such that q(g)is a degenerate edge, the essentially unique morphism idx g of O(Y/T )x is given bythe diagram

x xx

x y

gg

g

Dually, the identity map idy is the initial object of the fiber O(Y/T )y: the essentiallyunique morphism idy g of O(Y/T )y is given by the diagram

y xy

y y

g

g

The result now follows immediately.In light of Pr. 5.1, we deduce an identification

((Y ∨)op)t ' Y opt

that is functorial in t, as desired.

5.4. Construction. We now set about giving a precise construction of the∞-categoryO(Y/T ) and the left fibration M described in 5. We use very heavily the technology ofeffective Burnside ∞-categories from [Barwick, 2014].

We begin by identifying two subcategories of the arrow ∞-category O(Y ), each ofwhich contains all the objects. Suppose f : u v and g : x y morphisms of Y . Amorphism η : f g of O(Y ) given by a square

u v

x y

f

s(η) t(η)

g

lies in O(Y )† just in case q(s(η)) is an equivalence of T and t(η) is an equivalence of Y ;the morphism η lies in O(Y )† just in case s(η) is q-cocartesian.


Now form the effective Burnside ∞-categoriesO′(Y ) := Aeff (O(Y )op, (O(Y )†)op, (O(Y )†)op),O(T ) := Aeff (O(T )op, ιO(T )op,O(T )op),

(Y ∨)op := Aeff (Y op, Y op ×T op ιT op, (ιTY )op),Y := Aeff (Y op, ιY op, Y op),T := Aeff (T op, ιT op, T op).

The objects of O′(Y ) are thus morphisms f : u v of Y , and a morphism f g froman arrow f : u v to an arrow g : x y is a commutative diagram

u u′ x

v y′ y

φ

f

ξ

ψ

g

η

in which: φ is q-cocartesian, q(ψ) is an equivalence, and η is an equivalence.The source and target functors O(Y )op Y op along with the cocartesian fibration q

together induce a diagram of functorsO′(Y ) O(T )

(Y ∨)op × Y T × T

Observe that the omnibus theorem of the first author [Barwick, 2014, Th. 12.2] impliesthat all of the functors that appear in this quadrilateral are inner fibrations.

Furthermore, since the formation of the effective Burnside ∞-category respects fiberproducts, one may employ [Barwick, 2014, Th. 12.2] to show not only that the naturalmap

M ′ : O′(Y )(

(Y ∨)op × Y)×T×TO(T )

is an inner fibration, but also that every morphism of O′(Y ) is M ′-cocartesian. It is clearthat M ′ admits the right lifting property with respect to the inclusion ∆{0} ∆1, onededuces that M ′ is a left fibration.

As we see, the ∞-category O′(Y ) is much too large, but we now proceed to cut bothit and the left fibration M ′ down to size via pullbacks:(5.4.1) The first pullback in effect requires all equivalences in the description of the mor-

phisms of O′(Y ) above to be identities. We pull back M ′ along the inclusion

((Y ∨)op × Y ) ×T×TO(T )

((Y ∨)op × Y

)×T×TO(T )


(which is of course an equivalence) to obtain a left fibration

M ′ : O′(Y ) ((Y ∨)op × Y ) ×T×TO(T ).

(5.4.2) Second, we pull back the composite

O′(Y ) M ′ ((Y ∨)op × Y ) ×T×TO(T ) O(T )

along the inclusion T O(T ) of the degenerate arrows to obtain the desired leftfibration

M : O(Y/T ) (Y ∨)op ×T Y

It is now plain to see that O(Y/T ) is the∞-category described in 5, and M is the leftfibration described there.

A. Cartesian and cocartesian fibrationsA.1. Definition. Suppose p : X S an inner fibration of simplicial sets. Recall [HTT,Rk. 2.4.1.4] that an edge f : ∆1 X is p-cartesian just in case, for each integer n ≥ 2,any extension

∆{n−1,n} X,

Λnn

f

F

and any solid arrow commutative diagram

Λnn X

∆n S,

F

p

F

the dotted arrow F exists, rendering the diagram commutative.We say that p is a cartesian fibration [HTT, Df. 2.4.2.1] if, for any edge η : s t of S

and for every vertex x ∈ X0 such that p(x) = s, there exists a p-cartesian edge f : x ysuch that η = p(f).

Cocartesian edges and cocartesian fibrations are defined dually, so that an edge of Xis p-cocartesian just in case the corresponding edge of Xop is pop-cartesian, and p is acocartesian fibration just in case pop is a cartesian fibration.

A.2. Example. A functor p : D C between ordinary categories is a Grothendieckfibration if and only if the induced functor N(p) : ND NC on nerves is a cartesianfibration [HTT, Rk 2.4.2.2].


A.3. Example. For any ∞-category C, write O(C) := Fun(∆1, C). By [HTT, Cor.2.4.7.12], evaluation at 0 defines a cartesian fibration s : O(C) C, and evaluation at 1defines a cocartesian fibration t : O(C) C.

One can ask whether the functor s : O(C) C is also a cocartesian fibration. Onemay observe [HTT, Lm. 6.1.1.1] that an edge ∆1 O(C) is s-cocartesian just in casethe corresponding diagram

(Λ20)B ∼= ∆1 ×∆1 C

is a pushout square.Suppose S a simplicial set. Then the collection of cartesian fibrations to S with small

fibers is naturally organized into an ∞-category Catcart∞/S. To define it, let Catcart

∞ be thefollowing subcategory of O(Cat∞): an object X U of O(Cat∞) lies in Catcart

∞ if andonly if it is a cartesian fibration, and a morphism p q in O(Cat∞) between cocartesianfibrations represented as a square

X Y

U V

f

p q

lies in Catcart∞ if and only if f carries p-cartesian edges to q-cartesian edges. We now define

Catcocart∞/S as the fiber over S of the target functor

t : Catcart∞ ⊂ O(Cat∞) Cat∞.

Equivalently [HTT, Pr. 3.1.3.7], one may describe Catcart∞/S as the simplicial nerve of the

(fibrant) simplicial category of marked simplicial sets [HTT, Df. 3.1.0.1] over S that arefibrant for the cartesian model structure — i.e., of the form X\ S for X S acartesian fibration [HTT, Df. 3.1.1.8].

The straightening/unstraightening Quillen equivalence of [HTT, Th. 3.2.0.1] now yieldsan equivalence of ∞-categories

Catcart∞/S ' Fun(Sop,Cat∞).

So we obtain a dictionary between cartesian fibrations p : X S with small fibers andfunctors X : Sop Cat∞. For any vertex s ∈ S0, the value X(s) is equivalent to the fiberXs, and for any edge η : s t, the functor X(t) X(s) assigns to any object y ∈ Xt

an object x ∈ Xs with the property that there is a cocartesian edge x y that covers η.We say that X classifies p [HTT, Df. 3.3.2.2].

Dually, the collection of cocartesian fibrations to S with small fibers is naturally orga-nized into an∞-category Catcocart

∞/S , and the straightening/unstraightening Quillen equiv-alence yields an equivalence of ∞-categories

Catcocart∞/S ' Fun(S,Cat∞).


A.4. Notation. A cartesian (respectively, cocartesian) fibration with the property thateach fiber is a Kan complex — or equivalently, with the property that the functor thatclassifies it factors through the full subcategory Top ⊂ Cat∞ of Kan complexes — iscalled a right (resp., left) fibration. These are more efficiently described as maps thatsatisfy the right lifting property with respect to horn inclusions Λn

k ∆n such that1 ≤ k ≤ n (resp., 0 ≤ k ≤ n− 1) [HTT, Pr. 2.4.2.4].

For any cartesian (resp., cocartesian) fibration p : X S, one may consider thesmallest simplicial subset ιSX ⊂ X that contains the p-cartesian (resp., p-cocartesian)edges. The restriction ιS(p) : ιSX S of p to ιSX is a right (resp., left) fibration. Thefunctor Sop Top (resp., S Top) that classifies ιSp is then the functor given bythe composition ι ◦ X, where X is the functor that classifies p, and ι is the functorCat∞ Top that extracts the maximal Kan complex contained in an ∞-category.

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School of Mathematics, University of Edinburgh, James Clerk Maxwell Building,Peter Guthrie Tait Road, Edinburgh, EH9 3FD, United KingdomVincent Hall, 206 Church Street SE,Minneapolis, Minnesota 55455, USADepartement de Mathematiques, Institut Galilee, Universite Paris 13,99 av. J.B. Clement, FR-93430 Villetaneuse

Email: [email protected]@[email protected]

This article may be accessed at http://www.tac.mta.ca/tac/

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles thatsignificantly advance the study of categorical algebra or methods, or that make significant new contribu-tions to mathematical science using categorical methods. The scope of the journal includes: all areas ofpure category theory, including higher dimensional categories; applications of category theory to algebra,geometry and topology and other areas of mathematics; applications of category theory to computerscience, physics and other mathematical sciences; contributions to scientific knowledge that make use ofcategorical methods.Articles appearing in the journal have been carefully and critically refereed under the responsibility ofmembers of the Editorial Board. Only papers judged to be both significant and excellent are acceptedfor publication.Full text of the journal is freely available from the journal’s server at http://www.tac.mta.ca/tac/. Itis archived electronically and in printed paper format.Subscription information Individual subscribers receive abstracts of articles by e-mail as theyare published. To subscribe, send e-mail to [email protected] including a full name and postal address. For in-stitutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, [email protected] for authors The typesetting language of the journal is TEX, and LATEX2e isrequired. Articles in PDF format may be submitted by e-mail directly to a Transmitting Editor. Pleaseobtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor. Robert Rosebrugh, Mount Allison University: [email protected]

TEXnical editor. Michael Barr, McGill University: [email protected]

Assistant TEX editor. Gavin Seal, Ecole Polytechnique Federale de Lausanne:gavin [email protected]

Transmitting editors.Clemens Berger, Universite de Nice-Sophia Antipolis: [email protected] Blute, Universite d’ Ottawa: [email protected] Breen, Universite de Paris 13: [email protected] Brown, University of North Wales: ronnie.profbrown(at)btinternet.comValeria de Paiva: Nuance Communications Inc: [email protected] Getzler, Northwestern University: getzler(at)northwestern(dot)eduKathryn Hess, Ecole Polytechnique Federale de Lausanne: [email protected] Hyland, University of Cambridge: [email protected] Kock, University of Aarhus: [email protected] Lack, Macquarie University: [email protected]. William Lawvere, State University of New York at Buffalo: [email protected] Leinster, University of Edinburgh: [email protected] Moerdijk, Utrecht University: [email protected] Niefield, Union College: [email protected] Pare, Dalhousie University: [email protected] Rosicky, Masaryk University: [email protected] Rosolini, Universita di Genova: [email protected] Simpson, University of Ljubljana: [email protected] Stasheff, University of North Carolina: [email protected] Street, Macquarie University: [email protected] Tholen, York University: [email protected]. J. Wood, Dalhousie University: [email protected]

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