Leovigildo Alonso Tarr o - USCwebspersoais.usc.es/.../charlas/HHenPurdue09.ho.pdf · X !f Y !g Z...

Hochschild homology and Grothendieck Duality

Leovigildo Alonso Tarrıo

Universidade de Santiago de Compostela

Purdue UniversityJuly, 1, 2009

Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue – July 2009 1 / 48

Broad outline

1 Co/homology of singular spaces

2 Bivariant Hochschild theory

3 Bivariant Hochschild homology and cohomology

4 Orientation and fundamental class

This is joint work with A. Jeremıas (USC) & J. Lipman (Purdue).


Co/homology of singular spaces

Outline §1

1 Co/homology of singular spacesNonsingular spacesBivariant theories





Co/homology of singular spaces Nonsingular spaces

Co/homology of nonsingular spaces

Let X be a nonsingular space (technically an orientable topologicalmanifold —to fix ideas think on the underlying space of a complexmanifold).There are two theories, namely, homology and cohomology.

H∗(X ) and H∗(X )

By Poincare duality they convey the same information

Hi (X ) ∼= Hn−i (X )∨

However, if the space is singular these isomorphisms do not hold anymore.One is tempted to regard cohomology as the main invariant because of itsring structure and disregard homology.


Co/homology of singular spaces Bivariant theories

The starting point

This is not the right answer. MacPherson observed that cohomology andhomology played different roles, both important, in the case of singularspaces

1 Homology supports characteristic classes

2 Cohomology is the ring of operations of homology

It is desirable to get a general framework that makes sense of thisobservation.

This framework was developed by Fulton and MacPherson and christenedbivariant theories. They show how to make them work as a good languageto express Riemann-Roch type theorems for singular spaces.



Ingredients of a bivariant theory

A bivariant theory is not exactly a functor, it has some features related to(weak) 2-categorical ideas but this has not been pursued.In short it consists on

1 an underlying category, a category C with some extra structure,

2 the category of values, a graded counterpart GrA of a monoidal,abelian category A (usually graded modules over some ring),

3 and a map (the theory)

T : Arr(C) −→ GrA,

where Arr(−) denotes the class of arrows of a category,

subject to a list of conditions.



Structure of the underlying category I

The category C is endowed with two classes:

1 a class of maps in C called confined maps,

2 a class of diagrams in C, oriented commutative squares, calledindependent squares

Y ′ Y

X ′ Xg ′

g

f ′ fd

satisfying the following conditions.



Structure of the underlying category II

ucA The class of confined maps contains the identities and is stable forcomposition.

ucB The class of independent squares contains all squares d such thatf = f ′ and g = g ′ = id and is stable for vertical and horizontalcomposition.

ucC In a square d, if f (or g) is confined then so is f ′ (or g ′, respectively).In other words to be confined changes of base through independentsquares.

Y ′ Y

X ′ Xg ′

g

f ′ fd



Operations for a bivariant theory

Prod Given f : X → Y and g : Y → Z in C homomorphisms

· : T i (Xf−→ Y )⊗ T j(Y

g−→ Z ) −→ T i+j(Xgf−−→ Z ) (i , j ∈ Z).

PF For each f : X → Y confined and g : Y → Z in C a homomorphism

fT : T i (Xgf−−→ Z ) −→ T i (Y

g−→ Z ) (i ∈ Z).

PB For an independent square d, homomorphisms

Y ′ Y

X ′ Xg ′

g

f ′ fd

gT : T i (Xf−→ Y ) −→ T i (X ′

f ′−→ Y ′)



Axioms for the operations of a bivariant theory I

A1 Product is associative: Given Xf−→ Y

g−→ Zh−→W in C and

α ∈ T i (f ), β ∈ T j(g), γ ∈ T `(h) then

(α · β) · γ = α · (β · γ)

A2 Push-forward is functorial: Given Xf−→ Y

g−→ Zh−→W in C and

α ∈ T i (hgf ) with f and g confined, then

(gf )T (α) = gT fT (α)

A12 Product and push-forward commute: Given Xf−→ Y

g−→ Zh−→W in C

and α ∈ T i (gf ), β ∈ T j(h), with f confined, then

fT (α · β) = fT (α) · β



Axioms for the operations of a bivariant theory II

A3 Pull-back is functorial: Given independent squares

X ′′ X ′ X

Y ′′ Y ′ Y

d d′

h′ g ′

h g

f ′′ f ′ f

and α ∈ T i (f ) then

(gh)T (α) = hTgT (α)

in T i (f ′′)



Axioms for the operations of a bivariant theory III

Consider the diagram ofindependent squares

Z ′ Z

Y ′ Y

X ′ X

h′

h

g ′ g

h′′

f ′ f

A13 Product and pull-back commute:Given, in the diagram,α ∈ T i (f ), β ∈ T j(g), then

hT (α · β) = h′T

(α) · hT (β)

A23 Push-forward and pull-back commute:Given, in the diagram, α ∈ T i (gf ),with f confined, then

f ′T (hT (α)) = hT (fT (α))



Axioms for the operations of a bivariant theory IV

A123 Projection formula: Given the diagram

Y ′ Y Z

X ′ Xg ′

g h

f ′ fd

and α ∈ T i (f ), β ∈ T j(hg), with d independent and g confined, then

g ′T (gT (α) · β) = α · gT (β)


Bivariant Hochschild theory

Outline §2


2 Bivariant Hochschild theoryData for a bivariant Hochschild theoryDefinition of a bivariant Hochschild theoryChecking the compatibilities




Bivariant Hochschild theory Data for a bivariant Hochschild theory

The underlying category

Our base scheme will be a noetherian scheme S . The basic category isSS := Schfl

tf(S), the category of flat finite type separated schemes over S ,that we will denote simply S.

Note that all maps within S are separated and finite type, but notnecessarily flat.

Structure:

1 The proper maps of S constitute the class of confined maps.

2 The class of independent squares of S is formed by those orientedfiber squares in C such that the bottom is a etale morphism.

Observation

The axioms ucA, ucB and ucC hold by standard considerations.


Bivariant Hochschild theory Data for a bivariant Hochschild theory

A reminder on quasi-coherent cohomological operations.

To every scheme X in S we can associate its derived category Dqc(X ) ofcomplexes of sheaves with quasi-coherent homology. It is monoidal closedwith the derived tensor product denoted −⊗L −.Let f : X → Y a map in S. We have the usual adjunction

Lf ∗ a Rf∗

Another operation is the twisted inverse image f ! : Dqc(Y )→ Dqc(X ).In the proper case satisfies the adjunction

Rf∗ a f !

while in the etale case we have f ! := f ∗.That this notion makes sense as a pseudo functor and the basic propertiesof f ! is a non trivial theory developed by Grothendieck, Hartshorne,Deligne and Verdier and put up to date and clarified recently by Conrad,Neeman and Lipman (among others).


Bivariant Hochschild theory Definition of a bivariant Hochschild theory

The theory I

To define the theory we need a derived category incarnation of theHochschild complex.For each scheme X in S with structure map x : X → S take

δx = δ : X → X ×S X ,

the canonical diagonal embedding.We define the complex

HX := Lδ∗Rδ∗OX

its homologies are the sheafified Hochschild homology.



The theory II

As X is separated over S then δ is a closed embedding, therefore δ∗ isexact and δ∗ = Rδ∗.Note that the composition of δ∗ and Lδ∗ is not the derivative of thecomposition that is trivially the identity functor.

To relate this to the familiar Hochschild homology, consider the string ofisomorphisms

HX = Lδ∗δ∗OX = δ−1δ∗OX ⊗Lδ−1OX×SX

OX = OX ⊗Lδ−1OX×SX

OX

In the affine case with X = Spec(A) and S = Spec(R), A is flat over R, its−i th cohomology

H−i (HX ) = (H−i (A⊗LA⊗RA

A))∼ = (TorA⊗RAi (A,A))∼

corresponding to the usual identification HHi (A|R) = TorA⊗RAi (A,A).



The theory III

Let us recall some functorialproperties of the Hochschildcomplex. Let f : X → Y be amorphism in S and considerthe commutative square

X × X Y × Y

X Yf

f × f

δx δyd

We have the following composition of natural transformations

Lf ∗Lδ∗yδy ∗ −→Lδ∗xL(f × f )∗δy ∗ −→ Lδ∗xLδx∗Lf ∗

We apply it to OY and obtain the canonical morphisms

f ] : Lf ∗HY −→ HX and its adjoint f] : HY −→ Rf∗HX

Satisfying transitivity, i.e. (gf )] ∼= f ]Lf ∗(g ]) and (gf )] ∼= g]Rg∗(f]).

Proposition

If f : X → Y is etale then f ] : Lf ∗HY → HX is an isomorphism.



The theory IV

We associate to each map f : X → Y in S the graded module

HHi (Xf−→ Y ) := ExtiX (HX , f

!HY ) (i ∈ Z)

Note that ExtiX (HX , f!HY ) ∼= HomD(X )(HX , f

!HY [i ]).

We define the three operations:

1 The composition uses composition in the derived category.

2 The push forward denoted fH is defined through the functor Rf∗, thecovariant behavior of the Hochschild complex for a composition

Xf−→ Y

g−→ Z and the duality trace∫f .

3 The pull back, denoted gH is defined through the functor Lg∗, thecontravariant behavior of the Hochschild complex and the basechange isomorphism from duality g ′∗f ! → f ′!g∗.Its definition requires the etale condition on the independent squares.


Bivariant Hochschild theory Checking the compatibilities

Simple compatibilities

Proposition

The product is associative, i.e. A1 holds.

It follows from pseudo functoriality of the twisted inverse image.

Proposition

The push forward is functorial, i.e. A2 holds.

It follows from pseudo functoriality of f ! and Rf∗, functorial properties of∫f and the covariant behavior of the Hochschild complex.

Proposition

The pull back is functorial, i.e. A3 holds.

It follows from pseudo functoriality of f ! and Lg∗, functorial properties ofthe base change isomorphism from duality and the contravariant behaviorof the Hochschild complex.



Double compatibilities I

Proposition

Products and push forward commute, i.e. A12 holds.

It follows from the fact that the pseudo functoriality of f ! isomorphismand

∫f are natural transformations.

Proposition

Products and pull back commute, i.e. A13 holds.

It follows from the naturality of the base change isomorphism from dualityand the compatibility of the pseudo functoriality of f ! and the map thatexpresses the contravariant behavior of the Hochschild complex.



Double compatibilities II

Proposition

Push forward and pull back commute, i.e. A23 holds.

For α ∈ HHi (Xgf−→ Z ), to compare f ′H(hH(α)) and hH fH(α) we have to

express them through the commutativity of the corresponding diagrams.Next we construct a diagram that uses the naturality of the base changeisomorphism from duality, its compatibility with the pseudo functoriality off !, and the compatibility of the contravariant and covariant behavior ofHX together with the usual base change between ()∗ and ()∗.

Finally one appeals to thecommutative diagram of naturaltransformations. It expresses thecompatibility of

∫f with both

base-changes.

h′∗Rf∗f! h′∗

Rf ′∗h′′∗f ! Rf ′∗f

′!h′∗bch!

∫f

bch∗∫f ′



The projection formula

Proposition

The projection formula holds, i.e. A123 holds.

The result follows from the commutativity of a rather complicated diagramin which some parts commute due to naturalities and properties of thebehavior of the Hochschild complex as in the previous results.One concludes using the following key ingredient:

Lemma

In an independent square of S as before with f proper, the composition

f !g∗ −→ g ′∗g′∗f !g∗

bch!−→ g ′∗f′!g∗g∗ −→ g ′∗f

′!

is an isomorphism when restricted to D+qc(Y ′) and its inverse is

f !g∗

∫f ′←− f !g∗f

′∗f′! ←− f !f∗g

′∗f′! ←− g ′∗f

′!



Summing up

Theorem

Let S be a base scheme and R := Γ(S ,OS). The triple (S,GrR-Mod,HH)formed by

the underlying category S = Schfltf(S),

the category of values GrR-Mod,

the theory HH: Arr(S)→ GrR-Mod defined through the Hochschildcomplex

is a bivariant theory in the sense of Fulton-MacPherson.


Bivariant Hochschild homology and cohomology

Outline §3



3 Bivariant Hochschild homology and cohomologyHomology and cohomologyRelation to Caldararu’s theoryRelation to Hodge cohomology



Bivariant Hochschild homology and cohomology Homology and cohomology

Bivariant Hochschild cohomology modules

Let X ∈ S. The bivariant Hochschild cohomology modules or bivariantHochschild cohomology is defined as

HHi (X ) := HHi (Xid−→ X ) = ExtiX (HX ,HX )

The cup product

^ : HHi (X )⊗ HHj(X ) −→ HHi+j(X )

is associated to the composition Xid−→ X

id−→ X . There are pull backhomomorphisms

f H : HHi (X ) −→ HHi (X ′)

for every etale morphism f : X ′ → X .

These properties give HH∗ the structure of a ring-valued contravariantfunctor for etale morphisms.


Bivariant Hochschild homology and cohomology Homology and cohomology

Bivariant Hochschild homology modules

Let x : X → S be the structural map. The bivariant Hochschild homologymodules or bivariant Hochschild homology is defined as

HHi (X ) := HH−i (Xx−→ S) = Ext−iX (HX , x

!OS)

The cap product

_ : HHi (X )⊗ HHj(X ) −→ HHi+j(X )

is given by composition for the morphisms Xid−→ X

x−→ S .

Associated to the composition X ′f−→ X

x−→ S , with f proper (confined),there are push forward homomorphisms

fH : HHi (X ′) −→ HHi (X )

This gives HH∗ the structure of a covariant functor (for proper maps) thatis a module over HH∗ and satisfies the projection formula by A123:

fH(f H(β) _ α) = β _ fH(α) (f confined)Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue – July 2009 28 / 48

Bivariant Hochschild homology and cohomology Relation to Caldararu’s theory

Relation to Caldararu’s Hoschschild homology for schemes

According to Caldararu it makes sense to define Hoschschild homology forsmooth S-schemes X

x−→ S of relative dimension n, as follows

HHCali (X ) = HomD(X×X )(δ!OX , δ∗OX [i ])

Unravelling Caldararu’s definition δ!OX = δ∗ω−1X [−n]. Now we compute

HomD(X×X )(δ∗ω−1X [−n], δ∗OX [i ]) ∼= HomD(X )(OX , ωX [n]⊗ δ!δ∗OX [i ])

but ωX [n] = x !OS and so,

x !OS ⊗ δ!δ∗OX [i ] ∼= δ!OX×X ⊗ Lδ∗δ∗x!OS [i ] ∼= δ!δ∗x

!OS [i ]

Therefore, using the adjunctions,

HHCali (X ) = HomD(X )(OX , δ

!δ∗x!OS [i ]) ∼= HomD(X )(Lδ∗δ∗OX , x

!OS [i ])

and

HomD(X )(Lδ∗δ∗OX , x!OS [i ]) = HomD(X )(HX , x

!OS [i ]) = HHi (X )


Bivariant Hochschild homology and cohomology Relation to Caldararu’s theory

Relation to Caldararu’s Hoschschild cohomology forschemes

In the case of cohomology the situation is quite different. Caldararu’sdefinition (that in fact goes back to Kontsevich) is:

HHiCal(X ) = HomD(X×X )(δ∗OX , δ∗OX [i ])

Using the adjunctions,

HomD(X×X )(δ∗OX , δ∗OX [i ]) ∼= HomD(X )(Lδ∗δ∗OX ,OX [i ])

Now we recall that OX is direct summand of Lδ∗δ∗OX . This gives us asplit map

HHiCal(X ) = HomD(X )(HX ,OX [i ]) −→ HomD(X )(HX ,HX [i ]) = HHi (X )


Bivariant Hochschild homology and cohomology Relation to Hodge cohomology

The HKR isomorphism

Let now S = Spec(k) where k is a field of characteristic 0.Assume that X is smooth over k . Let n = dim(X ).We have the following

Theorem

There is a canonical quasi-isomorphism

HX∼=

n⊕p=0

ΩpX [p]

The second complex is understood as a zero differential complex.It was stated originally for affine schemes but it can be globalized bycompatibility with localization. The theorem is due to Hochschild,Konstant and Rosenberg.



Computation of HHi (X ), I

We start by recalling that in this case x !k ∼= ΩnX [n]. (Note that k = OS).

Also, there is a perfect pairing

ΩpX ⊗OX

Ωn−pX −→ Ωn

X

And as a consequence Ωn−pX∼= HomX (Ωp

X ,ΩnX ).

In our case

HHi (X ) = HomD(X )(HX , x!k[i ]) = HomD(X )(

n⊕p=0

ΩpX [p],Ωn

X [n + i ])

so

HHi (X ) =n⊕

p=0

HomD(X )(ΩpX [p],Ωn

X [n + i ])



Computation of HHi (X ), II

HHi (X ) =n⊕

p=0

HomD(X )(ΩpX [p],Ωn

X [n + i ])

=n⊕

p=0

Hi RΓ(X ,Hom•X (ΩpX ,Ω

nX [n − p]))

=n⊕

p=0

Hi RΓ(X ,Ωn−pX [n − p]))

=n⊕

q=0

Hq+i (X ,ΩqX )) (q := n − p)



Computation of HHi (X ), III

We got a relationship between bivarant Hochschild homology and Hodgehomology.

HHi (X ) =⊕

p−q=i

Hp(X ,ΩqX )

H0,0

H1,0 H0,1

H2,0 H1,1 H0,2

H2,1 H1,2

H2,2

With Hp,q = Hp(X ,ΩqX ).

Note:

The sums of the columnsof the Hodge diamondyields Hochschildhomology.

The sums of the rows ofthe Hodge diamond yieldsDe Rham cohomology.


Orientation and fundamental class

Outline §4




4 Orientation and fundamental classDefinition and meaning of the fundamental classOrientations in bivariant Hochschild HomologyA few words on proofs


Orientation and fundamental class Definition and meaning of the fundamental class

General definition of the fundamental class

Let f : X → Y be a map in S, we define a natural transformation

cf : Lδx∗Rδx∗Lf ∗ −→ f !Lδy

∗Rδy∗

Let Γ: X → X × Y be the graph of f , a closed immersion. The map cf isdefined as the composition of two natural maps

Lδ∗xδx∗Lf ∗af−→ LΓ∗Γ∗f

! bf−→∼

f !Lδ∗yδy∗

The map af is defined through a non trivial map

λ : δx∗Lf ∗ −→ L(id×f )∗Γ∗f!

using the duality trace, pseudo functorialities and base change.

The map bf is an isomorphism obtained using standard properties ofthe cohomological operations.



Meaning of the fundamental class I

If we apply cf to the sheaf OY , note that Lf ∗OY = OX , therefore wehave that

cf (OY ) : Lδx∗Rδx∗OX −→ f !Lδy

∗Rδy∗OY

or, otherwise saidcf (OY ) : HX −→ f !HY

This can be interpreted as saying that the fundamental class is a twistedcovariant functoriality of the Hochschild complex.



Meaning of the fundamental class II

Now we will look at the situation in which the morphism is the structuralmorphism of X , i.e. x : X → S , here HS = OS .The fundamental class becomes

cX : HX −→ f !OS

To grasp the significance of this map let us take −nth homology.H−n(HX ) = HHn(X ) and H−n(f !OS) = ωX , where ωX denotes thedualizing sheaf that can be charaterized through a universal property.We get

c−nX : HHn(X ) −→ ωX

Moreover, we may compose with the canonical map ΩnX → HHn(X ) and

obtain yet another version of the fundamental class

cX : ΩnX −→ ωX

It is an isomorphism precisely when X is smooth over S .Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality Purdue – July 2009 38 / 48

Orientation and fundamental class Orientations in bivariant Hochschild Homology

Gysin maps in bivariant theories

Let us discuss briefly Gysin maps (i.e. wrong-way functorialities) inbivariant Hochschild theory.

Let θ ∈ HHi (Xf−→ Y ), with i ∈ Z. There are two Gysin morphisms

1 θ∗ : HHj(Y ) −→ HHj−i (X ) (j ∈ Z).

2 θ∗ : HHj(X ) −→ HHj+i (Y ) (j ∈ Z and f confined)

Defined by

1 θ∗(α) = θ · α for α ∈ HHj(Y )

2 θ∗(β) = fH(β · θ) for α ∈ HHj(X ) (f confined).

These operations satisfy properties like functoriality, compatibility with pullback and with push forward, and certain mixed relations that followdirectly from the structure of bivariant theory.



Definition of orientation

Let, for a moment, (C,GrA,T ) be any bivariant theory. Let F be a classof maps in C stable for composition and containing the identity maps. Iffor every map f : X → Y in F there is given an element

c(f ) ∈ T ∗(Xf−→ Y )

such that

1 c(gf ) = c(f ) · c(g) for Xf−→ Y

g−→ Z in F.

2 c(idX ) = 1X ∈ T ∗(X ) for all X ∈ C

we say that c is a canonical orientation for the maps of F in thecorresponding bivariant theory. Sometimes we call the maps in F theorientable maps of C.



The fundamental class as a canonical orientation

Back to bivariant Hochschild theory (SS ,GrR-Mod,HH).Let f : X → Y be a flat morphism in S

Recall the fundamental class cf : HX → f !HY , i.e. cf ∈ HH0(Xf−→ Y ).

We have that

1 cidX = idHX∈ HH∗(X ).

2 Moreover:

Theorem

The fundamental class is transitive i.e. for flat maps Xf−→ Y

g−→ Z in S

cgf = cf · cg

Corollary

Flat morphisms constitute an orientable class of maps in bivariantHochschild theory and the orientation is given by the fundamental class.



Canonical Gysin maps in bivariant Hochschild homology

The fundamental class is an orientation for flat maps in bivariantHochschild homology, therefore we have wrong way functorialities, i.e.Gysin maps, defined for a flat map f : X → Y as follows

1 f H! : HH∗(Y )→ HH∗(X ) defined by f H! = (cf )∗.

2 fH! : HH∗(X )→ HH∗(Y ) defined by fH! = (cf )∗, where we assume inaddition f confined.

The properties of f H! and fH! follow from the structure of the bivarianttheory, let us spell them.



Properties of the canonical Gysin maps I

Let us discuss some properties of these morphisms

CG1 Functoriality: Let Xf−→ Y

g−→ Z be flat maps in S Then1 (gf )H!(α) = (f H!gH!)(α) for α ∈ HH∗(Z ).2 (gf )H!(β) = (gH!fH!)(β) for β ∈ HH∗(X ), if, in addition, f and g are

confined.

CG4 Mixed relations: Let Xf−→ Y be a flat map in S. Let α ∈ HH∗(X ),

β ∈ HH∗(Y ) and γ ∈ HH∗(Y ). Then1 fH!(f H(β) ^ α) = β ^ fH!(α), assuming f is etale.2 fH(α _ f H!(γ)) = fH!(α) _ γ, assuming f is confined.



Properties of the canonical Gysin maps II

CG3 Push forward: Let Xf−→ Y

g−→ Z , be flat maps in S, f is confined.Note that the following holds in HH∗(Y

g−→ Z )

fH(cgf ) = fH(cf · cg ) = fH(cf ) · cg

Then, we have the identities:1 fH(cgf ) · α = fH((gf )H!(α)) for α ∈ HH∗(Z ).2 gH(β · fH(cgf )) = (gf )H!(f H(β)) for β ∈ HH∗(Y ), where we assume

moreover that f is etale

An explicit computation of fH(cf ) would shed further light over theserelations.

What happened to CG2 (Pull back)?



Base change of the fundamental class

Let

Y ′ Y

X ′ Xg ′

g

f ′ fd

be an independent square in S.

Theorem

The fundamental class is compatible with base change, i.e.

gH(cf ) = cf ′



Pull back for the canonical Gysin maps

CG2 Pull back: Let d be an independent square as before

Y ′ Y

X ′ Xg ′

g

f ′ fd

with f (and therefore f ′) flat morphisms.By the previous theorem gH(cf ) = cf ′ . Then it follows that

1 g ′H(f ′H!(α)) = f H!(gH(α)) for α ∈ HH∗(Y ′), assuming g confined;

2 f ′H!(g ′H

(β)) = gH(fH!(β)) for β ∈ HH∗(X ), assuming f confined.


Orientation and fundamental class A few words on proofs

On the proofs of the structure I

The proofs of the existence of the bivariant Hochschild theory isbased on showing the commutativity of certain diagrams.

The constructions use properties of the cohomological operations,some formal properties and some further properties that have to bedeveloped from scratch.

The compatibilities need more and more complicated diagrams as wego from A1 all the way down to A123.


Orientation and fundamental class A few words on proofs

On the proofs of the structure II

The transitivity of the fundamental class amounts to saying that thediagram

δ∗xδx∗(gf )∗ (gf )!δ∗z δz∗

δ∗xδx∗f∗g∗ f !δ∗yδy∗g

∗ f !g !δ∗z δz∗

' '

cf g∗ f !cg

cgf

commutes.

This is achieved after decomposing it into diagram after diagram.This amounts to about 18 LATEXpages.

As Joe Lipman has remarked, advances in the problem of coherence incategories should provide a way to streamline the needed arguments.


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