Joyal's CatLab
Distributors and barrels

Please, feel free to leave a comment.

**Category theory** ##Contents * Contributors * References * Introduction * Basic category theory * Weak factorisation systems * Factorisation systems * Distributors and barrels * Model structures on Cat * Homotopy factorisation systems in Cat * Accessible categories * Locally presentable categories * Algebraic theories and varieties of algebras

Edit this sidebar

Contents

Barrels

Recall that a set of objects SS in category AA is said to be a sieve if the implication

target(f)Ssource(f)S\mathrm{target} (f)\in S \; \Rightarrow \; \mathrm{source}(f)\in S

is true for every arrow fAf\in A. Dually, a set of objects SAS\subseteq A is said to be a cosieve if the implication

source(f)Starget(f)S\mathrm{source}(f)\in S\; \Rightarrow \; \mathrm{target} (f)\in S

is true for every arrow fAf\in A. Notice that a subset SOb(A)S\subseteq Ob(A) is a sieve iff its complement is a cosieve. We shall often identify a sieve SOb(A)S\subseteq Ob(A) with the full subcategory of AA spanned by the objects in SS. If SAS\subseteq A is a sieve (resp. cosieve) then there exists a unique functor p:A[1]p:A\to [1] such that S=p 1(0)S=p^{-1}(0) (resp. S=p 1(1)S=p^{-1}(1)). This defines a bijection between the set of sieves (resp. cosieves) in AA and the set of functors A[1]A\to [1].

Definition

We shall say that an object of the category Cat/[1]\mathbf{Cat}/[1] is a barrel. The bottom of a barrel B=(B,p)B=(B,p) is the sieve B(0)=p 1(0)B(0)=p^{-1}(0) and its top is the cosieve B(1)=p 1(1)B(1)=p^{-1}(1).

The opposite of a barrel p:B[1]p:B\to [1] is a barrel p o:B o[1] o[1]p^o:B^o\to [1]^o\simeq [1]. Notice that B o(0)=B(1) oB^o(0)=B(1)^o and that B o(1)=B(0) oB^o(1)=B(0)^o. The opposition functor BB oB\mapsto B^o is an automorphism of the category Cat/[1]\mathbf{Cat}/[1].

Proposition

The category Cat/[1]\mathbf{Cat}/[1] is cartesian closed.

Proof

We have to show that every barrel (C,p)(C,p) is exponentiable. For this, it suffices to verify that the functor p:C[1]p:C\to [1] is a Conduché fibration. But the Conduché condition is trivially satisfied, since the arrow 010\to 1 in [1][1] has no nontrivial factorisation. Hence the category Cat/[1]\mathbf{Cat}/[1] is cartesian closed.

The internal hom Hom[B,C]Hom[B,C] between two barrels B=(B,p)B=(B,p) and C=(C,q)C=(C,q) can be described as follows.

We have Hom[B,C](0)=[B(0),C(0)]Hom[B,C](0)=[B(0),C(0)] and Hom[B,C](1)=[B(1),C(1)]Hom[B,C](1)=[B(1),C(1)]. Thus, an object of the category Hom[B,C]Hom[B,C] is either a functor B(0)C(0)B(0)\to C(0) or a functor B(1)C(1)B(1)\to C(1). The morphisms between two functors in Hom[B,C](0)Hom[B,C](0) is a natural transformation, and similarly for a morphism between two functors in Hom[B,C](1)Hom[B,C](1). If f:B(0)C(0)f:B(0)\to C(0) and g:C(1)D(1)g:C(1)\to D(1), then morphism fgf\to g in the category Hom[B,C]Hom[B,C] is a barrel map h:CDh:C\to D which is extending the functor fg:C(0)C(1)D(0)D(1)f\sqcup g:C(0)\sqcup C(1)\to D(0)\sqcup D(1). We leave to the reader the description of the composition law between the different kind of morphisms.

Consider the functor

(bot,top):Cat/[1]Cat×Cat(bot, top):\mathbf{Cat}/[1]\to \mathbf{Cat}\,\times\, \mathbf{Cat}

which associates to a barrel (C,p)(C,p) the pair of categories (C(0),C(1))(C(0),C(1)). The functor has a left adjoint and a right adjoint. The left adjoint associates to a pair of categories (A,B)(A,B) the category ABA\sqcup B equipped with the canonical functor AB11[1]A\sqcup B\to 1\sqcup 1 \to [1]. The right adjoint associates to a pair of categories (A,B)(A,B) their join ABA\star B equipped with the canonical functor AB11=[1]A\star B\to 1\star 1=[1].

Distributors

Recall that a distributor D:ABD\colon A ⇸B between two categories AA and BB is defined to be a functor D:A o×BSet.D\colon A^{o}\,\times\, B\to \mathbf{Set}. We shall regard the set D(a,b)D(a,b) as functor of two variables, contravariant in aAa\in A and covariant in bBb\in B. For example, the functor Hom A:A o×ASetHom_A \colon A^{o}\,\times\, A \to \mathbf{Set} is defining the unit distributor I A:AAI_A:A ⇸A. The distributors ABA ⇸B form a category

D(A,B)=[A o×B,Set].\mathbf{D}(A,B)=[A^o\,\times\, B, \mathbf{Set}].

The opposite of a distributor D:ABD\colon A ⇸B is the distributor D o:B oA oD^o\colon B^o ⇸A^o obtained by putting D o(b o,a o)=D(a,b)D^o(b^o,a^o)=D(a,b). We shall denote by f oD o(b o,a o)f^o\in D^o(b^o,a^o) the element corresponding to an alement fD(a,b)f\in D(a,b), so that

D o(b o,a o)={f o:fD(a,b)}={f:fD(a,b)} o=D(a,b) o.D^o(b^o,a^o)=\{f^o:f\in D(a,b)\}=\{f:f\in D(a,b)\}^o = D(a,b)^o.
Example

To every functor f:ABf:A\to B we can associate two distributors D(f):ABD(f)\colon A ⇸B and D o(f):BAD^o(f)\colon B ⇸A by putting

D(f)(a,b)=B(f(a),b)D o(f)(b,a)=B(b,f(a))D(f)(a,b)=B(f(a),b) \quad \quad D^o(f)(b,a)=B(b,f(a))

for every object aAa\in A and every object bBb\in B. Notice that D o(f)=D(f o) oD^o(f)=D(f^o)^o. An adjunction θ:fg\theta: f\vdash g between two functors f:ABf:A\to B and g:BAg:B\rightarrow A is exactly an isomorphism of distributors θ:D(f)D o(g)\theta:D(f)\simeq D^o(g).

Notation

it is often convenient to represent an element xD(a,b)x\in D(a,b) of a distributor D:ABD:A ⇸B by a dotted arrow or a wavy arrow

and more simply by a plain arrow x:abx:a\to b if the context is clear. If xD(a,b)x\in D(a,b) and vB(b,b)v\in B(b,b'), we shall write the element D(a,v)(x)D(a,b)D(a,v)(x)\in D(a,b') as a composite vx:abv x:a\to b',

Dually, if uA(a,a)u\in A(a',a), we shall write the element D(u,b)(x)D(a,b)D(u,b)(x)\in D(a',b) as a composite xux u,

Definition

To every distributor D:ABD\colon A ⇸B we can associate a category C=Col(D)=A DBC=Col(D)=A\star_D B called the collage of DD constructed as follows: Ob(C)=Ob(A)Ob(B)Ob(C)= Ob(A)\sqcup Ob(B) and for x,yOb(C)x,y\in Ob(C), we put

C(x,y) = D(x,y)ifxAandyB = A(x,y)ifxAandyA = B(x,y)ifxBandyB = ifxBandyA \begin{aligned} C(x,y) & = & D(x,y) \quad \mathrm{if} \quad x\in A \quad \mathrm{and} y\in B\\ & = & A(x,y) \quad \mathrm{if} \quad x\in A \quad \mathrm{and} y\in A\\ & = & B(x,y) \quad \mathrm{if} \quad x\in B \quad \mathrm{and} y\in B\\ & = & \emptyset \quad \mathrm{if} \quad x\in B \quad \mathrm{and} y\in A\\ \end{aligned}

The composition of arrows in A DBA\star_D B is defined as above. The associativity of composition is equivalent to the functorialty of DD.

There is a unique functor p:A DB[1]p:A\star_D B\to [1] such that p 1(0)=Ap^{-1}(0)=A and p 1(1)=Bp^{-1}(1)=B. This shows that the collage category Col(D)Col(D) carries the structure of a barrel. Notice the isomorphism Col(D) o=Col(D o)Col(D)^o=Col(D^o). We would like to say that collage operation is a functor

Col:DistCat/[1],Col:\mathbf{Dist} \to \mathbf{Cat}/[1],

but we have not yet defined the category of distributors Dist \mathbf{Dist}. A map DDD\to D' between two distributors D:ABD\colon A ⇸B and D:ABD'\colon A' ⇸B' is defined to be triple (f,α,g)(f,\alpha,g), where f:AAf:A\to A' and g:BBg:B\to B' are two functors and α:D(f o×g) *(D)\alpha:D\to (f^o\times g)^*(D') is a natural transformation,

α(a,b):D(a,b)D(fa,gb).\alpha(a,b):D(a,b)\to D'(f a, g b).

Maps of distributors can be composed and this defines the category Dist\mathbf{Dist}. It is obvious from this construction that the functor

(source,target):DistCat×Cat(source,target):\mathbf{Dist}\to \mathbf{Cat} \,\times\, \mathbf{Cat}

which associates to a distributor D:ABD\colon A ⇸B the pair of categories (A,B)(A,B) is a Grothendieck fibration. A map of distributors (f,g,α):DD(f,g,\alpha):D\to D' induces a functor between the collage categories,

f αg=Col(f,α,g):Col(D)Col(D).f\star_\alpha g=Col(f,\alpha,g):Col(D)\to Col(D').

This defines the collage functor

Col:DistCat/[1].Col:\mathbf{Dist} \to \mathbf{Cat}/[1].
Proposition

The collage functor is an equivalence of categories.

Proof

Let us define the inverse functor in the other direction

Γ:Cat/[1]Dist.\Gamma:\mathbf{Cat}/[1]\to \mathbf{Dist}.

It associates to a barrel C=(C,p)C=(C,p) the distributor Γ(C):C(0)C(1)\Gamma (C)\colon C(0) ⇸C(1) defined by putting Γ(C)(a,b)=C(a,b)\Gamma (C)(a,b)=C(a,b) for every pair of objects (a,b)C(0)×C(1)(a,b)\in C(0)\times C(1). It is easy to see that there is a natural isomorphism CCol(Γ(C))C\simeq Col(\Gamma (C)) for every barrel CC and a natural isomorphism DΓ(Col(D))D\simeq \Gamma (Col(D)) for every distributor DD.

Remark

The first of the following two triangles commutes strictly, whilst the second commutes only up to a natural isomorphism.

Definition

We shall say that a distributor D:ABD\colon A ⇸B is locally representable if the functor D(a,):BSetD(a,-):B\to \mathbf{Set} is representable for every object aAa\in A and say that DD is is representable if it is isomorphic to a distributor D(f)D(f) for a functor f:ABf:A\to B. Dually, we shall say that DD is locally corepresentable if the functor D(,b):A oSetD(-,b):A^o\to \mathbf{Set} is representable for every object bBb\in B and say that DD is corepresentable if it is isomorphic to a distributor D o(g)D^o(g) for a functor g:BAg:B\to A.

Proposition

A distributor D:ABD\colon A ⇸B is representable iff it is locally representable iff the full subcategory BA DBB\subseteq A \star_D B is reflexive. Dually, DD is corepresentable iff it is locally corepresentable iff the full subcategory AA DBA\subseteq A \star_D B is coreflexive.

Proof

(1\Rightarrow2) Let us suppose that DD is represented by a functor f:ABf:A\to B together with a natural isomorphism θ:D(f)D\theta:D(f)\simeq D. Then the map θ(a,):A(fa,)D(a,)\theta(a,-):A(f a,-)\simeq D(a,-) is a natural isomorphism
for every object aAa\in A and this shows that DD is locally representable. (2\Rightarrow3) If DD is locally representable, then for each object aAa\in A, there exists an object bBb\in B together with an element θ(a)D(a,b)\theta(a)\in D(a,b) which represents the functor D(a,)D(a,-). By definition, for every object cBc\in B and every element uD(a,c)u\in D(a,c), there exists a unique morphism v:bcv:b\to c such that vθ(a)=uv \theta(a)=u.

This means that the morphism θ(a):ab\theta(a):a\to b of the category A DBA \star_D B is reflecting the object aa into the subcategory BB. Thus, BB is a reflective subcategory of A DBA \star_D B. (3\Rightarrow1) If BB is a reflective subcategory of A DBA \star_D B, let us show that the distributor DD is representable. For each object aAa\in A, let us choose an object faBf a\in B together with a morphism θ(a):afa\theta(a):a\to f a which reflects the object aa into BB (we are using the axiom of choice here). Then for every morphism x:aax:a\to a' in AA, there exists a unique morphism y:fafay:f a \to f a such that θ(a)x=yθ(a)\theta(a')x =y\theta(a),

Let us put y=fxy=f x. This defines a functor f:ABf:A\to B which represents the distributor DD.

Remark

The proof that a locally representable distributor is representable depends on the axiom of choice. A locally representable distributor ABA ⇸B is called an anafunctor ABA\to B by Makkai here.

Remark

The functor D:[A,B]D(A,B)D:[A,B]\to \mathbf{D}(A,B) which associates to a functor f:ABf:A\to B the distributor D(f):ABD(f)\colon A ⇸B induces an equivalence between the category [A,B][A,B] and the full subcategory of D(A,B)\mathbf{D}(A,B) spanned by the representable distributors.

Remark

A functor f:ABf:A\to B has a right adjoint iff the distributor D(f):ABD(f)\colon A ⇸B is corepresentable, and it has a left adjoint iff the distributor D o(f)::BAD^o(f):\colon B ⇸A is representable.

The distributors also form a bicategory 𝒟𝒾𝓈𝓉\mathcal{Dist} whose objects are the small categories. The composite of a distributor X:ABX\colon A ⇸B with a distributor Y:BCY\colon B ⇸C is the distributor YX:ACY\circ X:A ⇸C defined by putting

where the tensor product between the contravariant functor Y(,c)Y(-,c) and the covariant functor X(a,)X(a,-) is used.

Proposition

The composition law

D(B,C)×D(A,B)D(A,C)\mathbf{D}(B,C)\,\times \, \mathbf{D}(A,B)\to \mathbf{D}(A,C)

is coherently associative and the distributor Hom A:A o×ASetHom_A:A^o\times A\to \mathbf{Set} is a unit I A:AAI_A\colon A ⇸A for this composition.

Idea of proof

This follows from the properties of the tensor product Y BXY\otimes_B X for YY a right BB-“module” and XX a left BB-“modules”. By definition, the tensor product is a quotient of the matrix-product

(Y× BX)(a,c)= bOb(B)Y(b,c)×X(a,b)(Y\,\times_B\, X)(a,c)=\bigsqcup_{b\in Ob(B)} Y(b,c)\,\times\, X(a,b)

and the canonical map

ϕ:Y× BXY BX\phi: Y\,\times_B \, X\to Y\otimes_B X

is universal among the maps satisfying the compatibility condition ϕ(y,fx)=ϕ(yf,x)\phi(y,f x)=\phi(y f,x) for (y,f,x)Y(b,c)×B(b,b)×X(a,b)(y,f,x)\in Y(b',c)\,\times\, B(b,b')\,\times\, X(a,b). If YY is a (B,C)(B,C)-“bimodule” and ZZ is a right CC-“module”, then the associativity isomorphism

Z C(Y BX)(Z CY) BXZ\otimes_C (Y\otimes_B X)\simeq (Z\otimes_C Y)\otimes_B X

can be obtained by showing that the two sides are actually isomorphic to the triple-tensor product Z CY BXZ\otimes_C Y\otimes_B X. By definition, the triple-tensor product of (Z,Y,X)(Z,Y,X) is a quotient of the triple matrix-product Z× CY× BXZ\,\times_C\, Y\,\times_B\, X, and the canonical map

ϕ:Z× CY× BXZ CY BX\phi: Z\,\times_C \,Y\,\times_B \, X\to Z\otimes_C Y\otimes_B X

is universal among the maps satisfying the “trilinearity” conditions ϕ(zh,y,x)=ϕ(z,hy,x)\phi(z h,y,x)=\phi(z,h y,x) and ϕ(z,yf,x)=ϕ(z,y,fx)\phi(z,y f,x)=\phi(z, y,f x). The unit isomorphism X AI AXX\otimes_A I_A\simeq X is obtained by showing that the right action X(a,b)×Hom(a,a)X(a,b)X(a',b)\,\times\, Hom(a,a')\to X(a,b) is a universal “bilinear” map.

The composition functor (E,D)ED(E,D)\mapsto E\circ D is divisible on both sides. Hence the functor E():D(A,B)D(A,C)E\circ (-): \mathbf{D}(A,B)\to \mathbf{D}(A,C) has a right adjoint SE\SS\mapsto E\backslash S for every distributor E:BCE\colon B ⇸C. By construction, we have

where the hom set of maps E(b,)T(a,)E(b,-)\to T(a,-) in the category [C,Set][C,\mathbf{Set}] is used. Dually the functor ()D:D(B,C)D(A,C)(-)\circ D:\mathbf{D}(B,C)\to \mathbf{D}(A,C) has a right adjoint SS/DS\mapsto S/D for every distributor D:ABD\colon A ⇸B. By construction, we have

where the hom set of maps D(,b)S(,c)D(-,b)\to S(-,c) in the category [A o,Set][A^o,\mathbf{Set}] is used.

Let us denote by CCAT\mathbf {CCAT} the category whose objects are the cocomplete locally small categories and whose morphisms are the cocontinuous functors. For any small category AA, we have

D(1,A)=[A,Set]=Set AandD(A,1)=[A o,Set]=Set A o\mathbf{D}(1,A)=[A,\mathbf{Set}]=\mathbf{Set}^A \quad \mathrm{and}\quad \mathbf{D}(A,\,1)=[A^o,\mathbf{Set}]=\mathbf{Set}^{A^o}

For a fixed D:ABD\colon A ⇸B, the composition functor

D !=D():D(1,A)D(1,B)D_!=D\circ (-): \mathbf{D}(1,A)\to \mathbf{D}(1,B)

is cocontinuous, since it has a right adjoint. This defines a functor DD !D\mapsto D_!,

() !:D(A,B)CCAT(Set A,Set B).(\,-\,)_!:\mathbf{D}(A,B) \to \mathbf {CCAT}\bigl(\mathbf{Set}^A,\mathbf{Set}^B\bigr).

Dually, the composition functor

D !=()D:D(B,1)D(A,1)D^!=(- )\circ D: \mathbf{D}(B,\,1)\to \mathbf{D}(A,\,1)

is cocontinuous, since it has a right adjoint. This defines a functor DD !D\mapsto D^!,

() !:D(A,B)CCAT(Set B o,Set A o).(\,-\,)^!:\mathbf{D}(A,B) \to \mathbf{CCAT}\bigl(\mathbf{Set}^{B^o},\mathbf{Set}^{A^o}\bigr).

Notice the canonical isomorphisms (ED) !=E !D !(E\circ D)_!=E_!D_! and (ED) !=D !E !(E\circ D)^!=D^!E^!.

Theorem

(Morita-Watts-Lawvere-Benabou) The functors () !(\,-\,)_! and () !(\,-\,)^! defined above are equivalence of categories.

To every functor f:ABf:A\to B in Cat\mathbf{Cat} is associated a pair of adjoint cocontinuous functors

f !:[A o,Set][B o,Set]:f *f_!:[A^o,\mathbf{Set}] \leftrightarrow [B^o,\mathbf{Set}]: f^*

where f *(G)=Gff^*(G)=G\circ f for a functor G:B oSetG:B^o\to \mathbf{Set}, and where f !(F)f_!(F) is the left Kan extension along ff of a functor F:A oSetF:A^o\to \mathbf{Set}. The two functors can represented by distributors. For every G:B1G\colon B ⇸1 we have

and this means that we have a canonical isomorphism f *=D(f) !f^*=D(f)^!, where D(f)D(f) is the distributor AB A ⇸B defined by putting D(f)(a,b)=B(fa,b). D(f) (a,b)=B(f a,b). For every F:A1F\colon A ⇸1 we have

and this means that we have a canonical isomorphism f !=D o(f) !f_!= D^o(f)^!, where D o(f) D^o(f) is the distributor BAB ⇸A obtained by putting D o(f)(b,a)=B(b,fa). D^o(f)(b,a)=B(b,f a). From the adjunction f !f *f_!\vdash f^*, we obtain an adjunction between distributors D(f)D o(f) D(f)\vdash D^o(f) (beware that (ED) !=D !E !(E\circ D)^!=D^!E^!). The unit of this adjunction is a map η:I AD o(f)D(f)\eta:I_A\to D^o(f) \circ D(f) in D(A,A)\mathbf{D}(A,A) and the counit is a map ϵ:D(f)D o(f)I B\epsilon: D(f)\circ D^o(f) \to I_B in D(B,B)\mathbf{D}(B,B). The counit ϵ\epsilon is the obvious map

defined by composing the pairs of arrows bfabb\to f a\to b'. The composite D o(f)D(f) D^o(f) \circ D(f) is the distributor

The unit η\eta is the map A(a,a)B(fa,fa)A(a,a')\to B(f a,f a') induced by the functor ff.

Duality

The bicategory of distributors 𝒟𝒾𝓈𝓉\mathcal{Dist} is symmetric monoidal. The tensor product of a distributor X:AAX\colon A ⇸A' with a distributor Y:BBY\colon B ⇸B' is the distributor X×Y:A×BA×B X\,\times\, Y\colon A\,\times\, B ⇸A'\,\times \,B' defined by putting

(X×Y)(a,b;a,b)=X(a,a)×Y(b,b).(X\,\times\, Y)(a,b;a',b')=X(a,a')\,\times\, Y(b,b').

The tensor product functor

×:D(A,A)×D(B,B)D(A×B,A×B)\times :\mathbf{D}(A,A')\, \times\, \mathbf{D}(B,B') \to \mathbf{D}(A\,\times\, B ,A'\, \times\, B')

is really a cartesian product in the category Dist\mathbf{Dist}. More precisely, we have a canonical isomorphism

Col(X×Y)=Col(X)× [1]Col(Y)Col(X\, \times\, Y)=Col(X)\,\times_{[1]}\, Col(Y)

in the category of cylinders Cat/[1]\mathbf{Cat}/[1], where Col(X)Col(X) is the collage barrel of a distributor XX.

If AA, BB and CC are small categories, then the categories D(A×B,C)\mathbf{D}(A\,\times\, B,C) and D(B,A o×C)\mathbf{D}(B,A^o\,\times\, C) are equivalent, since they are both isomorphic to the the category

[A o×B o×C,Set].[A^o\,\times\, B^o\,\times\, C, \mathbf{Set}].

The equivalence

Θ:D(A×B,C)D(B,A o×C)\Theta:\mathbf{D}(A\,\times\, B,C)\simeq \mathbf{D}(B,A^o\,\times\, C)

is actually natural when BB and CC are varying in the bicategory of distributors. In other words, the endo-functor A×()A\times (-) of the bicategory 𝒟𝒾𝓈𝓉\mathcal{Dist} is left adjoint to the endo-functor A o×()A^o\times (-). It follows that the objects AA and A oA^o are mutually dual in the symmetric monoidal bicategory 𝒟𝒾𝓈𝓉\mathcal{Dist}. Hence the bicategory 𝒟𝒾𝓈𝓉\mathcal{Dist} is compact closed. Let us examine this duality explicitly. If B=1B=1 and C=AC=A, then Θ(1 A)\Theta(1_A) is a distributor η A:1A o×A\eta_A\colon 1 ⇸ A^o\,\times\, A . If B=A oB=A^o and C=1C=1, then Θ 1(1 A o)\Theta^{-1}(1_{A^o}) is a distributor ϵ A:A×A o1.\epsilon_A\colon A\,\times\, A^o ⇸ 1. The distributor η A\eta_A is given by the functor Hom A:1 o×(A o×A)SetHom_A:1^o\,\times\, (A^o\,\times \, A)\to \mathbf{Set} and the distributor ϵ A\epsilon_A by the same functor Hom A:(A o×A)×1SetHom_{A}:(A^o\,\times\, A)\,\times\, 1\to \mathbf{Set}. Notice that η A(x 1,x 2)=A(x 1 o,x 2)\eta_A(x_1,x_2)=A(x_1^o,x_2) is a covariant functor of (x 1,x 2)A o×A(x_1,x_2)\in A^o\times A, whilst ϵ A(x 1,x 2)=A(x 1,x 2 o)\epsilon_A(x_1,x_2)= A(x_1,x_2^o) is a contravariant functor of (x 1,x 2)A×A o(x_1,x_2)\in A\times A^o. Notice also that ϵ=η o\epsilon=\eta^\o.

A monoidal bicategory is a tri-category. In this context, the adjunction identities are taking the form of a pair of isomorphisms between distributors,

α A:(ϵ A×A)(A×η A)I Aandβ A:(A o×ϵ A)(η A×A o)I A o.\alpha_A: (\epsilon_{A} \,\times\, A) \circ ( A \, \times\, \eta_A)\simeq I_{A}\quad \mathrm{and} \quad \beta_{A}:(A^o\, \times\, \epsilon_A ) \circ ( \eta_A\, \times \, A^o ) \simeq I_{A^o}.

The domain of α A\alpha_A is the composite DD of the distributors,

Thus, D(a,b)D(a,b) is a coend,

Notice that the product A(a,x 1)×A(x 2 o,x 3)A(a,x_1)\,\times \, A(x_2^o,x_3) in the integrant is covariant in (x 1,x 2,x 3)A×A o×A(x_1,x_2,x_3)\in A\,\times\, A^o\,\times\, A, whilst the product A(x 1,x 2 o)×A(x 3,b)A(x_1,x_2^o)\,\times \, A(x_3,b) is contravariant. The isomorphism α A:D(a,b)A(a,b)\alpha_A:D(a,b)\simeq A(a,b) is then induced by the map

A(x 1,x 2 o)×A(x 3,b)×A(a,x 1)×A(x 2 o,x 3)A(a,b) A(x_1,x_2^o)\, \times\, A(x_3,b)\,\times\, A(a,x_1)\,\times\, A(x_2^o,x_3)\to A(a,b)

which takes a quadruple (f 1,f 2,f 3,f 4)(f_1,f_2,f_3,f_4) to their composite f 2f 4f 1f 3:abf_2f_4f_1f_3:a\to b in AA,

Dually,the domain of β A\beta_A is defined to be the composite

This domain is isomorphic to D oD^o, since η A=ϵ A o\eta_A=\epsilon_A^o and ϵ A=η A o\epsilon_A=\eta_A^o. The isomorphism β A:D oI A o\beta_A:D^o\to I_{A^o} is induced by the isomorphism α A:DI A\alpha_A:D\to I_A.

Exercises

Exercise

Show that the functor

(bot,top):Cat/[1]Cat×Cat(bot,top):\mathbf{Cat}/[1]\to \mathbf{Cat}\, \times\, \mathbf{Cat}

is essentially a Grothendieck bifibration.

Exercise

In a compact monoidal category, every map f:XYf:X\to Y has a transpose tf:Y *X *{}^t f:Y^*\to X^*. In the monoidal bicategory of distributors, show that the transpose of a distributor D:ABD\colon A ⇸ B is its opposite D o:B oA oD^o\colon B^o ⇸ A^o.

References

Historical notes.

The notion of distributor was first introduced by Lawvere in a talk that he gave in 1966 at a meeting in Oberwolfach. They were used to represent cocontinuous functors between presheaf categories. The theory of distributors was later developed extensively by Bénabou who introduced also the notion of bicategory. The collage category is also due to him, but the terminology “collage category” was introduced by Street in his Rendiconti paper.

In a letter adressed to me (April 11,2010), Anders Kock has expressed his view on the invention of distributors. I find his opinion worth to be made public (with his permission):

The notion of profunctor/bimodule was not a creation of 1966, but is a result of an evolution, which includes for instance the section (p.22-23) in Cartan-Eilenberg on (the bicategory of) bimodules, and their tensor product (= composition in the bicategory), and the tensor product of a covariant and a contravariant functor on a category, cf. Watts’ contribution in the LaJolla volume. The important thing about the evolution of these notions was that they led to the notion of bicategory - likewise an evolution having many stages and inputs, including 2-categories, and monoidal categories. In this evolution, Benabou is a main actor. The identification of profunctors with cocontinuous functors between categories of presheaves is likewise part of an evolution, where ancestors are Morita’s characterization of equivalences between module categories, and Watts’ characterization of cocontinuous functors between module categories (Proc.Amer.Math.Soc, 1960).

I can agree with that. Most mathematical ideas are the result of an evolution. Mutations are playing an important role in biological evolution. What about mutations of mathematical ideas? What could be the role of mutations in the evolution of mathematical ideas?

The notion of symmetric monoidal bicategory has a complicated history. Let me quote Street:

An important point here was the realization that there are really three kinds of commutativity for monoidal bicategories and then it was important to find the correct definition of braided monoidal bicategory. Kapranov and Voevodsky made a definition of braiding on (I think) a slightly restricted notion of monoidal bicategory: and they left out an axiom. The missing axiom was corrected by Larry Breen and Baez-Neuchl. In my paper with Day we made up the term “syllepsis” for the extra notion of commutativity between “braiding” and “symmetry”.

Let me quote Breen:

Actually, I did not “correct” the Kapranov -Voevodsky definition, in fact I was aware of the full definition at least as far back as 1988. I wrote about this in a letter to Deligne, dated 4 Feb. 1988, (now on the website). I did not write this up as a paper at the time, but returned to this topic (in a more general topos context) in the last chapter of my Asterisque monograph (vol. 225, 1994), where I discuss these questions and in particular explicitly mention on page 148 the necessary correction to Kapranov-Voevodsky. In my letter the associativities are strict, so that only the commutativity conditions are considered. I had played around with the more complete diagrams involving both associativity and commutativity but I hadn’t at the time worked out the lax definition of the 2-categorical braiding axioms as Kapranov-Voevodsky did in their paper.

Many people have discovered independently that the symmetric monoidal bicategory 𝒟𝒾𝓈𝓉\mathcal{Dist} is compact closed. It was known to the Australian school for several decades. The general case of the symmetric monoidal bicategory of VV-distributors, for a closed symmetric monoidal category VV, is treated in the paper of Day and Street referred to below.

Books:

F. Borceux: Handbook of Categorical Algebra in 3 volumes, CUP, (1994)

Papers:

J. Baez, M. Neuchl Braided Monoidal 2-categories, Advances in Math, 121 (1996) 196-244

J. Bénabou: Introduction to bicategories, Springer Lecture Notes in Math 47, 1-17, (1967)

J. Bénabou: 2-Dimensional limits and colimits of distributors, Mathematisches Forschungsinstitut Oberwolfach, Tagungsbericht 30 (1972) 6–7.

J. Bénabou: Les Distributeurs, Univ. Catholique de Louvain, Séminaire de Mathématiques Pures, Rapport 33 (1973)

J. Bénabou: Distributors at work, (2000), pdf

B. Day, R. Street: Monoidal bicategories and Hopf algebroids, Advances in Math. 129 (1997) 99-157; MR99f:18013.

M. Kapranov and V.Voevodsky: 2-categories and Zamolodchikov tetrahedra equations, Proc. Symp. Pure Math. 56 (1994), part 2, p. 177-259.

M. Kapranov and V.Voevodsky: Braided monoidal 2-categories and Manin-Schechtman higher braid groups, J. Pure Appl. Algebra, 92 (1994), 241-267.

R. Street: Cauchy characterization of enriched categories, Rendiconti del Seminario Matematico e Fisico di Milano 51 (1981) 217-233; MR85e:18006. Reprints in Theory and Applications of Categories 4 (2004) 1-16, link

R. Street: Two constructions on lax functors, Cahiers topologie et géométrie différentielle 13 (1972) 217-264; MR50#436; numdam

Revised on September 10, 2012 06:29:59 by David Roberts