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June 27, 2014

Enriched Indexed Categories, Again

Posted by Emily Riehl

Guest post by Joe Hannon.

As the final installment of the Kan extension seminar, I’d like to take a moment to thank our organizer Emily, for giving all of us this wonderful opportunity. I’d like to thank the other participants, who have humbled me with their knowledge and enthusiasm for category theory and mathematics. And I’d like to thank the nCafé community for hosting us.

For the final paper of the seminar, we’ll be discussing Mike Shulman’s Enriched Indexed categories.

The promise of the paper is a formalism which generalizes ordinary categories and can specialize to enriched categories, internal categories, indexed categories, and even some combinations of these which have found use recently. In fact the paper defines three different notions of such categories, so-called small 𝒱\mathcal{V}-categories, indexed 𝒱\mathcal{V}-categories, and large 𝒱\mathcal{V}-categories, where 𝒱\mathcal{V} is an indexed monoidal category. For the sake of brevity, we’ll be selective in this blog post. I’ll quickly survey the background material, the three definitions, and their comparisons, and then I want to look at limits in enriched indexed categories. Note also that Mike himself made a post on this paper here on the nCafe in 2012, hence the title.

Indexed monoidal categories

An S\mathbf{S}-indexed monoidal category is pseudofunctor 𝒱:SMonCat\mathcal{V}\colon\mathbf{S}\to\text{MonCat}, where S\mathbf{S} is assumed to have finite products and be endowed with its cartesian monoidal structure, and MonCat\text{MonCat} is the 2-category of monoidal categories and strong monoidal functors (functors who preserve monoidal structure up to coherent isomorphism). We’ll use the script 𝒱\mathcal{V} for an enriched indexed monoidal category, and bold V\mathbf{V} for an ordinary monoidal category.

We will notate the image monoidal category of an object XSX\in\mathbf{S} as 𝒱 X\mathcal{V}^X and arrow f:XYf\colon X\to Y goes to f *:𝒱 Y𝒱 X.f^\ast\colon \mathcal{V}^Y\to\mathcal{V}^X. By the Grothendieck construction we may equivalently regard this as a fibration 𝒱S\int\mathcal{V}\to\mathbf{S} which is strict monoidal (preserves the monoidal structure on the nose), and the monoidal structure preserves the cartesian morphisms of the fibration. We think of 𝒱 1\mathcal{V}^1 (where 11 is the terminal object in S\mathbf{S}) as the underlying monoidal category of our indexed monoidal category, with the other fibers related by pullback by the terminal morphism.

The two principal examples of indexed monoidal categories are Fam(V)\mathit{Fam}(\mathbf{V}) and Self(S)\mathit{Self}(\mathbf{S}), out of which we will construct V\mathbf{V}-enriched categories and S\mathbf{S}-internal categories, respectively.

For Fam(V)\mathit{Fam}(\mathbf{V}), let S\mathbf{S} be the category of sets and V\mathbf{V} be an ordinary monoidal category, and to a set XX associate the category V X\mathbf{V}^X of XX-indexed objects in V\mathbf{V} and pointwise morphisms, and monoidal structure also given pointwise. If f:XYf\colon X\to Y then we have a functor f *:V YV Xf^\ast\colon\mathbf{V}^Y\to\mathbf{V}^X given by (f *B) x=B f(x),(f^\ast B)_x=B_{f(x)}, for B=(B y) yYV Y.B=(B_y)_{y\in Y}\in\mathbf{V}^Y.

And for Self(S)\mathit{Self}(\mathbf{S}) let S\mathbf{S} be any category with finite limits, and to each object XSX\in\mathbf{S} associate the category SX\mathbf{S}\downarrow X with its cartesian structure given by pullbacks. Then for BSYB\in\mathbf{S}\downarrow Y and f:XYf\colon X\to Y, we have f *B=X×YB,f^\ast B=X\underset{Y}{\times}B, the pullback again.

In any indexed monoidal category our fibers have a monoidal product by assumption, which we will call the fiberwise product, and denote A XBA\otimes_X B, for A,B𝒱 X.A,B\in\mathcal{V}^X. There is additionally an external product defined on 𝒱\int\mathcal{V}, as was implicit in our claim that the Grothendieck construction yields a strict monoidal fibration. We denote this by ABA\otimes B, for A𝒱 XA\in\mathcal{V}^X and B𝒱 YB\in\mathcal{V}^Y and call it the external product. It is related to the fiberwise product by the formula AB=π A *X A×Bπ B *Y,A\otimes B=\pi_A^\ast X\otimes_{A\times B}\pi_B^\ast Y, which is familiar from the theory of bundles or sheaves.

Additionally, if 𝒱\mathcal{V} satisfies some completeness properties (existence of S\mathbf{S}-indexed coproducts), then there is a third product structure called the canceling product. If every f *:𝒱 Y𝒱 Xf^\ast\colon \mathcal{V}^Y\to \mathcal{V}^X has a left adjoint f !:𝒱 X𝒱 Yf_!\colon \mathcal{V}^X\to \mathcal{V}^Y, and for any pullback square

h k f g \begin{matrix} & \overset{h}{\to} & \\ k \downarrow & & \downarrow f\\ & \underset{g}{\to} & \\ \end{matrix}

in S\mathbf{S} the Beck-Chevalley transformation k !h *g *f !k_!h^\ast\to g^\ast f_! is an isomorphism, then we say that 𝒱\mathcal{V} has S\mathbf{S}-indexed coproducts, and we define the canceling product in terms of the external tensor product as A [Y]B=π Y !Δ Y *(AB),A\otimes_{[Y]}B={\pi_Y}_!\Delta_Y^\ast (A\otimes B), for A𝒱 X×YA\in\mathcal{V}^{X\times Y} and B𝒱 Y×ZB\in\mathcal{V}^{Y\times Z}.

In Fam(V)\mathit{Fam}(\mathbf{V}), the external product is given by (AB) (x,y)X×Y=A xB y(A\otimes B)_{(x,y)\in X\times Y}=A_x\otimes B_y, has indexed coproducts if V\mathbf{V} has coproducts, and in that case the canceling product is given by “matrix multiplication” A [Y]B= yYA (x,y)B (y,z).A\otimes_{[Y]} B=\coprod_{y\in Y} A_{(x,y)}\otimes B_{(y,z)}.

In Self(S)\mathit{Self}(\mathbf{S}), the external product is just the cartesian product in S\mathbf{S}, has indexed coproducts, and the canceling product is given by a pullback which forgets the map to YY.

The various products can be combined; for AA in the fiber over X×Y×ZX\times Y\times Z and BB over Y×Z×WY\times Z\times W, then we can cancel the ZZ dependence and take the fiberwise product over YY, leaving an object over X×Y×W.X\times Y\times W.

Since we will be enriching over our indexed monoidal categories, we may also ask for an indexed version of closedness. If each fiber is closed as a monoidal category (meaning that the tensor product has a right adjoint), and in addition each pullback functor between fibers preserves this fiberwise hom, then we say the indexed monoidal category is closed. In that case, the other tensor products also admit adjoints: the canceling hom 𝒱 [Y](B,C)\mathcal{V}^{[Y]}(B,C) is the right adjoint of the external tensor, and the external hom 𝒱(B,C)\mathcal{V}(B,C) is the right adjoint of the canceling tensor.

In Fam(V)\mathit{Fam}(\mathbf{V}), the external hom is given by (AB) (x,y)X×Y=A xB y(A\otimes B)_{(x,y)\in X\times Y}=A_x\otimes B_y, and the canceling hom is given by “matrix multiplication” A [Y]B= yYA (x,y)B (y,z).A\otimes_{[Y]} B=\coprod_{y\in Y} A_{(x,y)}\otimes B_{(y,z)}.

In Self(S)\mathit{Self}(\mathbf{S}), the external hom is just the cartesian product in S\mathbf{S}, has indexed coproducts, and the canceling product is given by a pullback which forgets the map to YY.

Small 𝒱\mathcal{V}-categories


With the notion of an indexed monoidal category in hand, we may now meet the first of the three notions of an enriched indexed category:

A small 𝒱\mathcal{V}-category AA is an object ϵAS\epsilon A\in \mathbf{S} called the extent, an object A̲𝒱 ϵA×ϵA\underline{A}\in\mathcal{V}^{\epsilon A\times \epsilon A} which we think of as the arrows of AA, along with morphisms I ϵAidsA̲I_{\epsilon A}\overset{\text{ids}}{\to} \underline{A} and A̲ ϵAA̲A̲\underline{A}\otimes_{\epsilon A}\underline{A}\to \underline{A} satisfying the usual associativity and unital axioms:

A̲ϵA(A̲ϵAA̲) (A̲ϵAA̲)ϵAA̲ A̲ϵAA̲ A̲ϵAA̲ A̲ \begin{matrix} \underline{A}\underset{\epsilon A}{\otimes}(\underline{A}\underset{\epsilon A}{\otimes}\underline{A}) & \to & (\underline{A}\underset{\epsilon A}{\otimes}\underline{A})\underset{\epsilon A}{\otimes}\underline{A} & \to & \underline{A}\underset{\epsilon A}{\otimes}\underline{A}\\ \downarrow & & & & \downarrow \\ \underline{A}\underset{\epsilon A}{\otimes}\underline{A}& & \to & & \underline{A} \end{matrix}


A̲ I ϵA ϵAA̲ A̲ϵAA̲ I ϵA ϵAA̲ A̲ = = A̲ \begin{matrix} \underline{A} & \to & I_{\epsilon A}\otimes_{\epsilon A}\underline{A} & \to & \underline{A}\underset{\epsilon A}{\otimes}\underline{A} & \leftarrow & I_{\epsilon A}\otimes_{\epsilon A}\underline{A}& \leftarrow & \underline{A}\\ & & =\searrow & & \downarrow & & \swarrow = \\ & &&& \underline{A} && \end{matrix}


A functor f:ABf\colon A\to B of small 𝒱\mathcal{V}-categories is given by the data of a morphism of extents ϵf:ϵAϵB\epsilon f\colon \epsilon A\to \epsilon B in S\mathbf{S}, and a morphism of arrows A̲B̲\underline{A}\to\underline{B} in 𝒱\int\mathcal{V} such that I ϵA A̲ I ϵB B̲ \begin{matrix} I_{\epsilon A} & \to & \underline{A}\\ \downarrow & & \downarrow \\ I_{\epsilon B} & \to & \underline{B} \end{matrix} and A̲ ϵAA̲ A̲ar[d] B̲ ϵBB̲ B̲ \begin{matrix} \underline{A}\otimes_{\epsilon A}\underline{A} & \to & \underline{A}\ar[d]\\ \downarrow & & \downarrow \\ \underline{B}\otimes_{\epsilon B}\underline{B}& \to & \underline{B} \end{matrix} commute.

Natural transformations

A natural transformation α\alpha between functors f,g:ABf,g\colon A\to B of small 𝒱\mathcal{V}-categories is a morphism I ϵAB̲I_{\epsilon A}\to \underline{B} so that

A̲ A̲ϵAI ϵA fα B̲ ϵBB̲ I ϵAA̲ αg B̲ ϵBB̲ B̲ \begin{matrix} \underline{A}& \to & \underline{A}\underset{\epsilon A}\otimes I_{\epsilon A} & \overset{f\otimes\alpha}{\to} & \underline{B}\otimes_{\epsilon B} \underline{B}\\ \downarrow & & & & \downarrow \\ I_{\epsilon A}\otimes \underline{A} & \underset{\alpha\otimes g}{\to} & \underline{B}\otimes_{\epsilon B} \underline{B} & \to & \underline{B} \end{matrix}

With these definitions, 𝒱\mathcal{V}-categories, functors, and natural transformations constitute an ordinary 2-category.

Discrete enriched category

If 𝒱\mathcal{V} has S\mathbf{S}-indexed coproducts preserved by \otimes, then for any object XSX\in\mathbf{S} we have a small 𝒱\mathcal{V}-category δX\delta X with extent ϵ(δX)=X\epsilon(\delta X)=X and δX̲=(Δ X) !I X.\underline{\delta X}=(\Delta_X)_!I_X.


A profunctor H:ABH\colon A\nrightarrow B is an object H̲𝒱 ϵA×ϵB\underline{H}\in\mathcal{V}^{\epsilon A\times\epsilon B} with structure morphisms A̲ ϵAH̲H̲\underline{A}\otimes_{\epsilon A}\underline{H}\to\underline{H} and H̲ ϵBB̲H̲\underline{H}\otimes_{\epsilon B}\underline{B}\to\underline{H} so that the left and right actions are unital and associative and the left action commutes with the right action: A̲ ϵAA̲ ϵAH̲ A̲ ϵAH̲ A̲ ϵAH̲ H̲ \begin{matrix} \underline{A}\otimes_{\epsilon A}\underline{A}\otimes_{\epsilon A}\underline{H} & \to & \underline{A}\otimes_{\epsilon A}\underline{H}\\ \downarrow & & \downarrow \\ \underline{A}\otimes_{\epsilon A}\underline{H} & \to & \underline{H} \end{matrix} and H̲ ϵBB̲ ϵBB̲ H̲ ϵBB̲ H̲ ϵBB̲ H̲ \begin{matrix} \underline{H}\otimes_{\epsilon B}\underline{B}\otimes_{\epsilon B}\underline{B} & \to & \underline{H}\otimes_{\epsilon B}\underline{B}\\ \downarrow & & \downarrow \\ \underline{H}\otimes_{\epsilon B}\underline{B} & \to & \underline{H} \end{matrix} and H̲ A̲ ϵAH̲ H̲ \begin{matrix} \underline{H} & \to & \underline{A}\otimes_{\epsilon A}\underline{H}\\ & \searrow & \downarrow \\ & & \underline{H} \end{matrix} and H̲ H̲ ϵBB̲ H̲ \begin{matrix} \underline{H} & \to & \underline{H}\otimes_{\epsilon B}\underline{B}\\ & \searrow & \downarrow \\ & & \underline{H} \end{matrix} and A̲ ϵAH̲ ϵBB̲ H̲ ϵBB̲ A̲ ϵAH̲ H̲ \begin{matrix} \underline{A}\otimes_{\epsilon A}\underline{H}\otimes_{\epsilon B}\underline{B} & \to & \underline{H}\otimes_{\epsilon B}\underline{B}\\ \downarrow & & \downarrow \\ \underline{A}\otimes_{\epsilon A}\underline{H}& \to & \underline{H} \end{matrix}

We can restrict our profunctors as usual. Given H:ABH\colon A\nrightarrow B and f:AAf\colon A\to A' and g:BBg\colon B\to B', then we have a profunctor H(g,f)H(g,f) given by H(g,f)̲=(ϵf×ϵg) *H̲.\underline{H(g,f)}=(\epsilon f\times \epsilon g)^\ast\underline{H}. In particular for a functor ABA\to B, we have the representable profunctors B(1,f):ABB(1,f)\colon A\nrightarrow B and B(f,1):BA.B(f,1)\colon B\nrightarrow A.

If our indexed monoidal category has good completeness properties (S\mathbf{S}-indexed coproducts preserved by \otimes, and fiberwise coequalizers) then we define the composition of profunctors as (lemma 3.25) HK̲=coeq(H̲ [ϵB]B̲ [ϵB]K̲H̲ [ϵB]K̲). \underline{H\odot K}=\operatorname{coeq}\left(\underline{H}\otimes_{[\epsilon B]} \underline{B} \otimes_{[\epsilon B]}\underline{K}\rightrightarrows \underline{H}\otimes_{[\epsilon B]}\underline{K}\right).

If moreover the 𝒱\mathcal{V} is an S\mathbf{S}-indexed cosmos (i.e. closed as an indexed monoidal category, symmetric, complete and cocomplete with S\mathbf{S}-indexed products and coproducts), then \odot has left and right adjoints (lemma 3.27) 𝒱-Prof(A,C)(HK,L)𝒱-Prof(A,B)(H,KL)𝒱-Prof(B,C)(K,LH) \mathcal{V}\text{-}Prof(A,C)(H\odot K,L)\cong \mathcal{V}\text{-}Prof(A,B)(H,K\triangleright L) \cong \mathcal{V}\text{-}Prof(B,C)(K,L\triangleleft H) given by KL̲=eq(𝒱 [ϵC](K̲,L̲)𝒱 [ϵC](K̲,𝒱 [ϵC](C̲,L̲))) \underline{K\triangleright L}=\operatorname{eq}\left(\mathcal{V}^{[\epsilon C]}(\underline{K},\underline{L})\rightrightarrows\mathcal{V}^{[\epsilon C]}(\underline{K},\mathcal{V}^{[\epsilon C]}(\underline{C},\underline{L}))\right) and LH̲=eq(𝒱 [ϵA](H̲,L̲)𝒱 [ϵA](H̲,𝒱 [ϵA](A̲,L̲))). \underline{L\triangleleft H}=\operatorname{eq}\left(\mathcal{V}^{[\epsilon A]}(\underline{H},\underline{L})\rightrightarrows\mathcal{V}^{[\epsilon A]}(\underline{H},\mathcal{V}^{[\epsilon A]}(\underline{A},\underline{L}))\right).

Examples and two more definitions

As mentioned, the basic examples of monoidal indexed categories are 𝒱=Fam(V)\mathcal{V}=\text{Fam}(\mathbf{V}), in which a small 𝒱\mathcal{V}-category is a category enriched in V\mathbf{V}, and 𝒱=Self(S)\mathcal{V}=\text{Self}(\mathbf{S}), which gives a category internal to S\mathbf{S}. The paper also gives two alternate definitions of enriched indexed categories, which I will cite very briefly. An indexed 𝒱\mathcal{V}-category gives for each XSX\in\mathbf{S} a category enriched in 𝒱 X\mathcal{V}^X and functors relating the categories for each fiber (def 4.1), and this is seen to be a generalization of the ordinary indexed category and in fact every indexed 𝒱\mathcal{V}-category has a natural underlying ordinary S\mathbf{S}-indexed category (example 7.5).

And a large 𝒱\mathcal{V}-category is a kind of horizontal categorification of a small 𝒱\mathcal{V}-category, with collection of objects x,y,...x,y,... and for each object an extent ϵxS\epsilon x\in\mathbf{S} and for each pair of objects x,yx,y an object 𝒜(x,y)𝒱 ϵx×ϵy\mathcal{A}(x,y)\in\mathcal{V}^{\epsilon x\times\epsilon y} satisfying the usual axioms (def 5.1). Then there is a notion of 𝒱\mathcal{V}-fibrations and a Grothendieck-type construction which gives an equivalence between 𝒱\mathcal{V}-indexed categories and large 𝒱\mathcal{V}-categories (theorem 6.10).

The paper has many lovely examples, more than I want to discuss here. I’ll just mention one fun example from topology, one of the motivations for the paper (example 11.25): if we take for S\mathbf{S} the category of finite group objects in topological spaces denoted 𝒢\mathcal{G}, and for 𝒱\mathcal{V} the indexed monoidal category of based spaces with group actions, denoted Act(Top) *Act(\text{Top})_\ast. We have an Act(Top) *Act(\text{Top})_\ast-category 𝒢\mathcal{I}_\mathcal{G} with objects given by finite dimensional real representations ρ:GO(n)\rho\colon G\to O(n), with extent GG and hom-object 𝒢̲(G,G)\underline{\mathcal{I}_\mathcal{G}}(G,G') the space of linear isometric isomorphisms n n\mathbb{R}^n\to\mathbb{R}^{n'} plus basepoint with G×GG\times G' action by conjugation. The fiberwise monoidal structure is given by direct sum of representations. The presheaf category gives Anna Marie Bohmann’s global orthogonal spectra.

Center of the category of modules

This semester I attended a class at Boston University on noncommutative geometry by Ryan Grady. As a homework problem I was asked to show that the center of a category is isomorphic to the center of modules over that category, and then to generalize to the case of enriched categories and internal categories. Repeating a proof which is formally the same for three different contexts cries out for generalizing to a single unifying context, and enriched indexed categories promises to provide that context. So here is a fourth and (perhaps) final sketch of that proof.

The center Z(C)Z(C) of an ordinary category CC is defined to be the endomorphism monoid of the identity functor on CC. This is a construction worthy of being called the center since for a category with one object it produces the center of the endomorphism monoid. So it is the horizontal categorification of the classical notion. For an ordinary category a functor M:CSetM\colon C\to \text{Set} defines a notion of a (left) module MM over CC. We have an isomorphism between the center of the category of modules Z(C-Mod)Z(C\text{-Mod}) and Z(C)Z(C).

Briefly, in the case of an ordinary category, for each zZ(C)z\in Z(C) we obtain for each module MM a morphism of modules MzMz which is just the whiskered product MzMz of the functor MM with the natural transformation zz. It is a natural transformation on the identity functor on modules if it commutes with module morphism ff, which it does since each component of MzMz commutes with components of ff, which it does by centrality of zz. Conversely, given a central element ζ\zeta in Z(C-Mod)Z(C\text{-Mod}), for each module MM we have a morphism of modules ζ M:MM\zeta_M\colon M\to M. Any object aCa\in C may be viewed as a left module over CC by the contravariant Yoneda embedding, so we have ζ a:CC\zeta_a\colon C\to C. This module morphism as a natural transformation between functors has a component at aa, which is a function hom(a,a)hom(a,a).\hom(a,a)\to\hom(a,a). Then ζ a(1)\zeta_a(1) gives an element of Z(C)Z(C). By the Yoneda lemma all left-module morphisms CCC\to C are given by right multiplication, so we obtain an isomorphism of monoids Z(C)Z(C-Mod).Z(C)\cong Z(C\text{-Mod}).

Categories enriched over V\mathbf{V} naturally constitute a category enriched in V\mathbf{V}-categories, meaning instead of a hom-set of natural transformations between functors F,GF,G, we have an object of V\mathbf{V}, given by the end chom(Fc,Gc)\int_c \hom(Fc,Gc). If CC is a V\mathbf{V}-category, then so is C-Mod,C\text{-Mod}, and Z(C)Z(C-Mod)Z(C)\cong Z(C\text{-Mod}) is an isomorphism is of V\mathbf{V}-monoids.

In the case of a category C=(C 0;C 1;ids:C 0C 1;s,t:C 1C 0;comp:C 1× C 0C 1C 1)C=(C_0;C_1;\text{ids}\colon C_0\to C_1;s,t\colon C_1\to C_0;\text{comp}\colon C_1\times_{C_0}C_1\to C_1) internal to \mathcal{E}, a module MM over an internal category is also known as an internal diagram, and it is given by the data of a structure morphism MC 0M\to C_0 such that C 1×C 0M act M C 1 s C 0 \begin{matrix} C_1\underset{C_0}{\times} M & \overset{\text{act}}{\to}& M \\ \downarrow & & \downarrow \\ C_1 & \underset{s}{\to} & C_0 \end{matrix} commutes. This action is required to be associative and unital. The category of modules constitutes only an ordinary category, a subcategory of \mathcal{E}, and we obtain again an isomorphism of monoids.

Now let AA be a small 𝒱\mathcal{V}-category, for 𝒱\mathcal{V} an indexed monoidal category. Using the notation of the paper a one-sided left AA-module is a profunctor M:AI,M\colon A\nrightarrow I, where I=δ1I=\delta 1 is the discrete 𝒱\mathcal{V}-category whose extent is the terminal object of S\mathbf{S} and whose arrow object is the monoidal unit δI̲=I𝒱 1=V.\underline{\delta I}=I\in\mathcal{V}^1=\mathbf{V}.

A natural transformation of the identity functor AAA\to A is given by a morphism I ϵAzA̲I_{\epsilon A}\overset{z}{\to} \underline{A} in 𝒱\int\mathcal{V} so that A̲ A̲ϵAI ϵA A̲ϵAA̲ IϵAA̲ A̲ϵAA̲ A̲ \begin{matrix} \underline{A} & \to & \underline{A}\underset{\epsilon A}\otimes I_{\epsilon A} & \to & \underline{A}\underset{\epsilon A}\otimes \underline{A} \\ \downarrow & & & & \downarrow \\ I\underset{\epsilon A}\otimes \underline{A}& \to & \underline{A}\underset{\epsilon A}\otimes \underline{A}& \to & \underline{A} \end{matrix}

commutes. The collection of such morphisms is the center Z(A).Z(A). From such an arrow we obtain a morphism

MI ϵAϵAMz1A̲ϵAMM. M\to I_{\epsilon A}\underset{\epsilon A}\otimes M\overset{z\otimes 1}{\to} \underline{A}\underset{\epsilon A}\otimes M\to M.

This is a morphism of profunctors if it commutes with the profunctor action:

A̲ϵAM A̲ϵAI ϵAϵAM 1z1 A̲ϵAA̲ϵAM 1×act A̲ϵA act act M I ϵAϵAM z1 A̲ϵAM act M \begin{matrix} \underline{A}\underset{\epsilon A}{\otimes}M & \to & \underline{A}\underset{\epsilon A}{\otimes}I_{\epsilon A}\underset{\epsilon A}{\otimes}M & \overset{1\otimes z\otimes 1}{\to} & \underline{A}\underset{\epsilon A}{\otimes}\underline{A}\underset{\epsilon A}{\otimes}M & \overset{1\times\text{act}}{\to} & \underline{A}\underset{\epsilon A}{\otimes}\\ {\text{act}}\downarrow & & & & & & \downarrow{\text{act}} \\ M& \to & I_{\epsilon A}\underset{\epsilon A}\otimes M & \underset{z\otimes 1}{\to} & \underline{A}\underset{\epsilon A}\otimes M & \underset{\text{act}}{\to} & M \end{matrix} which commutes by a diagram chase.

Conversely, to every natural endomorphism ζ M:MM\zeta_M\colon M\to M of the category of profunctors M:AI,M\colon A\nrightarrow I, we associate an element of Z(A)Z(A). We notice that our small 𝒱\mathcal{V}-category AA may itself be viewed as a profunctor AI,A\nrightarrow I, giving us a component ζ A:A̲A̲.\zeta_A\colon \underline{A}\to \underline{A}. We obtain an element of Z(A)Z(A) by pre-composition with I ϵAA̲I_{\epsilon A}\to \underline{A}, since the following diagram commutes:

A̲ I ϵA ϵAA̲ ids1 A̲ ϵAA̲ ζ A1 A̲ ϵAA̲ A̲ ϵAI ϵA 1ids A̲ ϵAA̲ 1ζ A A̲ ϵAA̲ A̲ \begin{matrix} \underline{A} & \to & I_{\epsilon A}\otimes_{\epsilon A} \underline{A} & \overset{\text{ids}\otimes 1}{\to} & \underline{A}\otimes_{\epsilon A}\underline{A} & \overset{\zeta_A\otimes1}{\to} & \underline{A}\otimes_{\epsilon A}\underline{A}\\ \downarrow & & & & & & \downarrow \\ \underline{A}\otimes_{\epsilon A} I_{\epsilon A}& \underset{1\otimes\text{ids}}{\to} & \underline{A}\otimes_{\epsilon A} \underline{A} & \underset{1\otimes\zeta_A}{\to} & \underline{A}\otimes_{\epsilon A} \underline{A}& \to & \underline{A} \end{matrix} These maps Z(A)Z(Prof(A,I))Z(A)\leftrightarrow Z(\text{Prof}(A,I)) are inverse, which establishes the isomorphism, at least in the category of sets. Although this establishes the result for any small 𝒱\mathcal{V}-category, and hence for enriched, internal, indexed category, or combination categories, it is not an isomorphism of objects in the enrichment category. We have defined an ordinary 2-category of small 𝒱\mathcal{V}-categories, but for a full strength general result, we need instead a 𝒱\mathcal{V}-2-category, that is, a category enriched in 𝒱\mathcal{V}-categories, which would strengthen our result to an isomorphism of 𝒱\mathcal{V}-objects.


When I first read the abstract for this paper, I guessed naively that a framework that unified enriched categories with internal categories would use the language of monads, since both can be so succinctly described as monads in different 2-categories. Enriched categories are monads in the 2-category Mat(V)\text{Mat}(\mathbf{V}) of set indexed matrices with values in monoidal category V\mathbf{V}, and internal categories in \mathcal{E} are monads in the 2-category Span()\text{Span}(\mathcal{E}) of spans in .\mathcal{E}.

I was disappointed not to see 𝒱\mathcal{V}-categories as monads in the paper. But throughout the paper there are references to the technology of equipments, and one pleasant side effect of understanding enriched indexed categories in terms of equipments is that it becomes perfectly clear how to describe a 𝒱\mathcal{V}-category as a monad, in a way which includes matrices and spans as special cases.

Equipments have also been discussed here by Mike before, and so I want to rely on that background. But here is a brief recap of what is itself a “lightning-fast introduction to formal category theory”. Wood defined a 2-category with proarrow equipment, or an equipment for short, to axiomatize the properties of profunctors, as a functor between 2-categories which is bijective on objects, locally fully faithful, and taking each arrow to a left-adjoint. Such a structure enables the study of formal category theory, because objects, arrows, and 2-cells are not enough to reproduce all the constructions of 1-category theory. One needs something to abstract the behavior of hom-sets and profunctors.

Shulman argues persuasively that the more natural setting for this structure is not a 2-category, but rather a (pseudo) double category (ie a double category where composition of horizontal arrows is weak) whose vertical arrows are the arrows of our 2-category, and whose horizontal arrows are our profunctors. In this setting, here is the axiom that characterizes the profunctors. For any diagram “niche” of the form have also been discussed here

A B f g C K D \begin{matrix} A & & B\\ f \downarrow & & \downarrow g\\ C & \underset{K}{\nrightarrow} & D \end{matrix} there exists a filler K(f,g)K(f,g) 1-cell

A K(f,g) B f g C K D \begin{matrix} A & \overset{K(f,g)}{\nrightarrow} & B\\ f \downarrow & \Downarrow & \downarrow g\\ C & \underset{K}{\nrightarrow} & D \end{matrix} with the property that any other square whose vertical arrows factor through ff and gg

X Y fh gk C K D \begin{matrix} X & {\nrightarrow} & Y\\ fh \downarrow & \Downarrow & \downarrow gk\\ C & \underset{K}{\nrightarrow} & D \end{matrix}

itself also factors uniquely

X Y h ! k A K(f,g) B f g C K D \begin{matrix} X & {\nrightarrow} & Y\\ h \downarrow & \Downarrow \exists{!} & \downarrow k\\ A & \overset{K(f,g)}{\nrightarrow} & B\\ f \downarrow & \Downarrow & \downarrow g\\ C & \underset{K}{\nrightarrow} & D \end{matrix}

A double category satisfying this property is called a framed bicategory, which is equivalent to a 2-category with a proarrow equipment.

With this definition in hand, to an indexed monoidal category 𝒱:SMonCat\mathcal{V}\colon \mathbf{S}\to \text{MonCat} we associate a framed bicategory whose objects X,YX,Y are the objects of S\mathbf{S}, vertical arrows are the arrows f,gf,g of S\mathbf{S}, and whose horizontal arrows XYX\nrightarrow Y are objects in 𝒱 X×Y.\mathcal{V}^{X\times Y}. Composition of horizontal arrows is given by the canceling tensor product.

As we noted earlier, the canceling product is only defined under some a completeness criteria on 𝒱\mathcal{V} that are not always satisfied. In general, we would need to consider virtual double categories, which stand in the same relation to framed bicategories that multicategories stand to monoidal categories. In other words, instead of requiring that a composite 1-cell exist for any composible string of 1-cells, we simply consider squares whose top edge is a composible strings of 1-cells. A virtual double category is equivalent to Leinster’s notion of an fc-multicategory, which is a span of the form TABCTA\leftarrow B \to C in the category of graphs where TT is the free category monoid.

So in our two principal cases Fam(V)\text{Fam}(\mathbf{V}) and Self(S)\text{Self}(\mathbf{S}), we get double categories whose horizontal 2-categories are Mat(V)\text{Mat}(\mathbf{V}) and Span(S)\text{Span}(\mathbf{S})

And we can define a monoid in our double category. Following Shulman and Crutwell’s 2010 paper on generalized multicategories (also discussed on the nCafé here) we will not call them monads. A small 𝒱\mathcal{V}-category is a monoid in this virtual double category.

Let me also note that framed bicategories are not the only option for studying formal category theory. As was mentioned by Alex Campbell here on nCafé earlier in the seminar, Yoneda structures provide an alternate (equivalent?) setting for formal category theory.


Limits in a framed bicategory are defined in a way that generalizes weighted limits. Recall that for ordinary categories, given a functor J:KSetJ\colon K\to\text{Set} and a functor f:KCf\colon K\to C, the weighted limit is defined to be the representing object C(,lim Jf)=Set K(J,C(,f)).C(-,\lim^Jf)=\text{Set}^K(J,C(-,f-)). The motivation for this definition was discussed in Christina’s blog post here at the nCafé. In the context of formal category theory we can almost duplicate this definition, except the formalism of profunctors we are obliged to instead represent the limit with a vertical arrow: if J:AKJ\colon A\nrightarrow K and f:AC,f\colon A\to C, then (def 8.1) the JJ-weighted limit of ff is a vertical arrow :KC\ell\colon K\to C such that C(1,)C(1,f)J. C(1,\ell)\cong C(1,f) \triangleleft J. Similarly the JJ-weighted colimit is given by C(,1)JC(f,1). C(\ell,1)\cong J\triangleright C(f,1).

We can recover the classical example of weighted limits in an enriched category by taking 𝒱=Fam(V)\mathcal{V}=\text{Fam}(\mathbf{V}) as usual and setting K=δ1K=\delta 1, the unit 𝒱\mathcal{V}-category. In the general case, the generalization of the weights for weighted limits to profunctors (or bimodules) is forced on us because without S\mathbf{S}-indexed coproducts, δ1\delta 1 need not exist.

The generalization turns out to be quite useful, and leads to a more elegant and symmetrical statement of the adjunction between limits and colimits, which in the case of the large enriched indexed categories appears as proposition 8.5: 𝒱-CAT(colim Jf,g)𝒱-CAT(f,lim Jg). \mathcal{V}\text{-}CAT(\operatorname{colim}^J f,g)\cong \mathcal{V}\text{-}CAT(f,\lim{}^J g).

This is a point of view on weighted limits, that the weights should be bimodules, which I first learned of from Riehl’s lecture notes on weighted limits in the context of enriched categories. Those notes were apparently from a category theory seminar by Shulman, and now I think I know where Mike developed this point of view: from his work in formal category theory. Campbell argues the rightness of bimodule weights in his post on Yoneda structures as well.

Enrichment of the 2-category

In classical enriched category theory, we promote our ordinary 2-category of V\mathbf{V}-categories into a category enriched in V\mathbf{V}-categories by means of the venerable end: between any two V\mathbf{V}-categories, we have a V\mathbf{V}-category whose objects are V\mathbf{V}-functors F,GF,G and whose hom-objects are given by hom(F,G).\int\hom(F,G). I would have liked to have an analogous definition for our enriched indexed categories.

Here we have recalled a formal category theory definition of weighted limits. Can these be used to define an enriched indexed category of enriched indexed functors?

Posted at June 27, 2014 6:24 PM UTC

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Re: Enriched Indexed Categories, Again

Nice work, Joe. This paper is a rich and daunting one and you’ve done an excellent job of guiding us through it.

As Mike mentions in his paper, the notion of enriched indexed category has been discovered independently by a number of people including M. F. Gouzou and R. Grunig, Michal Przybylek, and Marta Bunge. Given the wide range of interesting concrete examples, it is not surprising that this definition has had a number of antecedents.

Posted by: Emily Riehl on June 27, 2014 7:05 PM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

One thing I’d like to see is a further generalisation so that we can take as our “base” not a (say) Set\mathbf{Set}-indexed monoidal category but rather a Cat\mathbf{Cat}-indexed monoidal category, possibly equipped with some bells and whistles. Of course, Cat\mathbf{Cat} is cartesian monoidal, so we could just take Mike’s definition as-is, but that wouldn’t work for my purposes. Rather, what I’d like to do is to take the Cat\mathbf{Cat}-indexed category \mathbb{H} whose fibre over a small category CC is C=Ho[C,sSet]\mathcal{H}^C = \operatorname{Ho} [C, \mathbf{sSet}] and have a good notion of \mathbb{H}-enriched category that provides a unified language for working with an (,1)(\infty, 1)-category of any flavour. In particular, \mathbb{H}-enriched categories should be generalisations of derivators, derivators should be the same thing as complete and cocomplete \mathbb{H}-enriched categories, and one should be able to easily (!) construct the \mathbb{H}-enriched homotopy category of a cofibrantly generated model category (i.e. without involving any notion of simplicial localisation).

Of course, one might ask why taking the Set\mathbf{Set}-indexed version of \mathbb{H} doesn’t work so well. After all, we can easily extend the canonical Set\mathbf{Set}-indexing of an ordinary category and recover the canonical Cat\mathbf{Cat}-indexing, essentially because [,𝒱][-, \mathcal{V}] is a right 2-adjoint. This fails miserably for \mathbb{H}: that is why we must take it as a Cat\mathbf{Cat}-indexed category. Quite remarkably, \mathbb{H} has enough information to reconstruct homotopy limits and colimits, at least for ordinary diagrams, so this should be an adequate base for enrichment.

The ultimate aim is to be able to define parametrised weighted homotopy limits and colimits for diagrams in a “\mathbb{H}-enriched category” (whatever that may be) in terms of representability of “\mathbb{H}-enriched profunctors”. This would let us understand homotopy limits and colimits in general (,1)(\infty, 1)-categories in terms of homotopy limits (and colimits) of spaces, instead of having to start from scratch (as is the case with many of the current definitions). More generally one hopes to make definitions for \mathbb{H}-enriched categories by mimicking the current practice of enriched category theory.

But I have no good idea for how to go about doing this.

Posted by: Zhen Lin on June 27, 2014 9:21 PM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

It’s very interesting that you bring this up! Moritz Groth and I have been working on a notion of “enriched derivator”, but our definition isn’t obviously the “complete and cocomplete objects” inside some larger collection of possibly-incomplete ones. We’ve been considering an enriched derivator to be an ordinary derivator equipped with the extra structure of a “closed module” over a monoidal derivator. This is good enough to talk about weighted limits and colimits, although not to characterize them in cases where they might not exist.

(Small correction: as far as I know, not every derivator in the ordinary sense is fully enriched over Ho(sSet)Ho(sSet): it has tensors and cotensors, but not necessarily homs. The tensors and cotensors give you a functor that the hom would have to represent if it existed, but I don’t know any way to show that it does exist in a general derivator. But I don’t know any examples of derivators where it doesn’t exist, either — because I don’t know any examples of derivators not arising from (,1)(\infty,1)-categories.)

A natural place to start trying to answer your question might be the bicategorical formulation. Joe mentioned that 𝒱\mathcal{V}-categories can be defined using monads in the (framed) bicategory constructed from an indexed monoidal category: specifically, small 𝒱\mathcal{V}-categories are monads, while large ones are “polyads” (“monads with many objects”).

Now any monoidal derivator gives rise to a bicategory of “profunctors” whose objects are small categories (and even to a framed bicategory whose vertical category is CatCat); see Theorem 5.9 of arXiv:1212.3277. Moreover, the construction which looks very similar to the construction of a bicategory from a general indexed monoidal category; you basically just have to insert some “op”s and replace some categories by twisted arrow categories. So we could consider polyads in this bicategory (starting from the derivator Ho(sSet)Ho(sSet), or more generally any monoidal derivator); might they be what you’re looking for?

Side comment: it’s also natural to wonder whether there is a single construction of a bicategory from an “indexed something” that generalizes both of these. I have an upcoming blog post (yes, I’m actually going to make a blog post again!) in which I was planning to speculate about that for other reasons.

Posted by: Mike Shulman on June 28, 2014 12:22 AM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Ah, I misremembered Cisinski’s result. I suppose it’s no great loss to ignore the derivators that cannot be enriched.

The construction of the framed bicategory of profunctors starting from a derivator is a good first step. (I was thinking this would have to be taken as extra data!) Is it really enough to just define \mathbb{H}-enriched categories to be monads/polyads in Prof()Prof(\mathbb{H})? There seems to be something strange about this, though: if I recall correctly, even in Prof(Set)Prof(\mathbf{Set}), there is more than one way to incarnate a small category as a monad. Also, how does one recover the prederivator associated with such a thing?

Returning to the point about weighted homotopy limits/colimits: one of my thoughts was that a \mathbb{H}-enriched category \mathbb{C} should have an underlying Cat\mathbf{Cat}-indexed \mathcal{H}-enriched category, so that we can at least talk about the homotopy type of the space of natural transformations of “diagrams in 𝒞\mathcal{C}”; this would be enough to define conical homotopy limits/colimits. However, in order to be able to define weighted homotopy limits/colimits à la enriched weighted limits/colimits, we would need to be able to define for an object AA in 𝒞 I\mathcal{C}^I and an object BB in 𝒞 J\mathcal{C}^J an object 𝒞(A,B)\mathcal{C}(A, B) in I op×J\mathcal{H}^{I^{op} \times J}. This seems to be something genuinely extra: playing around with universal properties only gets you the underlying “incoherent” diagram of 𝒞(A,B)\mathcal{C}(A, B).

I suppose what I want might be the framed bicategory of \mathbb{H}-enriched profunctors whose objects are \mathbb{H}-enriched categories (not just ordinary categories). Perhaps this might be obtained as some kind of completion of the previously-constructed framed bicategory of \mathbb{H}-enriched profunctors?

Posted by: Zhen Lin on June 28, 2014 2:30 AM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

how does one recover the prederivator associated with such a thing?

We need to play a game similar to that in section 6 of Enriched indexed categories, which is possible for polyads in any framed bicategory. An arbitrary polyad in Prof()Prof(\mathbb{H}) is analogous to a “large 𝒱\mathcal{V}-category”, so we can either allow functors between them that incorporate arbitrary functors between small categories on their objects, or require a fibrational-type condition that they have “restrictions”. Then we should be able to get out an underlying prederivator, just like in example 7.5 of the paper.

if I recall correctly, even in Prof(Set)Prof(\mathbf{Set}), there is more than one way to incarnate a small category as a monad.

That’s right; a monad in Prof(Set)Prof(\mathbf{Set}) consists of an identity-on-objects functor between two categories. Similarly, a monad in Prof()Prof(\mathbb{H}) will be something like a bijective-on-objects functor from an ordinary small category to an \mathcal{H}-enriched category. I say “something like” because I’m not sure exactly what kind of “functor” it will be; it seems sort of “semicoherent”.

In this case, I’m kind of inclined to chalk this up to the gap between prederivators and (,1)(\infty,1)-categories (which is rather larger than the gap between derivators and (,1)(\infty,1)-categories). Since axioms (Der1), (Der2), and (Der5) are designed to help close this gap, perhaps we ought to impose analogous axioms on a notion of “enriched prederivator”. At some point I think I suggested the term “semiderivator” for a prederivator satisfying (Der1) and (Der2).

we would need to be able to define for an object AA in 𝒞 I\mathcal{C}^I and an object BB in 𝒞 J\mathcal{C}^J an object 𝒞(A,B)\mathcal{C}(A, B) in I op×J\mathcal{H}^{I^{op} \times J}.

Isn’t this exactly what a polyad in Prof()Prof(\mathbb{H}) would give us? Its hom-object from an object AA of extent II and an object BB of extent JJ must be a 1-cell from JJ to II, which by definition of Prof()Prof(\mathbb{H}) is something in I op×J\mathcal{H}^{I^{op} \times J}. Or am I confused?

I suppose what I want might be the framed bicategory of \mathbb{H}-enriched profunctors whose objects are \mathbb{H}-enriched categories (not just ordinary categories). Perhaps this might be obtained as some kind of completion of the previously-constructed framed bicategory of \mathbb{H}-enriched profunctors?

Well, if you apply the ModMod construction to Prof()Prof(\mathbb{H}), you will get something whose objects are \mathbb{H}-enriched categories. But it won’t be a framed bicategory, since Prof()Prof(\mathbb{H}) doesn’t have local coequalizers — it’ll only be virtual. I’m not convinced this is a good thing to do; my inclination is to think that “incoherent” \mathbb{H}-enriched categories of this sort may be okay as representatives of the “large” (,1)(\infty,1)-categories that we take diagrams in, like derivators, but that for the small diagram shapes we want actual coherent things, like ordinary small categories, or perhaps small (,1)(\infty,1)-categories if we could get them.

By the way, I think there is a way to define the homotopy category of AA-diagrams in 𝒟\mathcal{D} for any derivator 𝒟\mathcal{D} and any small (,1)(\infty,1)-category AA. Present AA by a relative 1-category (B,W)(B,W), then consider the full subcategory of 𝒟(B)\mathcal{D}(B) on the objects whose underlying incoherent diagrams send the morphisms in WW to isomorphisms. I haven’t yet figured out how to extend things like Kan extensions and the bicategory of profunctors to these inputs, though.

Posted by: Mike Shulman on June 28, 2014 8:04 AM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

I’m afraid I don’t really know what a polyad is, but I think I see what you’re trying to say. I guess what I was thinking about is closer to the definition of “large \mathbb{H}-category” but what I was confused about was how to pass from an “indexed \mathbb{H}-category” to a “large \mathbb{H}-category”.

On coherence: Is your objection is that it is difficult to tell when a prederivator-like structure actually represents an (,1)(\infty, 1)-category? Then I guess I agree – it makes no sense to ask about diagrams of incoherent shape. I like your suggestion of using relative categories as proxies for (,1)(\infty, 1)-categories – it fits nicely with the idea that ordinary categories are (morally, if not actually) (,1)(\infty, 1)-dense in quasicategories.

Posted by: Zhen Lin on June 28, 2014 10:42 AM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Oops, what I wrote was a bit sloppy. There is no analogue of “indexed VV-categories” in this context, though we can still define “VV-fibrations” and show that functors between them are equivalent to indexed ones. So we can’t exactly follow example 7.5, but I think we can still recover a prederivator from a “Prof()Prof(\mathbb{H})-fibration” by taking ends of its hom-objects.

Now can we show that any (,1)(\infty,1)-category gives rise to such a thing?

Posted by: Mike Shulman on June 28, 2014 6:41 PM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

I don’t understand. What’s wrong with “Cat\mathbf{Cat}-indexed \mathbb{H}-categories”, and what are “Prof()Prof(\mathbb{H})-fibrations”? It seems to be a trivial matter to extract a prederivator from a Cat\mathbf{Cat}-indexed \mathbb{H}-category. Are you saying that if all we have are the various 𝒞(A,B):I op×J\mathcal{C} (A, B) : I^op \times J \to \mathcal{H}, then we should be able to build the underlying prederivator by taking 𝒞 I(F,G)= i:I𝒞(Fi,Gi)\mathcal{C}^I (F, G) = \int_{i : I} \mathcal{C} (F i, G i)? That sounds very plausible to me.

Any ordinary locally small category can be turned into a “Cat\mathbf{Cat}-indexed 𝕊et\mathbb{S}et-category”, or more plainly, that for any small category II and any locally small category 𝒞\mathcal{C}, the functor category [I,𝒞][I, \mathcal{C}] admits a natural [I,Set][I, \mathbf{Set}]-enrichment, viz.: [I,𝒞](F,G)= i:I[I(,i),𝒞(Fi,Gi)][I, \mathcal{C}](F, G) = \int_{i : I} [I (-, i), \mathcal{C} (F i, G i)] (Taking I=Δ opI = \mathbf{\Delta}^{op}, this is a special case of the totalisation of a cosimplicial simplicial set; taking 𝒞=Set\mathcal{C} = \mathbf{Set}, this is the usual formula for exponential objects.) Moreover, given diagrams A:I𝒞A : I \to \mathcal{C} and B:J𝒞B : J \to \mathcal{C}, we have 𝒞 I op×J(π *A,π *B)\mathcal{C}^{I^{op} \times J} (\pi^* A, \pi^* B) naturally isomorphic to 𝒞(A,B)\mathcal{C} (A, B) as diagrams I op×JSetI^{op} \times J \to \mathbf{Set}. I guess this should still work for simplicially enriched categories, but almost surely more effort be needed to get things homotopically correct.

I have no idea how to do the above for quasicategories, but it should be a straightforward matter to get the \mathcal{H}-profunctors we want in that context.

Posted by: Zhen Lin on June 28, 2014 11:39 PM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Oh, right, that’s the one that seems like it shouldn’t work but actually does (e.g. Example 8.14 of 1212.3277). So although it’s not a special case of section 6 of Enriched indexed categories, I guess there is also a sort of “indexed category” in addition to “enriched fibrations”, and you should be able to extract the former from the latter by a formula like yours, but perhaps not vice versa. No more time now, but maybe in the next few days I’ll be able to write up something more detailed.

Posted by: Mike Shulman on June 29, 2014 4:39 AM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Just this morning Dominic Verity announced that he and Emily Riehl have been able to construct a proarrow equipment using quasicategories and two-sided fibrations. It should be interesting to see how “\mathbb{H}-enriched categories” fits in with that picture.

Posted by: Zhen Lin on June 30, 2014 9:41 PM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Okay, here’s my proposal in a little more detail. Let 𝒱\mathcal{V} be a monoidal derivator, with associated bicategory Prof(𝒱)Prof(\mathcal{V}). A “Prof(𝒱)Prof(\mathcal{V})-category” 𝒞\mathcal{C} consists of

  1. For each small category AA, a class (not a category) 𝒞 0 A\mathcal{C}^A_0 of “𝒞\mathcal{C}-diagrams of shape AA”.

  2. For X𝒞 0 AX\in \mathcal{C}^A_0 and Y𝒞 0 BY\in \mathcal{C}^B_0, a hom-object 𝒞(X,Y)𝒱 A op×B\mathcal{C}(X,Y) \in \mathcal{V}^{A^{op}\times B}, i.e. a morphism BAB\to A in Prof(𝒱)Prof(\mathcal{V}).

  3. For XX, YY and Z𝒞 0 CZ\in \mathcal{C}^C_0, a map 𝒞(Y,Z)𝒞(X,Y)𝒞(X,Z)\mathcal{C}(Y,Z) \odot \mathcal{C}(X,Y) \to \mathcal{C}(X,Z) in Prof(𝒱)Prof(\mathcal{V}).

  4. For each XX, a map I B𝒞(X,X)I_B \to \mathcal{C}(X,X) in Prof(𝒱)Prof(\mathcal{V}).

  5. These maps are associative and unital.

Now a “Prof(𝒱)Prof(\mathcal{V})-fibration” is a Prof(𝒱)Prof(\mathcal{V})-category 𝒞\mathcal{C} such that for any functor f:ABf:A\to B and X𝒞 0 BX\in \mathcal{C}^B_0, there exists an f *X𝒞 0 Af^\ast X\in \mathcal{C}^A_0 and isomorphisms 𝒞(Z,f *X)(f×1) *𝒞(Z,X)\mathcal{C}(Z,f^\ast X) \cong (f\times 1)^\ast\mathcal{C}(Z,X) that are “natural in ZZ” in an appropriate sense.

In this case, if we define a 𝒱 A\mathcal{V}^A-enriched category 𝒞 A\mathcal{C}^A by an analogue of your formula, then we should be able to show that f:ABf:A\to B induces a functor 𝒞 B𝒞 A\mathcal{C}^B \to \mathcal{C}^A, yielding a CatCat-indexed 𝒱\mathcal{V}-category. However, I don’t understand your proposal for going backwards: if A𝒞 IA\in \mathcal{C}^I (rather than 𝒞 I op\mathcal{C}^{I^{op}}), then how can π *A\pi^\ast A be in 𝒞 I op×J\mathcal{C}^{I^{op}\times J}?

Posted by: Mike Shulman on July 1, 2014 11:48 PM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Argh, I must have overlooked the variances when doing my calculations. Although duality says that [I,𝒞][I, \mathcal{C}] is enriched over [I op,Set][I^{op}, \mathbf{Set}] as well as [I,Set][I, \mathbf{Set}], we’d have to combine these two somehow. Perhaps by thinking about (((𝒞 J) op) I op) op(((\mathcal{C}^J)^{op})^{I^{op}})^{op} instead of 𝒞 I op×J\mathcal{C}^{I^{op} \times J}…? That seems to suggest we’d need not just the indexed \mathbb{H}-category \mathbb{C} but also the indexed J\mathbb{H}^J-category ( J) op(\mathbb{C}^J)^op.

The point is that 𝒞(A,B) (i,j)[I(i,)×J(,j),𝒞(Ai,Bj)]\mathcal{C} (A, B) \cong \int_{(i, j)} [I (i, -) \times J (-, j), \mathcal{C} (A i, B j)] and by breaking this up a little it would seem to be expressible in terms of the I op×J\mathcal{H}^{I^{op} \times J}-enrichment of the I opI^{op}-fibre of ( J) op(\mathbb{C}^J)^op. I guess this isn’t recoverable from \mathbb{C} alone, but on the other hand, ( J) op(\mathbb{C}^J)^op can probably be constructed from the Prof()Prof(\mathbb{H})-category version of 𝒞\mathcal{C}.

Posted by: Zhen Lin on July 2, 2014 10:21 AM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Can you explain in more detail how to express that in terms of H I op×JH^{I^{op}\times J}-enrichment of the I opI^{op}-fiber of (C J) op(C^J)^{op}?

Posted by: Mike Shulman on July 3, 2014 4:07 AM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Since ( J) op(\mathbb{C}^{J})^{op} is supposed to be J\mathbb{H}^J-enriched, the fibre ((𝒞 J) op) I op((\mathcal{C}^{J})^{op})^{I^{op}} has an I op×J\mathcal{H}^{I^{op} \times J}-enrichment; and it seems reasonable to say ob((𝒞 J) op) I opob𝒞 I×J\operatorname{ob} ((\mathcal{C}^{J})^{op})^{I^{op}} \cong \operatorname{ob} \mathcal{C}^{I \times J}, in which case it makes sense to look at (((𝒞 J) op) I op) op(π *A,π *B)(((\mathcal{C}^{J})^{op})^{I^{op}})^{op} (\pi^* A, \pi^* B). The hope is that the hom-object in question has been defined as (((𝒞 J) op) I op) op(π *A,π *B) iI(i,)(𝒞 J) op(B,ΔAi)(((\mathcal{C}^{J})^{op})^{I^{op}})^{op} (\pi^* A, \pi^* B) \cong \int_i I (i, -) \pitchfork (\mathcal{C}^{J})^{op} (B, \Delta A i) where the hom-object on the RHS is in J\mathcal{H}^J (not \mathcal{H}); and if we also have (𝒞 J) op(B,ΔAi) jJ(,j)𝒞(Ai,Bj)(\mathcal{C}^{J})^{op} (B, \Delta A i) \cong \int_j J (-, j) \pitchfork \mathcal{C} (A i, B j) then putting the two formulae together, (((𝒞 J) op) I op) op(π *A,π *B) i j(I(i,)×J(,j))𝒞(Ai,Bj)(((\mathcal{C}^{J})^{op})^{I^{op}})^{op} (\pi^* A, \pi^* B) \cong \int_i \int_j (I (i, -) \times J (-, j)) \pitchfork \mathcal{C} (A i, B j) as desired. At least assuming the usual calculus of ends works, but the above calculation is just a heuristic anyway.

Posted by: Zhen Lin on July 3, 2014 8:19 AM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Very nice! I applaud your courage in taking on this paper, and not only due to its perhaps-excessive length; I well remember from my own days as a graduate student that giving a talk about a paper was always rather more intimidating with its author in the audience. (-:

It should certainly be possible to define enriched indexed functor categories. The only reason I didn’t include that in the paper was that I had to stop somewhere. If you want to work it out, more power to you!

You’re also absolutely right that it was in thinking about formal category theory that I realized the importance of weights being bimodules.

Finally, Yoneda structures are sort of equivalent to equipments as a context for formal category theory, but not exactly, because they treat size differently. In particular, it seems that a Yoneda structure must include “large but not locally small” objects, in order to have presheaf objects for “large and locally small” objects. I don’t like this, because in the enriched context, there is no very natural notion of “large but not locally small” category. The best you can do is to embed your enriching category VV into a bigger one VV', as Kelly does in his book, and talk about VV'-categories. This works, but I find it kind of kludgy, when (virtual) equipments allow you to talk about the locally small categories directly without needing to introduce the “locally large” ones. Moreover, constructing an analogue of VV' in the enriched indexed context would be trickier; it’s probably possible, but I think it would be even less natural.

Posted by: Mike Shulman on June 27, 2014 10:35 PM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Oh, and as long as we’re discussing this paper, I may as well point out that until recently there was a bug in its TeX code, causing the vertical \mapsto arrows to come out looking ugly. I noticed this when Joe asked me a question about the paper, and since TAC is fortunately run by a human being (Bob Rosebrugh), I was able to fix the bug not only on the arXiv but also in the officially posted journal version. So if you downloaded it a while ago, you may want to grab a new copy.

Posted by: Mike Shulman on June 28, 2014 12:42 AM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Yeah, the relationship between equipments and Yoneda structures is something I wanted to go into more depth in. I said something in the seminar that suggested that equipments and Yoneda structures were equivalent, and Alex Campbell refuted me. I should like to set the record straight.

Posted by: Joe Hannon on June 28, 2014 11:56 PM | Permalink | Reply to this

Re: Enriched Indexed Categories, Again

Yeah, the relationship between equipments and Yoneda structures is something I wanted to go into more depth in. I said something in the seminar that suggested that equipments and Yoneda structures were equivalent, and Alex Campbell refuted me. I should like to set the record straight.

Posted by: Joe Hannon on June 28, 2014 11:57 PM | Permalink | Reply to this

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