The n-Category Café

Indexed monoidal categories

An $\mathbf{S}$-indexed monoidal category is pseudofunctor $\mathcal{V}\colon\mathbf{S}\to\text{MonCat}$, where $\mathbf{S}$ is assumed to have finite products and be endowed with its cartesian monoidal structure, and $\text{MonCat}$ 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 $\mathcal{V}$ for an enriched indexed monoidal category, and bold $\mathbf{V}$ for an ordinary monoidal category.

We will notate the image monoidal category of an object $X\in\mathbf{S}$ as $\mathcal{V}^X$ and arrow $f\colon X\to Y$ goes to $f^\ast\colon \mathcal{V}^Y\to\mathcal{V}^X.$ By the Grothendieck construction we may equivalently regard this as a fibration $\int\mathcal{V}\to\mathbf{S}$ 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 $\mathcal{V}^1$ (where $1$ is the terminal object in $\mathbf{S}$) 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 $\mathit{Fam}(\mathbf{V})$ and $\mathit{Self}(\mathbf{S})$, out of which we will construct $\mathbf{V}$-enriched categories and $\mathbf{S}$-internal categories, respectively.

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

And for $\mathit{Self}(\mathbf{S})$ let $\mathbf{S}$ be any category with finite limits, and to each object $X\in\mathbf{S}$ associate the category $\mathbf{S}\downarrow X$ with its cartesian structure given by pullbacks. Then for $B\in\mathbf{S}\downarrow Y$ and $f\colon X\to Y$, we have $f^\ast B=X\underset{Y}{\times}B,$ 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 $A\otimes_X B$, for $A,B\in\mathcal{V}^X.$ There is additionally an external product defined on $\int\mathcal{V}$, as was implicit in our claim that the Grothendieck construction yields a strict monoidal fibration. We denote this by $A\otimes B$, for $A\in\mathcal{V}^X$ and $B\in\mathcal{V}^Y$ and call it the external product. It is related to the fiberwise product by the formula $A\otimes B=\pi_A^\ast X\otimes_{A\times B}\pi_B^\ast Y,$ which is familiar from the theory of bundles or sheaves.

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

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

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

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

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

The various products can be combined; for $A$ in the fiber over $X\times Y\times Z$ and $B$ over $Y\times Z\times W$, then we can cancel the $Z$ dependence and take the fiberwise product over $Y$, leaving an object over $X\times Y\times W.$

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 $\mathcal{V}^{[Y]}(B,C)$ is the right adjoint of the external tensor, and the external hom $\mathcal{V}(B,C)$ is the right adjoint of the canceling tensor.

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

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

Small $\mathcal{V}$-categories

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 $\mathcal{V}$-category $A$ is an object $\epsilon A\in \mathbf{S}$ called the extent, an object $\underline{A}\in\mathcal{V}^{\epsilon A\times \epsilon A}$ which we think of as the arrows of $A$, along with morphisms $I_{\epsilon A}\overset{\text{ids}}{\to} \underline{A}$ and $\underline{A}\otimes_{\epsilon A}\underline{A}\to \underline{A}$ satisfying the usual associativity and unital axioms:

$$\begin{matrix} \underline{A}\underset{\epsilon A}{\otimes}(\underline{A}\underset{\epsilon A}{\otimes}\underline{A}) & \to & (\underline{A}\underset{\epsilon A}{\otimes}\underline{A})\underset{\epsilon A}{\otimes}\underline{A} & \to & \underline{A}\underset{\epsilon A}{\otimes}\underline{A}\\ \downarrow & & & & \downarrow \\ \underline{A}\underset{\epsilon A}{\otimes}\underline{A}& & \to & & \underline{A} \end{matrix}$$

and

$$\begin{matrix} \underline{A} & \to & I_{\epsilon A}\otimes_{\epsilon A}\underline{A} & \to & \underline{A}\underset{\epsilon A}{\otimes}\underline{A} & \leftarrow & I_{\epsilon A}\otimes_{\epsilon A}\underline{A}& \leftarrow & \underline{A}\\ & & =\searrow & & \downarrow & & \swarrow = \\ & &&& \underline{A} && \end{matrix}$$

Functors

A functor $f\colon A\to B$ of small $\mathcal{V}$-categories is given by the data of a morphism of extents $\epsilon f\colon \epsilon A\to \epsilon B$ in $\mathbf{S}$, and a morphism of arrows $\underline{A}\to\underline{B}$ in $\int\mathcal{V}$ such that $$\begin{matrix} I_{\epsilon A} & \to & \underline{A}\\ \downarrow & & \downarrow \\ I_{\epsilon B} & \to & \underline{B} \end{matrix}$$ and $$\begin{matrix} \underline{A}\otimes_{\epsilon A}\underline{A} & \to & \underline{A}\ar[d]\\ \downarrow & & \downarrow \\ \underline{B}\otimes_{\epsilon B}\underline{B}& \to & \underline{B} \end{matrix}$$ commute.

Natural transformations

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

$$\begin{matrix} \underline{A}& \to & \underline{A}\underset{\epsilon A}\otimes I_{\epsilon A} & \overset{f\otimes\alpha}{\to} & \underline{B}\otimes_{\epsilon B} \underline{B}\\ \downarrow & & & & \downarrow \\ I_{\epsilon A}\otimes \underline{A} & \underset{\alpha\otimes g}{\to} & \underline{B}\otimes_{\epsilon B} \underline{B} & \to & \underline{B} \end{matrix}$$

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

Discrete enriched category

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

Profunctors

A profunctor $H\colon A\nrightarrow B$ is an object $\underline{H}\in\mathcal{V}^{\epsilon A\times\epsilon B}$ with structure morphisms $\underline{A}\otimes_{\epsilon A}\underline{H}\to\underline{H}$ and $\underline{H}\otimes_{\epsilon B}\underline{B}\to\underline{H}$ so that the left and right actions are unital and associative and the left action commutes with the right action: $$\begin{matrix} \underline{A}\otimes_{\epsilon A}\underline{A}\otimes_{\epsilon A}\underline{H} & \to & \underline{A}\otimes_{\epsilon A}\underline{H}\\ \downarrow & & \downarrow \\ \underline{A}\otimes_{\epsilon A}\underline{H} & \to & \underline{H} \end{matrix}$$ and $$\begin{matrix} \underline{H}\otimes_{\epsilon B}\underline{B}\otimes_{\epsilon B}\underline{B} & \to & \underline{H}\otimes_{\epsilon B}\underline{B}\\ \downarrow & & \downarrow \\ \underline{H}\otimes_{\epsilon B}\underline{B} & \to & \underline{H} \end{matrix}$$ and $$\begin{matrix} \underline{H} & \to & \underline{A}\otimes_{\epsilon A}\underline{H}\\ & \searrow & \downarrow \\ & & \underline{H} \end{matrix}$$ and $$\begin{matrix} \underline{H} & \to & \underline{H}\otimes_{\epsilon B}\underline{B}\\ & \searrow & \downarrow \\ & & \underline{H} \end{matrix}$$ and $$\begin{matrix} \underline{A}\otimes_{\epsilon A}\underline{H}\otimes_{\epsilon B}\underline{B} & \to & \underline{H}\otimes_{\epsilon B}\underline{B}\\ \downarrow & & \downarrow \\ \underline{A}\otimes_{\epsilon A}\underline{H}& \to & \underline{H} \end{matrix}$$

We can restrict our profunctors as usual. Given $H\colon A\nrightarrow B$ and $f\colon A\to A'$ and $g\colon B\to B'$, then we have a profunctor $H(g,f)$ given by $\underline{H(g,f)}=(\epsilon f\times \epsilon g)^\ast\underline{H}.$ In particular for a functor $A\to B$, we have the representable profunctors $B(1,f)\colon A\nrightarrow B$ and $B(f,1)\colon B\nrightarrow A.$

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

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

Examples and two more definitions

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

And a large $\mathcal{V}$-category is a kind of horizontal categorification of a small $\mathcal{V}$-category, with collection of objects $x,y,...$ and for each object an extent $\epsilon x\in\mathbf{S}$ and for each pair of objects $x,y$ an object $\mathcal{A}(x,y)\in\mathcal{V}^{\epsilon x\times\epsilon y}$ satisfying the usual axioms (def 5.1). Then there is a notion of $\mathcal{V}$-fibrations and a Grothendieck-type construction which gives an equivalence between $\mathcal{V}$-indexed categories and large $\mathcal{V}$-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 $\mathbf{S}$ the category of finite group objects in topological spaces denoted $\mathcal{G}$, and for $\mathcal{V}$ the indexed monoidal category of based spaces with group actions, denoted $Act(\text{Top})_\ast$. We have an $Act(\text{Top})_\ast$-category $\mathcal{I}_\mathcal{G}$ with objects given by finite dimensional real representations $\rho\colon G\to O(n)$, with extent $G$ and hom-object $\underline{\mathcal{I}_\mathcal{G}}(G,G')$ the space of linear isometric isomorphisms $\mathbb{R}^n\to\mathbb{R}^{n'}$ plus basepoint with $G\times G'$ 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 $Z(C)$ of an ordinary category $C$ is defined to be the endomorphism monoid of the identity functor on $C$. 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 $M\colon C\to \text{Set}$ defines a notion of a (left) module $M$ over $C$. We have an isomorphism between the center of the category of modules $Z(C\text{-Mod})$ and $Z(C)$.

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

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

In the case of a category $C=(C_0;C_1;\text{ids}\colon C_0\to C_1;s,t\colon C_1\to C_0;\text{comp}\colon C_1\times_{C_0}C_1\to C_1)$ internal to $\mathcal{E}$, a module $M$ over an internal category is also known as an internal diagram, and it is given by the data of a structure morphism $M\to C_0$ such that $$\begin{matrix} C_1\underset{C_0}{\times} M & \overset{\text{act}}{\to}& M \\ \downarrow & & \downarrow \\ C_1 & \underset{s}{\to} & C_0 \end{matrix}$$ 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 $A$ be a small $\mathcal{V}$-category, for $\mathcal{V}$ an indexed monoidal category. Using the notation of the paper a one-sided left $A$-module is a profunctor $M\colon A\nrightarrow I,$ where $I=\delta 1$ is the discrete $\mathcal{V}$-category whose extent is the terminal object of $\mathbf{S}$ and whose arrow object is the monoidal unit $\underline{\delta I}=I\in\mathcal{V}^1=\mathbf{V}.$

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

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

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

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

$$\begin{matrix} \underline{A}\underset{\epsilon A}{\otimes}M & \to & \underline{A}\underset{\epsilon A}{\otimes}I_{\epsilon A}\underset{\epsilon A}{\otimes}M & \overset{1\otimes z\otimes 1}{\to} & \underline{A}\underset{\epsilon A}{\otimes}\underline{A}\underset{\epsilon A}{\otimes}M & \overset{1\times\text{act}}{\to} & \underline{A}\underset{\epsilon A}{\otimes}\\ {\text{act}}\downarrow & & & & & & \downarrow{\text{act}} \\ M& \to & I_{\epsilon A}\underset{\epsilon A}\otimes M & \underset{z\otimes 1}{\to} & \underline{A}\underset{\epsilon A}\otimes M & \underset{\text{act}}{\to} & M \end{matrix}$$ which commutes by a diagram chase.

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

$$\begin{matrix} \underline{A} & \to & I_{\epsilon A}\otimes_{\epsilon A} \underline{A} & \overset{\text{ids}\otimes 1}{\to} & \underline{A}\otimes_{\epsilon A}\underline{A} & \overset{\zeta_A\otimes1}{\to} & \underline{A}\otimes_{\epsilon A}\underline{A}\\ \downarrow & & & & & & \downarrow \\ \underline{A}\otimes_{\epsilon A} I_{\epsilon A}& \underset{1\otimes\text{ids}}{\to} & \underline{A}\otimes_{\epsilon A} \underline{A} & \underset{1\otimes\zeta_A}{\to} & \underline{A}\otimes_{\epsilon A} \underline{A}& \to & \underline{A} \end{matrix}$$ These maps $Z(A)\leftrightarrow Z(\text{Prof}(A,I))$ are inverse, which establishes the isomorphism, at least in the category of sets. Although this establishes the result for any small $\mathcal{V}$-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 $\mathcal{V}$-categories, but for a full strength general result, we need instead a $\mathcal{V}$-2-category, that is, a category enriched in $\mathcal{V}$-categories, which would strengthen our result to an isomorphism of $\mathcal{V}$-objects.

Equipments

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 $\text{Mat}(\mathbf{V})$ of set indexed matrices with values in monoidal category $\mathbf{V}$, and internal categories in $\mathcal{E}$ are monads in the 2-category $\text{Span}(\mathcal{E})$ of spans in $\mathcal{E}.$

I was disappointed not to see $\mathcal{V}$-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 $\mathcal{V}$-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

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

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

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

itself also factors uniquely

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

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 $\mathcal{V}\colon \mathbf{S}\to \text{MonCat}$ we associate a framed bicategory whose objects $X,Y$ are the objects of $\mathbf{S}$, vertical arrows are the arrows $f,g$ of $\mathbf{S}$, and whose horizontal arrows $X\nrightarrow Y$ are objects in $\mathcal{V}^{X\times Y}.$ 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 $\mathcal{V}$ 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 $TA\leftarrow B \to C$ in the category of graphs where $T$ is the free category monoid.

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

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 $\mathcal{V}$-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

Limits in a framed bicategory are defined in a way that generalizes weighted limits. Recall that for ordinary categories, given a functor $J\colon K\to\text{Set}$ and a functor $f\colon K\to C$, the weighted limit is defined to be the representing object $C(-,\lim^Jf)=\text{Set}^K(J,C(-,f-)).$ 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 $J\colon A\nrightarrow K$ and $f\colon A\to C,$ then (def 8.1) the $J$-weighted limit of $f$ is a vertical arrow $\ell\colon K\to C$ such that $$C(1,\ell)\cong C(1,f) \triangleleft J.$$ Similarly the $J$-weighted colimit is given by $$C(\ell,1)\cong J\triangleright C(f,1).$$

We can recover the classical example of weighted limits in an enriched category by taking $\mathcal{V}=\text{Fam}(\mathbf{V})$ as usual and setting $K=\delta 1$, the unit $\mathcal{V}$-category. In the general case, the generalization of the weights for weighted limits to profunctors (or bimodules) is forced on us because without $\mathbf{S}$-indexed coproducts, $\delta 1$ 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: $$\mathcal{V}\text{-}CAT(\operatorname{colim}^J f,g)\cong \mathcal{V}\text{-}CAT(f,\lim{}^J g).$$

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 $\mathbf{V}$-categories into a category enriched in $\mathbf{V}$-categories by means of the venerable end: between any two $\mathbf{V}$-categories, we have a $\mathbf{V}$-category whose objects are $\mathbf{V}$-functors $F,G$ and whose hom-objects are given by $\int\hom(F,G).$ 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?