Section 5. (0MLT) · ScalingStacks

5. Correspondences and the category $\gaunt_{n}$ [0MLT]

The category $\cat_{n}$ is cartesian closed, which means that for each strict $n$ -category $X$ , the endo-functor $(-)\times X$ admits a right adjoint, $\uHom(X,-)$ . The strict $n$ -categories $\uHom(X,Y)$ make $\cat_{n}$ enriched in itself, and thus it can be regarded as a large strict $(n+1)$ -category.

The category $\gaunt_{n}$ , as a full subobject of $\cat_{n}$ , is likewise a large strict $(n+1)$ -category. Moreover the $n$ -category $\uHom(X,Y)$ is gaunt whenever $Y$ is gaunt, making $\gaunt_{n}$ itself cartesian closed. In fact, if we replace $\gaunt_{n}$ by a skeleton, then it is a large gaunt $(n+1)$ -category, though we will not need this observation.

If $X$ is a strict $n$ -category (possibly large) then we may consider the slice category $(\gaunt_{n}/X)$ of gaunt $n$ -categories equipped with a functor to $X$ . This is naturally enriched in $\cat_{n}$ as well and hence also a (large) $(n+1)$ -category. If $p_{0}:A_{0}\to X$ and $p_{1}:A_{1}\to X$ are in $(\gaunt_{n}/X)$ , then the $\cat_{n}$ -enriched hom from $p_{0}$ to $p_{1}$ is given by the following pull-back:

[0MI9]

Definition 5.1. A correspondence (or $k$ -correspondence, for clarity) of gaunt $n$ -categories is an object of the overcategory $(\gaunt_{n}/C_{k})$ . In other words, a $k$ -correspondence of gaunt $n$ -categories is a gaunt $n$ -category $M$ along with a functor $M\to C_{k}$ .

[0MIA]

Remark 5.2. By Lemma 3.5 $\gaunt_{n}$ is locally finitely presentable. It follows (see [1, Corollary 2.44, 2.47]) that each of the categories of $k$ -correspondence is also locally finitely presentable.

Given two $k$ -correspondences $M\to C_{k}$ and $N\to C_{k}$ , we may form the product correspondence $M\times_{C_{k}}N$ . This is the product in the category $(\gaunt_{n}/C_{k})$ of $k$ -correspondences.

The terminology and connection to the classical theory of correspondences is made more clear by introducing the following category.

[0MIB]

Definition 5.3. Define $\corr_{n}^{0}\mathrel{\mathop{:}}=\gaunt_{n}$ , and for $k>0$ , define $\corr_{n}^{k}$ recursively as follows. The objects are triples $(X_{0},X_{1},F)$ consisting of two gaunt $n$ -categories $X_{0}$ and $X_{1}$ and a functor

F\colon X_{0}^{\mathrm{op}}\times X_{1}\to\corr_{n-1}^{k-1}.

(Here $\mathrm{op}=\rho(1,0,\dots,0)$ is the opposite obtained by reversing just the 1-morphisms of $X_{0}$ .) A morphism $(X_{0},X_{1},F)\to(Y_{0},Y_{1},G)$ is a triple $(f_{0},f_{1},\alpha)$ consisting of functors $f_{0}\colon X_{0}\to Y_{0}$ and $f_{1}\colon X_{1}\to Y_{1}$ and a natural transformation

\alpha\colon F\to G\circ(f_{0}^{\mathrm{op}}\times f_{1}).

[0MIC]

Lemma 5.4. There is a natural equivalence of categories

\phi_{n}^{k}\colon(\gaunt_{n}/C_{k})\simeq\corr_{n}^{k}.

[0MID]

Proof. The functor $\phi_{n}^{0}$ is simply the identity. We now define $\phi_{n}^{k}$ recursively. Suppose that $k>0$ , and assume that the equivalence $\phi_{n-1}^{k-1}$ has been defined.

Let us define the functor $\phi_{n}^{k}\colon(\gaunt_{n}/C_{k})\to\corr_{n}^{k}$ . For any object $p\colon X\to C_{k}$ of $(\gaunt_{n}/C_{k})$ , let $\phi_{n}^{k}(p)$ be the triple $(X_{0},X_{1},F)$ , where $X_{0}$ and $X_{1}$ are the fibers of $p$ over $0$ and $1$ , respectively, and the functor $F\colon X_{0}^{\mathrm{op}}\times X_{1}\to\corr_{n-1}^{k-1}$ is the composite $\phi_{n-1}^{k-1}\circ F^{\prime}$ , where the functor

F^{\prime}\colon X_{0}^{\mathrm{op}}\times X_{1}\to(\gaunt_{n-1}/C_{k-1})

is defined as follows:

•

For any objects $(x_{0},x_{1})\in X_{0}^{\mathrm{op}}\times X_{1}$ , let $F(x_{0},x_{1})$ be the $(n-1)$ -category $X(x_{0},x_{1})$ , equipped with the functor $X(x_{0},x_{1})\to C_{k}(0,1)=C_{k-1}$ induced by $p$ .
•

For any objects $(x_{0},x_{1})$ and $(y_{0},y_{1})$ of $X_{0}^{\mathrm{op}}\times X_{1}$ , the functor

$X_{0}(y_{0},x_{0})\times X_{1}(x_{1},y_{1})\to\Fun_{/C_{k-1}}(X(x_{0},x_{1}),X(y_{0},y_{1}))$

is simply composition.

For any commutative triangle

of gaunt $n$ -categories, we define

\phi_{n}^{k}(f)\mathrel{\mathop{:}}=(f_{0},f_{1},\alpha)\colon\phi_{n}^{k}(p)=(X_{0},X_{1},F^{\prime}\circ\phi_{n}^{k})\to(Y_{0},Y_{1},G^{\prime}\circ\phi_{n}^{k})=\phi_{n}^{k}(q),

where $f_{0}$ and $f_{1}$ are the restrictions of $f$ to the fibers, and $\alpha$ is the composite $\phi_{n-1}^{k-1}\ast\alpha^{\prime}$ , in which the natural transformation $\alpha^{\prime}$ is the one whose components are given by the functor $X(x_{0},x_{1})\to Y(f(x_{0}),f(x_{1}))$ induced by $f$ .

We now construct a quasi-inverse $\psi_{n}^{k}\colon\corr_{n}^{k}\to(\gaunt_{n}/C_{k})$ to $\phi_{n}^{k}$ . Again, when $k=0$ , we let $\psi_{n}^{0}$ be the identity, and we proceed recursively. We assume $k>0$ and that the quasi-inverse $\psi_{n-1}^{k-1}$ to $\phi_{n-1}^{k-1}$ has been defined.

For any object $(X_{0},X_{1},F)\in\corr_{n}^{k}$ , define a gaunt $n$ -category $U(X_{0},X_{1},F)$ with object set $\mathrm{ob}X_{0}\sqcup\mathrm{ob}X_{1}$ and

U(X_{0},X_{1},F)(a,b)\mathrel{\mathop{:}}=\begin{cases}X_{0}(a,b)&\textrm{ if }a,b\in X_{0}\\ X_{1}(a,b)&\textrm{ if }a,b\in X_{1}\\ \psi_{n-1}^{k-1}(F(a,b))&\textrm{ if }a\in X_{0},b\in X_{1}\\ \varnothing&\textrm{else}\end{cases}

The composition in $U(X_{0},X_{1},F)$ is the obvious one, and it is clear that this defines a functor $U\colon\corr_{n}^{k}\to\gaunt_{n}$ . We now apply this functor to the terminal object of $\corr_{n}^{k}$ , namely the triple $(C_{0},C_{0},\phi_{n-1}^{k-1}(C_{k-1}))$ . Since $U(C_{0},C_{0},\phi_{n-1}^{k-1}(C_{k-1}))=C_{k}$ , it follows that $U$ factors through a functor $\psi_{n}^{k}\colon\corr_{n}^{k}\to(\gaunt_{n}/C_{k})$ .

It is now a simple matter to observe that $\psi_{n}^{k}$ is indeed quasi-inverse to $\phi_{n}^{k}$ . ∎

[0MIE]

Remark 5.5. Unwinding the functor $\phi_{n}^{k}$ in the argument above, one finds that the product in the category $\corr_{n}^{k}$ may be written recursively in the following manner:

(X_{0},X_{1},F)\times(Y_{0},Y_{1},G)\cong(X_{0}\times Y_{0},X_{1}\times Y_{1},F\otimes G)

where $F\otimes G$ is defined as the composite:

(X_{0}\times Y_{0})^{\mathrm{op}}\times(X_{1}\times Y_{1})\cong(X_{0}^{\mathrm{op}}\times X_{1})\times(Y_{0}^{\mathrm{op}}\times Y_{1})\stackrel{{\scriptstyle(F,G)}}{{\longrightarrow}}\corr_{n-1}^{k-1}\times\corr_{n-1}^{k-1}\stackrel{{\scriptstyle\times}}{{\longrightarrow}}\corr_{n-1}^{k-1}.

[0MIF]

Lemma 5.6. The category $(\gaunt_{n}/C_{k})$ of $k$ -correspondences is cartesian closed.

[0MIG]

Proof. The claim is that for any $k$ -correspondence $N\to C_{k}$ , the functor

-\times_{C_{k}}N\colon(\gaunt_{n}/C_{k})\to(\gaunt_{n}/C_{k})

admits a right adjoint $\uHom_{C_{k}}(N,-)$ .

Since $\gaunt_{n}/C_{k}$ is locally finitely presentable, it is cocomplete and admits a strong generator [1, Th. 1.20] and is co-wellpowered [1, Th. 1.58]. Thus by the special adjoint functor theorem [30, Sect. V.8] the functor

-\times_{C_{k}}N\colon(\gaunt_{n}/C_{k})\to(\gaunt_{n}/C_{k})

admits a right adjoint precisely if it commutes with colimits.

When $k=0$ , this follows from the fact that $\gaunt_{n}$ itself is cartesian closed.

For $k>0$ , suppose $(\gaunt_{n-1}/C_{k-1})$ is cartesian closed. To prove that the category $(\gaunt_{n}/C_{k})$ is cartesian closed, we require a description of colimits in terms of the equivalent category $\corr_{n}^{k}$ . For any small category $\Lambda$ and any diagram $X\colon\Lambda\to\corr_{n}^{k}$ with

X_{\lambda}=(X_{\lambda,0},X_{\lambda,1},F_{\lambda}),

it is easy to see that the colimit is given by the triple $(X_{0},X_{1},F)$ where

X_{0}=\colim_{\lambda\in\Lambda}X_{\lambda,0}\text{\quad and\quad}X_{1}=\colim_{\lambda\in\Lambda}X_{\lambda,1},

and $F\colon X_{0}^{\mathrm{op}}\times X_{1}\to\corr_{n-1}^{k-1}$ is the enriched left Kan extension of

\colim_{\lambda\in\Lambda}F_{\lambda}\colon\colim_{\lambda\in\Lambda}(X_{\lambda,0}^{\mathrm{op}}\times X_{\lambda,1})\to\corr_{n-1}^{k-1}

along the diagonal

\colim_{\lambda\in\Lambda}(X_{\lambda,0}^{\mathrm{op}}\times X_{\lambda,1})\to X_{0}^{\mathrm{op}}\times X_{1}.

Now for any object $Y=(Y_{0},Y_{1},G)\in\corr_{n}^{k}$ , we wish to compare $(\colim_{\lambda\in\Lambda}X_{\lambda})\times Y$ and $\colim_{\lambda\in\Lambda}(X_{\lambda}\times Y)$ . In light of our descriptions of products in $\corr_{n}^{k}$ , we see that the former is $(X_{0}\times Y_{0},X_{1}\times Y_{1},F\otimes G)$ , and the latter is the colimit of the diagram $Z\colon\Lambda\to\corr_{n}^{k}$ that carries $\lambda$ to

(X_{\lambda,0}\times Y_{1},X_{\lambda,1}\times Y_{1},F_{\lambda}\otimes G).

Note that, since $\gaunt_{n}$ is cartesian closed, one has

\colim_{\lambda\in\Lambda}(X_{\lambda,0}\times Y_{0})\cong X_{0}\times Y_{0}\text{\quad and\quad}\colim_{\lambda\in\Lambda}(X_{\lambda,1}\times Y_{1})\cong X_{1}\times Y_{1};

hence our description of colimits in $\corr_{n}^{k}$ exhibits the colimit of $Z$ as $(X_{0}\times Y_{0},X_{1}\times Y_{1},(F\otimes G)^{\prime})$ , where $(F\otimes G)^{\prime}$ is the enriched left Kan extension of

\colim_{\lambda\in\Lambda}(F_{\lambda}\otimes G)\colon\colim_{\lambda\in\Lambda}((X_{\lambda,0}\times Y_{0})^{\mathrm{op}}\times(X_{\lambda,1}\times Y_{1}))\to\corr_{n-1}^{k-1}

along the diagonal

\colim_{\lambda\in\Lambda}((X_{\lambda,0}\times Y_{0})^{\mathrm{op}}\times(X_{\lambda,1}\times Y_{1}))\to(X_{0}\times Y_{0})^{\mathrm{op}}\times(X_{1}\times Y_{1}).

Our induction hypothesis is that $\corr_{n-1}^{k-1}$ is cartesian closed; so this enriched left Kan extension can be identified with the composition of the enriched left Kan extension of

\colim_{\lambda\in\Lambda}(F_{\lambda},G)\colon\colim_{\lambda\in\Lambda}(X_{\lambda,0}^{\mathrm{op}}\times X_{\lambda,1})\times(Y_{0}^{\mathrm{op}}\times Y_{1})\to\corr_{n-1}^{k-1}\times\corr_{n-1}^{k-1}

along the diagonal

\colim_{\lambda\in\Lambda}(F_{\lambda},G)\colon\colim_{\lambda\in\Lambda}(X_{\lambda,0}^{\mathrm{op}}\times X_{\lambda,1})\times(Y_{0}^{\mathrm{op}}\times Y_{1})\to(X_{0}^{\mathrm{op}}\times X_{1})\times(Y_{0}^{\mathrm{op}}\times Y_{1}).

But now this left Kan extension is simply the product of $G$ with the left Kan extension that defines $F$ . In other words, we have an isomorphism $(F\otimes G)^{\prime}\cong F\otimes G$ , whence the proof is complete. ∎

Clark Barwick, Christopher Schommer-Pries

Original source: arXiv:1112.0040v6

Original source · 1112.0040v6

5. Correspondences and the category Gauntn\gaunt_{n}[0MLT]

5. Correspondences and the category $\gaunt_{n}$ [0MLT]