Extending 2-variable adjunctions to diagram categories

We’d like to understand the following proposition, due (as near as I can tell) to Quillen in Homotopical Algebra:

Proposition If $\calC$ is a category, then the category of simplicial $\calC$-objects is enriched, tensored, and cotensored over $\sSet$.

In fact, this is a special case of the following proposition

Proposition If $-\otimes- : \calC\times \calD\to \calE$ forms a 2-variable adjunction with $[-,-] : \calC^\op\times\calE \to \calD$ and $\{-,-\} : \calD^\op\times \calE\to \calC$, $I$ is a small category, and $\calC$ and $\calD$ are complete and cocomplete, then there is a 2-variable adjunction $-\widehat{\otimes} - : \calC^I \times \calD^I \to \calE^I$, with left and right adjoints $\widehat{[-,-]}$ and $\widehat{\{-,-\}}$ respectively, defined by the following formulae, where $F:I\to \calC$, $G:I\to\calD$, $H:I\to \calE$:

$$F\widehat{\otimes}G = I\toby{\Delta} I\times I \toby{F\times G} \calC\times\calD \toby{\otimes} \calE,$$ $$\widehat{[F,H]} = \int_{i\in I} \left[ I(-,i) \otimes Fi,Hi \right],$$ $$\widehat{\{G,H\}} = \int_{i\in I} \left\{ I(-,i) \otimes Gi,Hi \right\}$$

Proof.

The proof is a standard exercise in manipulating ends, we’ll give the natural isomorphism for $\widehat{\otimes}$ and $\widehat{[-,-]}$, since the other natural isomorphism is the symmetric.

$$\newcommand\of[1]{\left({#1}\right)} \newcommand\hattimes{\widehat{\otimes}} \newcommand\hatbrak[2]{\widehat{\left[{#1},{#2}\right]}} \newcommand\hatbrace[2]{\widehat{\left\{ {#1},{#2}\right\}}} \begin{aligned} \calE^I \of{ G, \hatbrak{F}{H} } &\simeq \int_{j\in I} \calE \of{ Gj, \int_{i\in I} [I(j,i)\otimes Fi, Hi] } \\ &\simeq \int_{i,j\in I} \calE \of{ Gj, [I(j,i)\otimes Fi, Hi] } \\ &\simeq \int_{i,j\in I} \calE \of{ \of{I(j,i)\otimes Fi}\otimes Gj, Hi } \\ &\simeq \int_{i,j\in I} \calE \of{ I(j,i)\otimes \of{Fi\otimes Gj}, Hi } \\ &\simeq \int_{i,j\in I} \Set\of{ I(j,i), \calE \of{ Fi\otimes Gj, Hi } } \\ &\simeq \int_{i\in I} \calE \of{ Fi\otimes Gi, Hi } \\ &\simeq \calE^I\of{F\hattimes G,H} \qquad \blacksquare \end{aligned}$$

The result itself is fairly mundane to prove, but the ideas behind it and the consequences are quite interesting.

Some Consequences

Let’s first observe some corollaries.

Corollary (a) If $\calV$ is closed (symmetric) monoidal, complete, and cocomplete, then $\calV^I$ is also closed (symmetric) monoidal. (b) $\sSet$ is closed symmetric monoidal. (c) If $\calC$ is $\calV$-enriched, tensored, and cotensored, complete and cocomplete as an ordinary category, and $\calV$ is as in (a), then $\calC^I$ is $\calV^I$-enriched, tensored, and cotensored. (d) If $\calC$ is complete and cocomplete, then $\calC^{\Delta^\op}$ is $\sSet$-enriched, tensored, and cotensored.

Proof.

(a) Since the multiplication on the functor category is the pointwise tensor product, it’s easy to check that we get monoidal structure natural isomorphisms on the functor category.

(b) This follows from (a), since $\sSet=\Set^{\Delta^\op}$.

(c) Directly analagous to (a), since being enriched, tensored, and cotensored can be expressed as there being a tensor $\calV\times \calC \to \calC$, which has appropriate unitality and associativity isomorphisms which are compatible with those of the monoidal structure on $\calV$, and admitting the appropriate adjoints to form a 2-variable adjunction.

(d) Follows from (c) in exactly the same manner as (a). We note that the tensor product of a simplicial set $K_\bullet$ and a simplicial object $C_\bullet$ is given by

$$(K_\bullet\otimes C_\bullet)_n = K_n\otimes C_n,$$

the hom simplicial set between simplicial objects $C_\bullet$ and $D_\bullet$ is given by

$$\underline{[\Delta^\op,\calC]}(C_\bullet,D_\bullet)_n = [\Delta^\op,\calC](\Delta^n \otimes C_\bullet, D_\bullet),$$

and the cotensor of a simplicial set $K_\bullet$ with a simplicial object $C_\bullet$ is given by

$$[K_\bullet,C_\bullet]_n = \int_{m\in \Delta^\op} [\Delta^n_m\times K_m, C_m].$$

Some Thoughts and Observations

Firstly, we can exploit the symmetry of 2-variable adjunctions to get 2-variable adjunctions $(\calC^\op)^I\times \calE^I \to \calD^I$ given by $(F,H)\mapsto [F-,H-]$ and $(\calD^\op)^I\times \calE^I\to \calC^I$ given by $(G,H)\mapsto \{G-,H-\}$. The left mapping adjoint in these adjunctions gives us 2-variable adjunctions $\calC^{I^\op}\times \calD^I\to\calE^I$ and $\calD^{I^\op}\times \calC^I\to \calE^I.$

In particular, this should give us a simplicial enrichment on the category of cosimplicial objects.

The first functor defined is the cotensor, $[K_\bullet,C_\bullet]_n=[K_n,C_n],$ as described above, since it is of the form $\sSet^\op\times \calC^\Delta\to \calC^\Delta$. By symmetry, we can view it as a tensor for the purposes of the proposition above. When we think of $[-,-]:\Set^\op \times\calC \to \calC$ as the tensor, then the right mapping adjoint is of the form $\calC^\op\times \calC\to \Set^\op$, which is equivalent to $\calC\times \calC^\op\to\Set .$ Thus the right mapping adjoint should be given by $\calC^\op :\calC\times\calC^\op\to \Set.$ The right mapping adjoint of the cotensor of simplicial objects will thus be of the form $(\calC^\Delta)^\op\times \calC^\Delta \to (\Set^\op)^\Delta \simeq \sSet^\op,$ and be computed by

$$(F,G)\mapsto \int^{m\in\Delta} \calC^\op([\Delta^{-}_{m},Fm],Gm),$$

where the end should have become a coend, since we swapped $\Set^\op$ with $\Set$, but it’s going to need to turn back into an end for the final formula, and I can’t figure out where the calculation went wrong. Then (identifying the functor with its opposite) this is of the form $\calC^\Delta\times(\calC^\Delta)^\op\to \sSet.$ Note that the arguments to the right mapping functor are reversed from what we expect for the enriched hom functor. Swapping these gives the formula:

$$\underline{\calC^\Delta}(F,G)_n = \int_{m\in \Delta} \calC(Fm, [\Delta^n_m,Gm]).$$

As a quick check, computing

$$\begin{aligned} \sSet(K_\bullet,\underline{\calC^\Delta}(F,G)_\bullet) &\simeq \int_{n\in\Delta^\op}\int_{m\in\Delta} \Set(K_n,\calC(Fm,[\Delta^n_m,Gm])) \\ &\simeq \int_{n\in\Delta^\op}\int_{m\in\Delta} \Set(\Delta^{n_m},\calC(Fm,[K_n,Gm])) \\ &\simeq \int_{n\in\Delta^\op}\calC(Fn,[K_n,Gn]) \\ &\simeq \calC^{\Delta^\op}(F_\bullet,[K_\bullet,G_\bullet]), \\ \end{aligned}$$

as required.

It turns out that we’ve recovered the construction of the simplicial enriching on cosimplicial objects described by $n$Lab here, which we obtain by regarding the category of cosimplicial objects in $\calC$ as the opposite of the category of simplicial objects in $\calC^\op.$

We can compute the hom simplicial sets of the construction on $n$Lab as well to verify this. We get

$$\underline{\calC^\Delta}(F,G)_n = \int_{m\in\Delta}\calC^\op([\Delta^n_m,Gm],Fm)= \int_{m\in \Delta} \calC(Fm, [\Delta^n_m,Gm]).$$