Adjoint of simplicial left Kan extension

Question

Let $\mathcal{C}$ be a small simplicial category and let $F\: : \:\mathcal{C}^{op}\to sSet$ be a simplicial functor, we denote with $$ \int_{\mathcal{C}}F $$ the category where objects are triples $\left( x,\left[n \right],C \right)$ such that $x\in F_{n}\left(C \right) $ and maps $$ \left(\phi, f \right)\: : \: \left( y,\left[m \right],C' \right)\to\left( x,\left[n \right],C \right) $$ where $\phi\in \operatorname{Hom}_{\Delta}\left(\left[ n\right], \left[m \right] \right) $ and $f\in \operatorname{Hom}_{\mathcal{C}}\left(C, C' \right) $ such that $$ F\left(f \right)_{n}\circ \phi\left(y \right)=x. $$ There is a natural functor $$ \pi_{F}\: : \: \int_{\mathcal{C}}F\to \mathcal{C}\times \Delta $$ My goal is try to understand the proof of

Theorem

Let $\mathcal{C},\mathcal{E}$ be simplicial tensored and cotensored categories (tensored and cotensored over finite simplicial sets). Assume that $\mathcal{E}$ is cocomplete and $\mathcal{C}$ is small. Let $sPh\left( \mathcal{C}\right)$ be the category of simplicial presheaves over $\mathcal{C}$ (where each functors is a simplicial functor). Let $H\: : \: \mathcal{C}\to \mathcal{E}$ be a simplicial functor. Then the simplicial functor \begin{eqnarray*} R\: : \: \mathcal{E}&\to& sPh\left( \mathcal{C}\right)\\ E&\to& R\left( E\right), \end{eqnarray*} where $R\left( E\right)\left(C \right):=\operatorname{Map}_{\mathcal{E}}\left(G \left(C \right), E \right)$ has a left adjoint \begin{eqnarray*} L\: : \: sPh\left( \mathcal{C}\right)&\to& \mathcal{E}\\ \end{eqnarray*} given by $$ L\left( F\right):=\operatorname{colim}_{\int_{\mathcal{C}}F}\left(H\left(-\right) \otimes \Delta\left[- \right]\circ \pi_{F}\left(x, \left[n\right],C \right) \right) $$ Moreover we have that the functor $L$ preserves tensor objects and the adjunction is simplicially enriched, i.e. $$ \operatorname{Map}_{\mathcal{E}}\left(L\left(F \right) ,E \right)\cong \operatorname{Map}_{sPh\left( \mathcal{C}\right)}\left(F,R\left( E\right) \right). $$ This is theorem 2.3.21 of (Franz Vogler, Derived Manifolds from Functors of Points, logos Verlag Berlin). I understand why there is an ordinary adjunction but I have some difficult to understand why it is simplicial. Could you help me please?

Answer 1

It is generally true that if $\mathcal{E}$ is cotensored, then the underlying adjunction $L \dashv R$ lifts to enriched adjunction iff $R$ preserves cotensors. Let us assume that $\mathbb{V}$ is our base of enrichment. By classical Yoneda lemma, the objects: $$\operatorname{Map}_{\mathcal{E}}\left(L\left(F \right) ,E \right)\cong \operatorname{Map}_\mathcal{F}\left(F,R\left( E\right) \right)$$ are isomorphic iff for every object $X \in \mathbb{V}$, the following sets are naturally isomorphic: $$\hom(X, \operatorname{Map}_{\mathcal{E}}\left(L\left(F \right) ,E \right))\cong \hom(X, \operatorname{Map}_\mathcal{F}\left(F,R\left( E\right) \right))$$ Using the definition of the cotensor, the above may be rewritten as: $$\operatorname{Map}_{\mathcal{E}}\left(L\left(F \right) ,E\pitchfork X \right)_0\cong \operatorname{Map}_\mathcal{F}\left(F,R\left( E\right) \pitchfork X \right)_0$$ where the subscript $0$ denotes taking the underlying set. Because $R$ preserves cotensors, there is a canonical isomorphism $R(E) \pitchfork X \approx R(E \pitchfork X)$, and the above reduces to: $$\operatorname{Map}_{\mathcal{E}}\left(L\left(F \right) ,E\pitchfork X \right)_0\cong \operatorname{Map}_\mathcal{F}\left(F,R\left( E\pitchfork X\right) \right)_0$$ which holds as an instance of the underlying adjunction. (The other direction is trivial, because enriched right adjoint preserves all enriched limits that exist.)

The above may be strengthen in the obvious way --- it suffices to consider cotensors with objects from a dense subcategory $\mathbb{D} \subseteq \mathbb{V}$.

In case the base of the enrichment is a category of presheaves, $\mathbb{V} = \mathbf{Set}^{\mathbb{C}^{op}}$, one may choose for a dense subcategory the category of representables $\mathbb{D} = \mathbb{C}$ (again by Yoneda, representables form a dense subcategory).

Therefore, for simplicial categories the following holds: if $\mathcal{E}$ is cotensored over representable simplicial sets and $R$ preserves cotensors with representable simplicial sets, then the underlying adjunction $L \dashv R$ lifts to enriched adjunction.

Because functor $R$ from your definition clearly preserves every limit that exists in $\mathcal{E}$ (so, in particular, cotensors), the underlying adjunction: $$\operatorname{Map}_{\mathcal{E}}\left(L\left(F \right) ,E \right)_0\cong \operatorname{Map}_{sPh\left( \mathcal{C}\right)}\left(F,R\left( E\right) \right)_0$$ lifts to enriched adjunction: $$\operatorname{Map}_{\mathcal{E}}\left(L\left(F \right) ,E \right)\cong \operatorname{Map}_{sPh\left( \mathcal{C}\right)}\left(F,R\left( E\right) \right)$$