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1. • CommentRowNumber1.
   • CommentAuthoradeelkh
   • CommentTimeMay 2nd 2014
   • (edited May 3rd 2014)
   • PermaLink
   Author: adeelkh
   Format: MarkdownItexLet C be a category admitting colimits and limits. It is well known that the category of simplicial objects sC in a category C has an action by sSet: for $X \in sC$, $K \in sSet$, $X \otimes K$ is the simplicial object whose nth component is the coproduct $\coprod_{K_n} X_n$. One can verify that 1. For a fixed $K$, the functor $- \otimes K : sC \to sC$ admits a right adjoint $(-)^K$. 2. For $X \in C$, $X \otimes -$ preserves colimits and $X \otimes \Delta^0 = X$. 3. For $X \in C$ and $K,L \in sSet$, $A \otimes (K \times L) = (A \otimes K) \otimes L$. These facts imply that with the mapping spaces $Map_C(X, Y)$ defined by $$ Map_C(X,Y)_n = Hom_C(X \otimes \Delta^n, Y), $$ sC becomes an sSet-enriched category. All this is well-documented, for example in the book of Goerss-Jardine. More generally, let S be a small category. Let $PSh(S, C)$ denote the category of presheaves on S with values in C. Define an action of $PSh(S) = PSh(S, Set)$ on $PSh(S, C)$ in the same way as above: $$ (F \otimes K)(X) = \coprod_{K(X)} F(X) $$ Presumably, the above three facts are still true and one can define a $PSh(S)$-valued Hom by $$ \mathbf{Hom}(F, G)(X) = Hom_{PSh(S,C)}(F \otimes h(X), G) $$ where $h(X)$ is the presheaf of sets represented by $X \in S$. I imagine that this makes $PSh(S, C)$ into a $PSh(S)$-enriched category. Are these things written down somewhere (maybe on the nLab)? The arguments written in Goerss-Jardine probably work without modification, but I am wondering if there is a simpler description of the exponential objects $X^K$ than what they do.

   Let C be a category admitting colimits and limits. It is well known that the category of simplicial objects sC in a category C has an action by sSet: for $X \in sC$, $K \in sSet$, $X \otimes K$ is the simplicial object whose nth component is the coproduct $\coprod_{K_n} X_n$. One can verify that

   1. For a fixed $K$, the functor $- \otimes K : sC \to sC$ admits a right adjoint $(-)^K$.
   2. For $X \in C$, $X \otimes -$ preserves colimits and $X \otimes \Delta^0 = X$.
   3. For $X \in C$ and $K,L \in sSet$, $A \otimes (K \times L) = (A \otimes K) \otimes L$.

   These facts imply that with the mapping spaces $Map_C(X, Y)$ defined by

   $$Map_C(X,Y)_n = Hom_C(X \otimes \Delta^n, Y),$$

   sC becomes an sSet-enriched category. All this is well-documented, for example in the book of Goerss-Jardine.

   More generally, let S be a small category. Let $PSh(S, C)$ denote the category of presheaves on S with values in C. Define an action of $PSh(S) = PSh(S, Set)$ on $PSh(S, C)$ in the same way as above:

   $$(F \otimes K)(X) = \coprod_{K(X)} F(X)$$

   Presumably, the above three facts are still true and one can define a $PSh(S)$-valued Hom by

   $$\mathbf{Hom}(F, G)(X) = Hom_{PSh(S,C)}(F \otimes h(X), G)$$

   where $h(X)$ is the presheaf of sets represented by $X \in S$. I imagine that this makes $PSh(S, C)$ into a $PSh(S)$-enriched category.

   Are these things written down somewhere (maybe on the nLab)? The arguments written in Goerss-Jardine probably work without modification, but I am wondering if there is a simpler description of the exponential objects $X^K$ than what they do.

2. • CommentRowNumber2.
   • CommentAuthorZhen Lin
   • CommentTimeMay 2nd 2014
   • PermaLink
   Author: Zhen Lin
   Format: MarkdownItexYou don't need colimits or limits. There's a end formula for the hom-spaces: see [here](http://zll22.user.srcf.net/EnrichedFunctorCategory.pdf).

   You don’t need colimits or limits. There’s a end formula for the hom-spaces: see here.

3. • CommentRowNumber3.
   • CommentAuthoradeelkh
   • CommentTimeMay 2nd 2014
   • PermaLink
   Author: adeelkh
   Format: MarkdownItexInteresting, thanks!

   Interesting, thanks!