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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJan 29th 2019
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   Author: Mike Shulman
   Format: MarkdownItexI have a silly question. Suppose $E$ is a simplicially enriched category (or maybe a category enriched over any cartesian monoidal category, or maybe even any monoidal category, though that seems more dubious). Then its slice categories $E/x$ are also simplicially enriched, where for $f:y\to x$ and $g:z\to x$ the hom-object $(E/x)(f,g)$ is the fiber of $(g\circ -) : E(y,z) \to E(y,x)$ over $f$. Simplicial copowers in these slice categories are, like ordinary colimits, created in $E$, which means in particular that for $f:x \to y$ the induced functor $f_! : E/x \to E/y$ preserves copowers. Equivalently, the adjunction $f_! \dashv f^*$ is simplicially enriched, and the pullback functor $f^*$ preserves simplicial powers. Now what if $E$ is also locally cartesian closed, so that $f^*$ has a right adjoint $f_*$? Is the adjunction $f^* \dashv f_*$ also necessarily simplicially enriched? Does pullback $f^*$ preserve copowers, and dependent product $f_*$ preserve powers? I feel like I should know the answer, and I know that in the past I've believed that this was true; but I also can't see how it to prove it right now.

   I have a silly question. Suppose $E$ is a simplicially enriched category (or maybe a category enriched over any cartesian monoidal category, or maybe even any monoidal category, though that seems more dubious). Then its slice categories $E/x$ are also simplicially enriched, where for $f:y\to x$ and $g:z\to x$ the hom-object $(E/x)(f,g)$ is the fiber of $(g\circ -) : E(y,z) \to E(y,x)$ over $f$. Simplicial copowers in these slice categories are, like ordinary colimits, created in $E$, which means in particular that for $f:x \to y$ the induced functor $f_! : E/x \to E/y$ preserves copowers. Equivalently, the adjunction $f_! \dashv f^*$ is simplicially enriched, and the pullback functor $f^*$ preserves simplicial powers.

   Now what if $E$ is also locally cartesian closed, so that $f^*$ has a right adjoint $f_*$? Is the adjunction $f^* \dashv f_*$ also necessarily simplicially enriched? Does pullback $f^*$ preserve copowers, and dependent product $f_*$ preserve powers? I feel like I should know the answer, and I know that in the past I’ve believed that this was true; but I also can’t see how it to prove it right now.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJan 30th 2019
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   Author: Mike Shulman
   Format: MarkdownItexActually I think the question can be reduced to the simpler case of ordinary cartesian closure. That is, if $E$ is an enriched category with products (in the enriched sense) whose underlying category is cartesian closed, is the cartesian-closure adjunction necessarily an enriched adjunction?

   Actually I think the question can be reduced to the simpler case of ordinary cartesian closure. That is, if $E$ is an enriched category with products (in the enriched sense) whose underlying category is cartesian closed, is the cartesian-closure adjunction necessarily an enriched adjunction?