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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJan 29th 2019
   • PermaLink
   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 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 are also simplicially enriched, where for and the hom-object is the fiber of over . Simplicial copowers in these slice categories are, like ordinary colimits, created in , which means in particular that for the induced functor preserves copowers. Equivalently, the adjunction is simplicially enriched, and the pullback functor preserves simplicial powers.

   Now what if is also locally cartesian closed, so that has a right adjoint ? Is the adjunction also necessarily simplicially enriched? Does pullback preserve copowers, and dependent product 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 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?