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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeDec 7th 2013
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am feeling a bit silly about the following, as this is probably easy and I must be being dense, but I'll throw this out as a question now anyway: given an $\infty$-functor $f : X \longrightarrow Y$ between $\infty$-groupoids, we get an induced pullback $$ f^\ast : E Mod(Y) \longrightarrow E Mod(X) $$ for $E$ any $E_\infty$-ring and $E Mod(-) \coloneqq Func(-, E Mod)$ the $\infty$-category of $E$-$\infty$-modules on $X$. This $f^\ast$ should be _closed_ monoidal, I suppose. I can see a pseudo-proof, but I am a bit stuck with making it a rigorous proof. Can anyone help?

   I am feeling a bit silly about the following, as this is probably easy and I must be being dense, but I’ll throw this out as a question now anyway:

   given an $\infty$-functor $f : X \longrightarrow Y$ between $\infty$-groupoids, we get an induced pullback

   $$f^\ast : E Mod(Y) \longrightarrow E Mod(X)$$

   for $E$ any $E_\infty$-ring and $E Mod(-) \coloneqq Func(-, E Mod)$ the $\infty$-category of $E$-$\infty$-modules on $X$.

   This $f^\ast$ should be closed monoidal, I suppose. I can see a pseudo-proof, but I am a bit stuck with making it a rigorous proof. Can anyone help?

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 7th 2013
   • (edited Dec 7th 2013)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAh, I realize that I am looking here exactly for the $\infty$-version of Mike's article * Mike Shulman _Enriched indexed categories_ ([arXiv:1212.3914](http://arxiv.org/abs/1212.3914)) The context is example 2.2 there, with $\mathbf{V}$ now the $\infty$-category $E Mod$, and I am after the $\infty$-analog of the consequence example 2.17 of theorem 2.14 there. With a minimum of enriched $\infty$-category theory all this should just go through verbatim...

   Ah, I realize that I am looking here exactly for the $\infty$-version of Mike’s article

   • Mike Shulman Enriched indexed categories (arXiv:1212.3914)

   The context is example 2.2 there, with $\mathbf{V}$ now the $\infty$-category $E Mod$, and I am after the $\infty$-analog of the consequence example 2.17 of theorem 2.14 there.

   With a minimum of enriched $\infty$-category theory all this should just go through verbatim…