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A $\ast$-autonomous category is a biclosed monoidal category together with a dualizing object. An object $\bot$ in a biclosed monoidal category $(\mathcal{C},\otimes)$ with left internal hom $[-,-]$ is called a dualizing object if the functor $[-,\bot] \colon \mathcal{C}^{op}\rightarrow \mathcal{C}$ is an equivalence.

This immediately suggests the notion of a $\ast$-autonomous $(\infty,1)$-category:

Call an object $\bot$ in a biclosed monoidal $(\infty,1)$-category $\mathcal{C}$ with left internal hom $[-,-]$ a dualizing object if the $(\infty,1)$-functor $[-,\bot]:\mathcal{C}^{op} \rightarrow \mathcal{C}$ is an equivalence. A $\ast$-autonomous category is a biclosed monoidal $(\infty,1)$-category together with a dualizing object.

As I learned from this answer, a certain subcategory of the stable $(\infty,1)$-category of spectra seems to be an example of a $\ast$-autonomous $(\infty,1)$-category. The Anderson spectrum is a dualizing object.

Are there other naturally occurring $\ast$-autonomous $(\infty,1)$-categories?

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  • $\begingroup$ I assume you mean other than the "trivial" case of compact closed $(\infty,1)$-categories. In that direction, I presume there must be an $\infty$-Chu construction... $\endgroup$
    – Mike Shulman
    Commented Jan 17 at 17:59
  • $\begingroup$ @MikeShulman By a "compact closed $(\infty,1)$-category" do you mean a symmetric monoidal $(\infty,1)$-category whose homotopy category is compact closed? If so, you assume correctly :-) Your comment makes me wonder whether your results from $\ast$-Autonomous Envelopes and Conservativity extend to the $(\infty,1)$-categorical setting … $\endgroup$
    – Max Demirdilek
    Commented Jan 31 at 16:09
  • $\begingroup$ Good question; I haven't thought about it. $\endgroup$
    – Mike Shulman
    Commented Feb 2 at 12:15

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