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1. Definition
Firstly, recall the following nLab-definition of a $\ast$-autonomous category:

A $\ast$-autonomous category is a symmetric closed monoidal category $(C,\otimes,I,\multimap)$ with a global dualizing object: An object $\bot$ such that the canonical morphism $$d_A: A \rightarrow (A \multimap \bot) \multimap \bot$$ which is the transpose of the evaluation map is an isomorphism for all $A \in C$.

Secondly, call an object $A$ in a (symmetric) monoidal category $(M,\otimes,\top)$ invertible if there exist an object $B \in M$, and two isomorphisms $\eta: \top\rightarrow A \otimes B$ and $\epsilon: B \otimes A \rightarrow \top$ satisfying the two zig-zag-identities. (In fact, by an argument of Saavedra Rivano it suffices to require that only one zig-zag identity holds, but that is beside the point.)

2. Question
What are ('real-world') examples of $\ast$-autonomous categories with non-invertible global dualizing object?

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  • $\begingroup$ What unit and counit are you asking to be invertible? $\endgroup$
    – Max New
    Commented Sep 13, 2022 at 16:48
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    $\begingroup$ The OP is referring to the general notion of a dualizable object in a monoidal category (which is not the same as a dualizing object): ncatlab.org/nlab/show/dualizable+object $\endgroup$
    – Qiaochu Yuan
    Commented Sep 13, 2022 at 17:02
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    $\begingroup$ The constructible category of sheaves on a singular variety might be an example, there doesn’t seem to be obvious dualising maps for the dualising sheaf. $\endgroup$
    – Chris H
    Commented Sep 13, 2022 at 21:34
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    $\begingroup$ @ChrisH Cool! That might be the most "naturally-occurring" $\ast$-autonomous category I've ever encountered. It would be great to have that as an example on the nLab (hint, hint)... $\endgroup$
    – Mike Shulman
    Commented Sep 16, 2022 at 16:08
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    $\begingroup$ Well, if you quote a definition in your question saying that it's symmetric, you shouldn't be surprised if people only give you answers that are symmetric... (-: $\endgroup$
    – Mike Shulman
    Commented Jan 16 at 17:46

1 Answer 1

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The dualizing object of a star-autonomous category is invertible iff the category is compact closed. See here.

There are lots of examples of star-autonomous categories which are not compact closed. In algebraic geometry, one systematically constructs dualizing sheaves which are not generally invertible -- some of this is alluded to in the comments.

In homotopy theory, the category of spectra of finite type is a good example. The dualizing object is the Anderson dualizing spectrum $I_{\mathbb Z}$.

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  • $\begingroup$ The first sentence is not quite true. There are $r$-categories (i.e. (symmetric) monoidal categories where the monoidal unit is a dualizing object) which are not rigid (aka compact closed). One example is the category $\mathsf{SupLat}$ of sup-lattices. Others are given in Examples 0.9 and 2.3 in Drinfeld's and Boyarchenko's paper. $\endgroup$
    – Max Demirdilek
    Commented Jul 31 at 13:31
  • $\begingroup$ What is true (with your first link or Proposition 1.3 in the above paper by Boyarchenko-Drinfeld) is that the dualizing object of a star-autonomous category is invertible iff the category is an r-category. $\endgroup$
    – Max Demirdilek
    Commented Jul 31 at 13:45
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    $\begingroup$ Thanks for the heads-up. I will have to work out if I’ve messed up in that other answer I linked to. $\endgroup$
    – Tim Campion
    Commented Aug 1 at 5:26

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