4
$\begingroup$

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?

$\endgroup$
7
  • $\begingroup$ What unit and counit are you asking to be invertible? $\endgroup$
    – Max New
    Sep 13, 2022 at 16:48
  • 3
    $\begingroup$ 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 $\endgroup$ Sep 13, 2022 at 17:02
  • 2
    $\begingroup$ 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
    Sep 13, 2022 at 21:34
  • $\begingroup$ @ChrisH Is that category $\ast$-autonomous? I didn't know that. $\endgroup$ Sep 14, 2022 at 22:54
  • 1
    $\begingroup$ @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)... $\endgroup$ Sep 16, 2022 at 16:08

1 Answer 1

3
$\begingroup$

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 $I_{\mathbb Z}$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.