$V\times V^{op}$ is a special case of the Chu construction.

]]>Added the example of $V \times V^{op}$.

]]>Added pointer to:

- Jürgen Fuchs, Gregor Schaumann, Christoph Schweigert, Simon Wood,
*Grothendieck-Verdier duality in categories of bimodules and weak module functors*[arXiv:2306.17668]

added the publication data:

- Albrecht Dold, Dieter Puppe,
*Duality, Trace and Transfer*, Proceedings of the Steklov Institute of Mathematics**154**(1984) 85–103 [mathnet:tm2435, pdf, pdf]

Added a reference to Dold and Puppe’s work which implies that a $\ast$-autonomous category with a natural isomorphism $(A \otimes B)^\ast \simeq A^\ast \otimes B^\ast$ is a compact closed category.

]]>I think the generalization from posets (as in Girard’s phase semantics) to more general idempotent adjunctions was original. I haven’t written it or seen it anywhere else, so yes, I guess you could cite the nLab page.

]]>Incidentally, Mike has a corresponding MathOverflow question/comment at MO:343167.

]]>Okay, so is this observation due to Mike Shulman? Should I cite this nLab entry as the original reference on “predualizing objects”?

]]>I don’t know off the top of my head. But that paragraph was added in revision 39 by Mike Shulman.

]]>One of the examples is

Suppose $\langle C,\otimes, I,\multimap\rangle$ is a closed symmetric monoidal category equipped with a “pre-dualizing object” $\bot$, in the sense that the contravariant self-adjunction $(-\multimap\bot) \dashv (-\multimap\bot)$ is idempotent, i.e. the double-dualization map $A \to (A \multimap \bot) \multimap \bot$ is an isomorphism whenever $A$ is of the form $B\multimap\bot$. (Note that idempotence is automatic if $C$ is a poset.) Then the category of fixed points of this adjunction, i.e. the full subcategory of objects of the form $B\multimap\bot$, is $*$-autonomous. For it is closed under $\multimap$, as $(A\multimap (B\multimap \bot)) \cong (A\otimes B\multimap \bot)$, and reflective with reflector $(-\multimap\bot)\multimap\bot$, and it contains $\bot$ since $\bot\cong (I\multimap \bot)$. Hence it is closed symmetric monoidal with tensor product $((A\otimes B)\multimap\bot)\multimap\bot$, and all its double-dualization maps are isomorphisms by assumption. A historically important example is Girard’s phase semantics of linear logic. Note that this category is a full subcategory of $Chu(C,\bot)$ closed under duality — indeed, it is the intersection of the two embeddings of $C$ and $C^{op}$ therein — but its tensor product is

notthe restriction of the tensor product of $Chu(C,\bot)$.

Is there a citeable reference where this example is presented? Who should be credited for this observation?

]]>Added definition of the internal hom from $\parr$.

]]>Suggest a new notion of weak $*$-autonomous category: « We can weaken the requirement that the canonical morphism $d_A: A \to (A \multimap \bot) \multimap \bot ,$ is an isomorphism to the one that it is a monomorphism. »

$Vec_{\mathbb{K}}$ is a weak $*$-autonomous category. It allows to speak about duality in more general situations.

I would be glad to have returns on this.

]]>Added an example of a subcategory of $Chu(Vect,K)$ which is a *-autononomous category and the reference $On *-autonomous categories of topological vector spaces$ by Michael Barr which develops the idea.

]]>Change the link from Vect to FinVect category in examples

Oleg Nizhnik

]]>Added a list of relations to other structures, including the fact that a traced $\ast$-autonomous category must be compact closed (Hajgató and Hasegawa).

]]>Still a bug, though, even if we can work around it easily. (-:

]]>That must be it. Have shifted to round ones.

]]>Same here. Perhaps related to the square brackets around that sentence?

]]>The links aren’t appearing for me at star-autonomous category in this sentence

Regarding the use of “autonomous”, this was once used as a bare adjective to describe a closed monoidal category, or sometimes a compact closed monoidal category,…

Anyone else?

]]>Thought I’d add Mike’s explanation of “autonomous”.

]]>Added Hyland-Schalk reference

]]>Ok, finally got my copy of Barr’s original monograph “$\ast$-autonomous categories” from ILL.

Firstly I was not quite right in #8 that our Definition 2 is that of Day-Street, since the former requires a priori only that $(-)^\ast$ is fully faithful, whereas Day and Street require it to be an equivalence (as does Definition C of Barr’s non-symmetric paper). But that doesn’t affect its coherence or lack thereof.

Our Definition 2 actually does appear in Barr’s monograph in paragraph (4.3). It’s not his *definition* of $\ast$-autonomous category, but he proves that this data “suffices to determine” a $\ast$-autonomous category, which isn’t quite the same as claiming that it’s an equivalent *definition*; so that could be compatible with what we’re seeing.

However, his actual definition doesn’t seem quite right to me either. He defines a $\ast$-autonomous category to be a closed symmetric monoidal category $\mathcal{C}$ together with a $\mathcal{C}$-enriched functor (which he calls a “closed functor” for some reason) $(-)^\ast : \mathcal{C}^{op}\to \mathcal{C}$ and a $\mathcal{C}$-enriched natural isomorphism $A \cong A^{\ast\ast}$. It seems like this would have the same problem of being a torsor over the center of $\mathcal{C}$.

]]>Oh, silly me: of course $(-)^{\ast\ast}$ is $\mathcal{C}$-enriched, since it’s isomorphic to the identity.

But I still don’t quite understand #9: I see that the isomorphism $\mathcal{C}(A \otimes B, C) \cong \mathcal{C}(A, (B \otimes C^\ast)^\ast)$ determines an isomorphism from $(-)^{\ast\ast}$ to the identity, and also a universal property of $(-)^\ast$, as special cases, but how do these special cases in turn determine it?

]]>Re #11: yes, I mean enriched. Admittedly I didn’t check carefully yet that $(-)^\ast$ is enriched, but let’s see: a contravariant strength would take the form $(B \otimes A)^\ast \otimes B \to A^\ast$ which we can derive from the identity $(B \otimes A)^\ast \to (B \otimes A)^\ast$, so I think it would work out.

]]>