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Stream: learning: questions

Topic: "Higher" dagger categories

Paolo Perrone (Jul 18 2025 at 10:25):

Some bicategories, like the one of profunctors and its subcategory of adjoint functors, seem to be similar to dagger categories, but with an involution on objects as well. (Profunctors categorify relations, etc.)
Is there a name for this higher analogue of a dagger category?

David Corfield (Jul 18 2025 at 10:39):

I remember last year Dagger n-Categories appeared. Not sure if that might be of use.

Morgan Rogers (he/him) (Jul 18 2025 at 11:09):

Weaker than a 2-categorical version of [[star-autonomous category]], I guess?

Mike Shulman (Jul 18 2025 at 14:37):

Prof is actually stronger than $\ast$-autonomous, it's a [[compact closed category]].

John Baez (Jul 18 2025 at 15:48):

But Prof is not like a dagger-category, or dagger-2-category, in that a profunctor from $C$ to $D$ doesn't give a profunctor from $D$ to $C$ - unless e.g. $C$ and $D$ are dagger categories!

Amar Hadzihasanovic (Jul 18 2025 at 16:01):

I think that's what Paolo meant when he said that there is an involution on objects as well. But it does seem to me that if there is no ability to compose a morphism with its dagger, the "flavour" of dagger-categories is lost, and we are (even in cases when it is not literally true as it is for Prof) closer to categories-with-duals land.

Paolo Perrone (Jul 18 2025 at 16:20):

Wait, why can't we compose them?

Kevin Carlson (Jul 18 2025 at 17:23):

Well, the dagger of a profunctor $C\to D$ runs $D^{\mathrm{op}}\to C^{\mathrm{op}},$ no?

Paolo Perrone (Jul 18 2025 at 17:38):

I see. Right.

Mike Shulman (Jul 18 2025 at 18:49):

Although we don't usually call it the "dagger" for this reason.