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Stream: learning: questions

Topic: (ID my structure) dagger transpose double category

Nathaniel Virgo (Jul 07 2024 at 09:25):

A dagger category is a category $\mathcal{C}$ with an involutive identity-on-objects functor $\dagger:\mathcal{C}^{op}\to \mathcal{C}$.

A variation on this concept would be a strict double category $\mathbb{C}$ with an involutive identity-on-objects functor $\dagger : \mathbb{C}^\top\to\mathbb{C}$, where ${}^\top$ is the transpose operator, which swaps the horizontal and vertical morphisms.

So this a category where there are two kinds of morphism (horizontal and vertical), and each morphism has an adjoint, but the adjoint is always a morphism of the opposite kind.

Does this concept have a name, and are there any interesting known examples?

Bryce Clarke (Jul 07 2024 at 09:40):

This is an interesting idea. As far as I know, it doesn’t have a name. Can you see ordinary dagger categories as an example?

Nathaniel Virgo (Jul 07 2024 at 09:46):

Every dagger category gives rise to one of these things: for a dagger category $\mathcal{C}$, form the double category $\mathbb{C}$ where the horizontal is $\mathcal{C}$, the vertical one is $\mathcal{C}^{op}$ and for any given frame there is a unique square iff the frame commutes in $\mathcal{C}$. Then there is a functor $\mathbb{C}^\top\to\mathbb{C}$ that maps a horizontal morphism $f$ to $f^\dagger$ as a vertical morphism, and vice versa.

Nathaniel Virgo (Jul 07 2024 at 09:48):

But there's a similar construction where $\mathcal{C}$ is just an ordinary category and the functor $\mathbb{C}^\top \to \mathbb{C}$ just maps $f$ to the other version of $f$.

Nathaniel Virgo (Jul 07 2024 at 10:45):

(I was missing an ${}^{op}$ above, I've edited.)

This isn't super satisfying though - it doesn't really feel like dagger categories are an examples of these things.

Matteo Capucci (he/him) (Jul 08 2024 at 06:59):

The idea of a transpositional symmetry is very intriguing! Transposition and reversal are two different symmetries though, so it feels to me such a symmetry would be different than the one for dagger cats. @Evan Patterson recently put forward a proposal for compact double categories, which are somewhat dagger-like, using the reversal symmetry.