Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.


Stream: learning: questions

Topic: Generalization of isometry


view this post on Zulip Sascha Haupt (May 08 2025 at 10:45):

I am looking for a name for the following generalization of the concept of isomtery in a dagger category:
In a [[dagger category]], an isomtery is a morphism ABA\rightarrow B is called an isometry if ff=idAf^{\dagger}\circ f = id_A and a partial isometry when fff^{\dagger}\circ f is dagger idempotent.
Is there a name for a morphism ff with the property that fff^{\dagger}\circ f is an isomorphism?

view this post on Zulip Amar Hadzihasanovic (May 08 2025 at 11:25):

Just some comments as to “why maybe there isn't a name already”:
typically in dagger-categories it is “wrong” to talk about isomorphisms which are not unitary isomorphisms, so the more likely concept is: ff such that fff^\dagger \circ f is unitary.
Now, fff^\dagger \circ f is also self-adjoint, so in particular being unitary would mean it is an involution.
Both in Rel\mathbf{Rel} and in Hilb\mathbf{Hilb} there are no examples in which ff is not already an isometry:

view this post on Zulip Amar Hadzihasanovic (May 08 2025 at 11:28):

Now I think pretty much any “named concept” in dagger category theory is an extrapolation from Rel\mathbf{Rel} and Hilb\mathbf{Hilb}, so if there are no examples in these two I see it as unlikely that it would have shown up!

view this post on Zulip Amar Hadzihasanovic (May 08 2025 at 11:29):

But there seems to be no strong reason for this to happen in general... Do you have an example in mind?

view this post on Zulip Aaron David Fairbanks (May 08 2025 at 15:53):

It is called a "closed monomorphism" by Matt Di Meglio in R*-categories, because in Hilb\mathbf{Hilb} it corresponds to a monomorphism whose image is closed. It's also equivalent to being monic and pseudoinvertible (a fact that appears in the extended version of my paper about pseudoinverses now available here).

view this post on Zulip Sascha Haupt (May 08 2025 at 19:09):

Thanks @Amar Hadzihasanovic and @Aaron David Fairbanks for your remarks, those are helpful to me.

I am thinking about different daggers in the category of affine relations (where the morphosis are affine subspaces of some vector space). I don't have a concrete example for a closed morphism in that context yet.