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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 is called an isometry if and a partial isometry when is dagger idempotent.
Is there a name for a morphism with the property that is an isomorphism?
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: such that is unitary.
Now, is also self-adjoint, so in particular being unitary would mean it is an involution.
Both in and in there are no examples in which is not already an isometry:
Now I think pretty much any “named concept” in dagger category theory is an extrapolation from and , so if there are no examples in these two I see it as unlikely that it would have shown up!
But there seems to be no strong reason for this to happen in general... Do you have an example in mind?
It is called a "closed monomorphism" by Matt Di Meglio in R*-categories, because in 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).
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.