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Stream: learning: questions

Topic: Generalization of isometry

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 $A\rightarrow B$ is called an isometry if $f^{\dagger}\circ f = id_A$ and a partial isometry when $f^{\dagger}\circ f$ is dagger idempotent.
Is there a name for a morphism $f$ with the property that $f^{\dagger}\circ f$ is an isomorphism?

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: $f$ such that $f^\dagger \circ f$ is unitary.
Now, $f^\dagger \circ f$ is also self-adjoint, so in particular being unitary would mean it is an involution.
Both in $\mathbf{Rel}$ and in $\mathbf{Hilb}$ there are no examples in which $f$ is not already an isometry:

• in $\mathbf{Rel}$ because for $R^\dagger \circ R$ to be invertible, $R$ would need to be total, in which case $R^\dagger \circ R$ always contains the identity relation, and so in order to be an involution, it has to be the identity relation;
• in $\mathbf{Hilb}$ because $A^\dagger A$ is positive, so its spectrum is positive, but an involution must have spectrum in $\{+1, -1\}$, so a positive involution has spectrum $\{+1\}$ and then it is the identity

Amar Hadzihasanovic (May 08 2025 at 11:28):

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

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?

Aaron David Fairbanks (May 08 2025 at 15:53):

It is called a "closed monomorphism" by Matt Di Meglio in R*-categories, because in $\mathbf{Hilb}$ 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).

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.