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Joe Moeller (Jun 26 2024 at 18:12):I'm wondering if anybody's ever categorified the notion of a positive definite map? It seems like the sort of thing Lawvere would have done, but I didn't immediately find it on a quick skim. I have my own ideas and purposes, but I'm interested in seeing what's already out there.
Cole Comfort (Jun 27 2024 at 10:36):There is the obvious dagger categorical generalization of a positive semidefinite matrix: ie one which admits a decomposition of the form $ f;f^\dag $ . But I don't think this counts as categorification. I would be interested to know as well!
John Baez (Jun 27 2024 at 11:14):So @Joe Moeller: do you really mean "categorification" in the officially sense of boosting things from sets to categories, and vector spaces to 2-vector spaces, and so on - or do you mean "generalization to other categories"? Since you were my student I hope you'd never become one of those terrible people who uses "categorification" in the other sense, but sometimes it happens.
John Baez (Jun 27 2024 at 11:15):Both categorification and generalization to other categories are interesting here, and I think @Cole Comfort nailed it for the latter.
Joe Moeller (Jun 27 2024 at 18:38):My intention is to categorify in the sense of going up a level, but I'd be happy to hear about "horizontal categorifications" as well.
Mike Stay (Jun 27 2024 at 18:39):You might want the hom functor of a compact closed rig category. A matrix is positive definite iff it defines an inner product. The inner product satisfies
<x, 0> = 0 = <0, x>
<x + y, z> = <x, z> + <y, z>
<x, y + z> = <x, y> + <x, z>
<cx, y> = c̅ <x, y>
<x, cy> = c <x, y>
<x, y> = bar{<y, x>}
∑ᵢ <x, eᵢ> <eᵢ, y> = <x, y>
<T† x, y> = <x, Ty>
where bar/overline is conjugation, † is conjugate transpose and eᵢ is some orthonormal basis.
In every category, the hom satisfies properties analogous to some of these:
∫ᵢ hom(x, i) hom(i, y) ≅ hom(x, y)
hom(Lx, y) ≅ hom(x, Ry) if L, R are adjoint functors.
In a compact closed category, where is isomorphic to , it also satisfies
hom(x, y) ≅ hom(y, x)*.
In a compact closed rig category like FinVect having a monoidal operation + with unit 0 over which tensor distributes, the rest of the laws hold up to isomorphism.
Joe Moeller (Jun 27 2024 at 18:40):This dagger categorical form makes sense to me, thanks!
Joe Moeller (Jun 27 2024 at 18:41):Thanks Mike, this looks super interesting. I'll stew on it a bit.
Cole Comfort (Jun 27 2024 at 21:51):Dominic Verdon appears to have categorified completely positive maps, which is a closely related concept, so maybe you will find something useful in this paper:
Cole Comfort (Jun 27 2024 at 21:51):