You said that you thought that this could be done in a categorical way. I attempted something which appears to describe the same thing when applied to the category FinSet , but I’m not sure it’s the sort of thing you meant by when you suggested that the combinatorial part could potentially be done in a categorical way instead, and I’m not sure that it is fully categorical.
Let S be an object. For i from 1 to k, let Ai be an object, (which is not anything isomorphic to the product of itself with itself, or at least is not the terminal object) . Let f:∏iAi→S be an isomorphism. Then, say that ((Ai)i,f,S) is a representation of a factorization of S. If ((Ai)i,f,S) and ((A′i)i,f′,S) are each a representative of a factorization of S, then say that they represent the same factorization of S iff there exist isomorphisms gi:Ai→A′i such that , where ⟨g1,g2,...,gk⟩:∏iAi→∏iA′i is the isomorphism obtained from the gi with the usual product map, the composition of it with f’ is equal to f, that is, ⟨g1,g2,...,gk⟩;f′=f .
Then say that a factorization is, the class of representative of the same factorization. (being a representation of the same factorization is an equivalence relation).
For FinSet , the factorizations defined this way correspond to the factorizations as originally defined.
However, I’ve no idea whether this definition remains interesting if applied to other categories.
For example, if it were to be applied to the closed disk in a category of topological spaces and continuous functions, it seems that most of the isomorphisms from [0,1] * [0,1] to the disk would be distinct factorizations, even though there would still be many which are identified, and I don’t really see talking about the different factorizations of the closed disk as saying much of note. I guess the factorizations using [0,1] and [0,1] correspond to different cosets of the group of automorphisms of the closed disk by a particular subgroup, but I’m pretty sure it isn’t a normal subgroup, so no luck there. If instead we try the category of vector spaces and linear maps over a particular field, then I guess it looks more potentially interesting. I guess things over sets having good analogies over vector spaces is a common occurrence. But here still, the subgroups of the automorphism groups given largely by the products of the automorphism groups of the things in the product, seems like they still usually fail to be a normal subgroup, I think. But regardless, it still looks like there’s some ok properties to them, something kinda Grassmannian-ish ? idk. Better properties than in the topological spaces case anyway.
I have not thought much about applying to things other than finite sets. (I looked at infinite sets enough to know there is nontrivial work to be done.) I do think it is good that you are thinking about it, but I don’t have any promises that it will work out.
What I meant when I think that this can be done in a categorical way is that I think I can define a nice symmetric monodical category of finite factored sets such that things like orthogonality can be given nice categorical definitions. (I see why this was a confusing thing to say.)
You said that you thought that this could be done in a categorical way. I attempted something which appears to describe the same thing when applied to the category FinSet , but I’m not sure it’s the sort of thing you meant by when you suggested that the combinatorial part could potentially be done in a categorical way instead, and I’m not sure that it is fully categorical.
Let S be an object.
For i from 1 to k, let Ai be an object, (which is not anything isomorphic to the product of itself with itself, or at least is not the terminal object) .
Let f:∏iAi→S be an isomorphism.
Then, say that ((Ai)i,f,S) is a representation of a factorization of S.
If ((Ai)i,f,S) and ((A′i)i,f′,S) are each a representative of a factorization of S, then say that they represent the same factorization of S iff there exist isomorphisms gi:Ai→A′i such that , where ⟨g1,g2,...,gk⟩:∏iAi→∏iA′i is the isomorphism obtained from the gi with the usual product map, the composition of it with f’ is equal to f, that is, ⟨g1,g2,...,gk⟩;f′=f .
Then say that a factorization is, the class of representative of the same factorization. (being a representation of the same factorization is an equivalence relation).
For FinSet , the factorizations defined this way correspond to the factorizations as originally defined.
However, I’ve no idea whether this definition remains interesting if applied to other categories.
For example, if it were to be applied to the closed disk in a category of topological spaces and continuous functions, it seems that most of the isomorphisms from [0,1] * [0,1] to the disk would be distinct factorizations, even though there would still be many which are identified, and I don’t really see talking about the different factorizations of the closed disk as saying much of note. I guess the factorizations using [0,1] and [0,1] correspond to different cosets of the group of automorphisms of the closed disk by a particular subgroup, but I’m pretty sure it isn’t a normal subgroup, so no luck there.
If instead we try the category of vector spaces and linear maps over a particular field, then I guess it looks more potentially interesting. I guess things over sets having good analogies over vector spaces is a common occurrence. But here still, the subgroups of the automorphism groups given largely by the products of the automorphism groups of the things in the product, seems like they still usually fail to be a normal subgroup, I think. But regardless, it still looks like there’s some ok properties to them, something kinda Grassmannian-ish ? idk. Better properties than in the topological spaces case anyway.
I have not thought much about applying to things other than finite sets. (I looked at infinite sets enough to know there is nontrivial work to be done.) I do think it is good that you are thinking about it, but I don’t have any promises that it will work out.
What I meant when I think that this can be done in a categorical way is that I think I can define a nice symmetric monodical category of finite factored sets such that things like orthogonality can be given nice categorical definitions. (I see why this was a confusing thing to say.)