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Stream: learning: questions

Topic: formalization of partially ordered category?

Sean Maley (Nov 15 2024 at 18:37):

Hi all, I am in a category theory cohort, and one of my classmates had a question that I am posting on his behalf.

Hi there, recently I was digging into an old paper [1]. My question is on the "Representation of categories" Appendix. There, it is shown that any abstract category A can be embedded as a subcategory of the category S, the objects of which consist of all sets and the mappings of which are the totality of all set-theoretical many-one mappings of sets. I hope I am not misunderstanding this part. Correct me if I am wrong. 

At the end of such subsection, the author pointed out that even though most of the categories have an intrinsic partial order, there are others where finding such intrinsic partial order it is not trivial. The authors claim: "The problem of getting 'order preserving representations' would require probably a suitable formalization of the concept of a partially ordered category". After that they illustrate the type of arguments which may be involved by considering the category of discrete groups.

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This paper was published almost 80 years ago, and I would like to know if any of you have some more recent references on such "formalization of the concept of a partially ordered category". Has this already been done? Thank you so much in advance!

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[1] Eilenberg, S., & MacLane, S. (1945). General theory of natural equivalences. Transactions of the American Mathematical Society, 58(2), 231-294.

Eilenberg-GeneralTheoryNatural-1945.pdf

Mike Shulman (Nov 15 2024 at 21:18):

It's so fun to try to decipher the archaic language in old papers and see what it corresponds to in modern vocabulary.

First on the category $\mathfrak{S}$, their definition (p239) is "a mapping $\sigma$ of $\mathfrak{S}$ is determined by a pair of sets $S_1$ and $S_2$ and a many-one correspondence between $S_1$ and a subset of $S_2$, which assigns to each $x\in S_1$ a corresponding element $\sigma x\in S_2$." In modern terminology this is just "a function from $S_1$ to $S_2$". It's a little surprising to me that they apparently didn't have that terminology yet in 1945.

Second, you're right that that's what the first part of the appendix shows, but the proof only works when the abstract category $\mathfrak{A}$ is small. I'm also surprised they didn't notice this, since they were careful to say in their definition of category that it has an "aggregate" of objects and morphisms rather than a set, but then in this appendix they say "we shall denote by $R(A)$ the set of all [morphisms] $\alpha\in \mathfrak{A}$ such that [$A$ is the codomain of $\alpha$]", even though this is clearly also just an "aggregate" (a.k.a. class) and not a set.

Finally, as to your question about partially ordered categories, by which they mean categories with a partial order on the objects such as "subset" or "subgroup", I think for the most part category theorists have decided that this is not a very useful concept. To say that a group $G$ is a subgroup of a group $H$ is not invariant under isomorphism, so it's not a "category-theoretic" concept. Instead we usually talk about injective homomorphisms between groups, which are the same thing as subgroups up to isomorphism, but which are preserved by equivalences of categories such as this representation.

With that said, there does exist in the literature a formalization of the notion of "partially ordered category": in Awodey-Butz-Simspon-Streicher's paper Relating first-order set theories, toposes and categories of classes this was called a "system of inclusions" on a category. However, not much has been done with it for the reasons I mentioned, and in particular I'm not aware of any "representation theorem" for such.

Kevin Carlson (Nov 15 2024 at 22:56):

This partially-ordered-objects notion also reminds me of the proof that, what is it...every locally small category is a quotient of a concrete one, I think that's the statement? Can't remember the author without digging around, but I remember it involves a kind of delicate sorting of the objects by ordinals, so actually eventually a total order. But having a nice partial order related to cardinality or something probably helps you get started.