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Stream: deprecated: combinatorics

Topic: Topologies on finite sets

JR (Oct 24 2023 at 15:39):

Transitive digraphs determine categories such that if there is a morphism between a given source and target, it is unique. A theorem of Evans, Harary, and Lynn from 1967 states that when everything is finite, (labeled) transitive digraphs are in bijection with (labeled) topologies, cf. https://oeis.org/A000798. A couple of questions: 1. Is this fact discussed anywhere in a category-theoretical context? 2. Is there a known Grothendieck-topological-analogue?

JR (Oct 24 2023 at 15:50):

As an aside, the result of Evans, Harary and Lynn was anticipated and generalized to the infinite case by Ahlborn in Prop. 2.5.1 of a 1964 technical report

Graham Manuell (Oct 25 2023 at 00:12):

The "transitive digraph" terminology strikes me as pretty unusual. Usually, these would be called preorders. That the category of finite topological spaces and the category of finite preorders are equivalent is well known, though I do not know a source. The infinite version is the equivalence between Alexandrov topological spaces and preorders.

I don't know how relevant Grothendieck topologies per se are here, but finite $T_0$ spaces = finite sober spaces are equivalent to the opposite category of finite distributive lattices = finite frames as a restriction of the the general Stone duality between topological spaces and frames, but I think this special case was probably known earlier than that.

John Baez (Oct 25 2023 at 07:56):

Oh, I was wondering what a "transitive digraph" is. The equivalence between finite topological spaces and finite preorder goes like this, for anyone curious: given points $x,y$ in a finite topological space, we say $x \le y$ iff $x$ is contained in the closure of $y$ . The proof that this works - in particular, how you get back from preorders to topological spaces - is outlined in the Wikipedia article on a finite topological spaces.

John Baez (Oct 25 2023 at 08:04):

Key buzzwords for learning more are

specialization preorder: the preorder you get from a topological space, in the way I just described
Alexandrov topology: the topological space you get from a preorder
Birkhoff's representation theorem: a theorem closely connected to the equivalence between finite topological spaces and finite preorders.

Eric Forgy (Oct 25 2023 at 17:02):

I love this stuff :heart_eyes:
Probably not what motivated this topic, but it relates to some early tantalizing stuff from Rafael Sorkin (Perimeter Institute) and others, e.g.

Spacetime as a Causal Set, Sorkin

Love this classic website: https://www2.perimeterinstitute.ca/personal/rsorkin/

Eric Forgy (Oct 25 2023 at 17:18):

(Sidenote: One of my mentors, Terrence Honan, was a colleague of Sorkin (and student of Charles Misner). Also interested in discrete foundations for physics. His dissertation is one of the most beautiful papers I've ever read, but it does not exist in electronic form. If my life ever settles down enough, I'd like to digitize it.)

Jens Hemelaer (Oct 25 2023 at 17:30):

If you have a finite category $\mathcal{C}$ together with a Grothendieck topology $J$ on it, then the category of sheaves $\mathbf{Sh}(\mathcal{C},J)$ is equivalent to a category of presheaves $\mathbf{PSh}(\mathcal{D})$ , for $\mathcal{D}$ another finite category. This is a generalization of the situation for topological spaces, for topological spaces $\mathcal{C}$ would be the frame of open subsets and $\mathcal{D}$ would be the specialization preorder.