Matthew Doty wrote:
>Hey Owen,
This reminds me a lot of [formal concept analysis]( Did you have this in mind when you presented your galois connections above?

No, I didn't know this kind of thing had a name! Although I imagine whoever I first heard talking about examples like these probably did know—thanks so much for providing the reference!

[Imagine the squiggly "challenges ahead" road sign here]

I think the reason this kind of example stuck in my head was that I learned it at about the same time I was learning about the Zariski topology on \\(\mathbb{C}^n\\). As the collection of objects, take the set of points of \\(\mathbb{C}^n\\), and have the collection of "attributes" be the ring \\(\mathbb{C}[x_1,\dots,x_n]\\) of polynomials in \\(n\\) variables, where a point \\((c_1,\dots,c_n)\\) has attribute \\(f\\) if and only if \\(f(c_1,\dots,c_n) = 0\\). Then the "closure" operation on subsets of \\(\mathbb{C}^n\\) is the literal closure in the Zariski topology, and one way of thinking about Hilbert's Nullstellensatz is that the corresponding closure operator on subsets of \\(\mathbb{C}[x_1,\dots,x_n]\\) is the radical of the ideal generated by the subset.