Categorical version of the idea "intersection of all X property sets containing a set"

Question

Consider affine hull, convex hull, closure, interior.. all these concepts seem to have a similar deifnition that we have to intersect all the sets with a property containing our set under consideration. Is there a categorical idea behind htis?

Answer 1

Yes!

These are all examples of closure operators on a lattice, and there's some very general theorems describing when we can expect closure operators to exist. These theorems are often special cases of categorical theorems (viewing our lattice as a thin category). For example, anytime we have a galois connection between two posets (a special case of adjoint functors) we get a closure operator on each lattice by going back and forth. Conversely, every closure operator comes from a galois connection where one map is just the inclusion of the sublattice of closed objects into the full lattice.

What do these have to do with your observation that we usually construct the closure (resp. convex hull, etc.) of some set $X$ by taking the intersection of all the closed (resp. convex, etc.) sets containing $X$?

It turns out there's a third way to view this construction! There's a notion of a "topped $\bigcap$-structure" (somtimes also called a "closure system", iirc), which is a super common way for complete lattices to arise in practice! The idea is that as long as we know how to intersect big enough things, we also know how to take joins. You can see that all of your examples are of this form, and this is no accident -- these topped $\bigcap$-structures are extremely common in lattice theory, universal algebra, etc. There's more to say here, since a theory with universal axioms is "closed downwards" under substructure, and so the ubiquity of universally quantified axioms is some moral reason we see $\bigcap$-structures so often. For instance, there's a special case of closure oprators/complete lattices called "algebraic" which come from algebraic theories.

Now, remarkably, a topped $\bigcap$-structure is the same data as a closure operator! So we now have three ways of viewing the same construction, in varying levels of category-theoretic, haha. You can read more about this in Davey and Priestley's excellent book Introduction to Lattices and Order, particularly in Chapter 7. The authors go out of their way to not use the word "category", but they do a great job making it so that those in the know can recognize the category theory anyways.

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I hope this helps ^_^