Not everything can be thought of as functions
This is a stand-alone version of a post that was split in two reblog chains under a different mostly unrelated post.
There are non-concretizable categories, aka categories which can not be thought of as sets and functions.
A concrete category is a category with a faithful (= injective on each hom-set, but not necessarily on objects) functor to the category of sets.
An classical example of a non-concretizable category is Ho(Top), the homptopy category of Topological spaces.
But why is this relevant?
Like with any size issue you can trick your way into ignoring it when you have large enough cardinals with respect to the category you are working with and are willing to step up in the size hierarchy. Similarily there are trivial counterexamples (if you allow categories with proper hom-classes).
Whenever we have a category K and a universe of set theory V in which this category is set-sized we have an easy concreatization F: K→V which maps objects X to ∪{K(A,X) | A∈K} and morphisms f∈K(X,Y) to -○f.
And if a category has a proper class sized hom-class there cannot be a concretization.
However there is the relevant case of proper class sized categories with set-sized hom-sets. For example the category of topologigal spaces. If we care about "all" topological spaces the trick of taking an universe of set theory with larger cardinals is no longer viable, since this would mean we can also have more topological spaces. Morally similar to how circumventions of various other size issues don't fundamentally invalidate those size issues.
But topological spaces are still concretizable, by simply forgetting the topological structure. The interesting insight is that this can not be done for all categories. Not all categories with set sized hom-sets can be thought of as sets with additional structure and morphisms which are particular functions, hence my original claim.
Ho(Top) has set sized hom-sets, since those are surjected on from the hom-sets of Top. Yet it can not be thought of as sets and functions.
In some way you can reinterpret the fact that not all locally small categories are concretizable as "the hom-sets being small does not imply that the objects can be represented via small sets".
How to prove this?
Theorem (Isbell, Freyd). A category which has all finite limits is concretizable if and only if it is well-powered when restricted to regular subobjects.
In fact this theorem holds for a weakening of the finite limits condition, but that basically just adds notational overhead.
Here an object being well-powered means that it has a partially ordered set of subobjects (= monomorphisms to the object). (This corresponds to the intuition "the powerset of the object exists. Ofc monomorphisms to an object are always partially ordered up to isomorphisms by monomorphisms between their domains, the important part here is that it is only set-sized. Also you can only take the domains of those monomorphisms as the subobjects, which also works.)
A monomorphism (or subobject) is regular if it is the equalizer of two morphisms in the category. (So intuetively if it can be defined as "the part of the domain of two morphisms on which they agree". This has little to do with the notion of regularity in set theory.)
The "⇒" of the theorem is due to Isbell and is quite easy to prove by
just applying the definitions to get that faithful functors are injective on equivalence classes of regular subobjects,
and the fact that sets have power-sets.
Try it yourself! (The "⇐" direction is due to Freyd and more complicated.)
Freyd discovered that we can now use this argument by Isbell to prove that Ho(Top) is not concretizable.
