In math there's an ordered hierarchy of concepts: stuff, structure and properties. This is because categories have 3 things: objects, morphisms between objects, and equations between morphisms. But we see it in the usual form of math definitions, like:
"A group is a set (that's the stuff ) with some operations (that's the structure) obeying some equations (those are the properties)."
To build a group we start with an object in the category of sets - that's the stuff. Then we give it some operations, i.e. some morphisms - that's the structure. Then we impose some equations - those are the properties.
These ideas let us talk about how much a forgetful functor forgets. There are 3 main options. Some functors forget nothing at all. Some forget properties but not structure or stuff. Some forget properties and structure, but not stuff. And some forget properties, structure and stuff. We can get more fancy than this, but this is a good place to start.
Let me say all this more technically, since the point of this junk is that we can make it all precise and prove theorems with it.
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An equivalence of categories is full, faithful and essentially surjective. It forgets nothing at all. It preserves everything.
A full and faithful functor forgets at most properties. In other words, it doesn't forget structure or stuff. In other words, it preserves structure and stuff. For example: the forgetful functor from abelian groups to groups. Its failure to be essentially surjective is how it's forgetting the property of being abelian.
A faithful functor forgets at most structure. In other words, it can forget properties and structure, but not stuff. In other words, it preserves stuff. For exampe: the forgetful functor from groups to sets. Its failure to be full is how it's forgetting the structure of being a group.
In general a functor can forget properties, structure and stuff. For example, the functor from pairs of sets to sets, that discards the second set. Its failure to be faithful is how it's forgetting the stuff of the second set.
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@johncarlosbaez Thanks for this! I enjoy reading your posts, but having never really learned about Category Theory a lot of them end up being pretty opaque to me. This helps explain at least one of the basic concepts. I keep thinking I need to go do some reading about Category Theory so that I can better appreciate the stuff you talk about.
I guess next I need to figure out what a monad is (outside the context of Leibniz's metaphysics). 😁
@johncarlosbaez It's funny because, of course, as a theoretical physicist I took a fair amount of math, including graduate courses in the math department, but not one of them even mentioned the existence of Category Theory. I only ever heard about it from random folks I talked to online.
To me it mostly seemed like a curiosity, a different way to formalize mathematics (rather than the set-based approach I was used to), sort of like using an algebraic axiomatization of probability theory rather than the traditional set-based Kolmagorov approach. Of course, I guess even that leads to interesting things like being able to define non-commutative probability theory, so I probably should have known that Category Theory could lead useful places. Seeing that you seem to consider it so important definitely has made me think I must be missing a really useful tool.
@internic @johncarlosbaez In the recent years there've been some efforts in theoretical physics to utilise categories for generalised symmetries, which roughly consider conserved higher dimensional objects rather than point charges. I'm not very familiar with its usefulness. The Simons Foundation has information on its recent activities on the funding org page https://scgcs.berkeley.edu/.