>>10925615Sets are collections of thing. They are small: there is no set of all sets. The only "real" property a set has is its size, so set theory deals with this a lot. We can use sets to build other mathematical structures by adding more stuff to sets.
A category is a collection of things along with maps between those things satisfying various axioms. They can be large: there is a category of all sets. There is a category of all categories if you're careful (see: Grothendieck universes), though these size issues are typically ignored.
Type theory is an alternative to set theory. It axiomatises the notion of one thing being of a particular type, rather than an element of a collection ("x has the type of a natural number", as opposed to "x is an element of the set of natural numbers"). You still don't have a type of all types, but size issues seem clearer than in set theory: you can sweep them under the rug using universes a la category theory.
Higher category theory is like category theory with morphisms between morphisms, and morphisms between morphisms between morphisms and so on. This is related to homotopy type theory, which is like type theory, but we thing of types as more topological.
