Category theory

Remarks

The three levels are tied together: small categories and functors form a category, and functors from one category to another with natural transformations between them form a category again. Example: the statement "a functor sends isomorphisms to isomorphisms" belongs to category theory, since it only uses composition and identities. Indeed, if $g \circ f$ and $f \circ g$ are identities, then so are $F(g) \circ F(f)$ and $F(f) \circ F(g)$. Non-example: the statement "every finite group of prime order is cyclic" is not a categorical statement, since it refers to the elements of a group. Its main constructions are limits and colimits, adjoint functors, and the Yoneda lemma.