What is Category Theory?

2-category 2-morphisms A-object abstract sets adjoint functors adjunctive square algebraic geometry algebraic topology anti-diagonal map applied arrows axioms basic bifunctor Cartesian category of sets Category Theory chimera chimera morphisms codomain cohomology comma category commutative composition concepts construction context corresponding counit defined definition determination diagram category domain Eilenberg and Mac elements equivalence example existence factorization finite function functor categories functor F G.Sica give given Grothendieck groupoid het-bifunctor Het(x heteromorphisms Hilbert homology homology theory homomorphism identity invariant Jackendoff knot language Lawvere left adjoint LimD logic Mac Lane Math mathematicians morphisms natural isomorphism natural transformations NFUP notion operations over-and-back pair paper philosophical proof properties quandle quantum relation representation right adjoint role satisfies self-determination semantics set theory Sica structure subcategory subset surface tensor theoretical topological spaces topos topos theory toposes underlying set uniformly slowly growing unique variable