Algebraic topology

cats, functors (morphisms between cats), natural transformations (morphisms between functors), adjoint pairs, limits (product, pull-back, projective limit), colimits (push-out, coproduct)

k-spaces, k-ifications, weak Hausdorff spaces, compactly generated spaces

the pointed cat; wedges, smashes, suspensions; homotopy groups

fundamental groupoid and its action on homotopy groups; weak equivalences; relative homotopy groups and the long exact sequence of a pair

cofibrations (homotopy extension property, neighborhood deformation retract pairs); mapping cones and mapping cylinders; cofibers; cofiber sequence

fibrations (homotopy lifting property); homotopy fiber; fiber sequence; Hopf fibration; Hopf invariant; application to the classification of real division algebras

compression lemma; cellular approximation; Whitehead theorem