homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
An equivalence in an (∞,1)-category in ∞Grpd, hence ann equivalence of (∞,1)-categories between (∞,1)-categories that happen to be ∞-groupoids.
In terms of presentation by topological spaces or simplicial sets (see at homotopy hypothesis) this is a weak homotopy equivalence.
In terms of presentation by CW-complexes or Kan complexes, it is a homotopy equivalence.