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
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
A homotopy type is called finite if it is presented (via the discussion at homotopy hypothesis) by either
a simplicial set with a finite number of non-degenerate simplices;
a CW-complex with a finite number of cells.
Similarly a spectrum (stable homotopy type) given by a sequence of finite homotopy types is called a finite spectrum.
Beware that a finite homotopy type in general does not have finite and finitely many homotopy groups (see e.g. at homotopy groups of spheres). Homotopy types with finite and finitely many homotopy groups have alternatively been called $\pi$-finite, or tame, or (adapted from homological algebra) “of finite type” (which needs to be carefully distinguishes, therefore, from “finite homotopy type”). See at homotopy type with finite homotopy groups.
The compact objects in ∞Grpd are the retracts of finite homotopy types. Not every such retract is itself a finite homotopy type; the vanishing of Wall's finiteness obstruction is a necessary and sufficient condition for this to happen.
p-local homotopy type, p-complete homotopy type, rational homotopy type
(infinity,1)-presheaves on $\infty Grpd_{fin}^{op}$ form the classifying (infinity,1)-topos for objects.
Last revised on July 8, 2021 at 19:36:43. See the history of this page for a list of all contributions to it.