nLab homotopy type with finite homotopy groups



Homotopy theory

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



Paths and cylinders

Homotopy groups

Basic facts


Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type
falseinitial objectempty type
proposition(-1)-truncated objecth-proposition, mere proposition
proofgeneralized elementprogram
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
logical conjunctionproductproduct type
disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)
implicationinternal homfunction type
negationinternal hom into initial objectfunction type into empty type
universal quantificationdependent productdependent product type
existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)
equivalencepath space objectidentity type/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
completely presented setdiscrete object/0-truncated objecth-level 2-type/set/h-set
setinternal 0-groupoidBishop set/setoid
universeobject classifiertype of types
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels




The concept of a homotopy type (homotopy n-type) all of whose homotopy groups are finite groups does not have an established name. Sometimes it is called π\pi-finiteness. In the context of groupoid cardinality “tameness” is used. In homological algebraof finite type” is used, which in homotopy theory however badly clashes with the concept of finite homotopy type which is crucially different from homotopy type with finite homotopy groups. (Anel 21) uses “truncated coherent spaces”.


(Co-)Limits indexed by homotopy types with finite homotopy groups

see at K(n)-local stable homotopy theory (…)

Relation to coherent objects


∞Grpd is a coherent (∞,1)-topos and a locally coherent (∞,1)-topos. An object XX, hence an ∞-groupoid, is an n-coherent object precisely if all its homotopy groups in degree knk \leq n are finite. Hence the fully coherent objects here are the homotopy types with finite homotopy groups.

(Lurie SpecSchm, example 3.13)


The (,1)(\infty, 1)-category of homotopy types with finite homotopy groups is the initial (∞,1)-pretopos.

(Anel 21, Theorem 2.6.6).

Codensity monad of the inclusion into all homotopy types

The inclusion of homotopy types with finite groups into all homotopy types generates a codensity monad whose algebras lie somewhere between totally disconnected compact Hausdorff condensed \infty-groupoids, and all compact Hausdorff condensed \infty-groupoids. (See Scholze for more details.)


  • G. J. Ellis, Spaces with finitely many non-trivial homotopy groups all of which are finite, Topology, 36, (1997), 501–504, ISSN 0040-9383.

  • Jean-Louis Loday, Spaces with finitely many non-trivial homotopy groups (pdf)

  • Peter Scholze, Infinity-categorical analogue of compact Hausdorff, MO answer

Discussion as an elementary (∞,1)-topos:

Last revised on July 8, 2021 at 15:53:39. See the history of this page for a list of all contributions to it.