nLab homotopy type with finite homotopy groups

Contents

Context

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

Introductions

Definitions

Paths and cylinders

Homotopy groups

Basic facts

Theorems

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

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
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
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
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

semantics

Contents

Idea

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”.

Properties

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

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

Relation to coherent objects

Proposition

∞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)

Proposition

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.)

References

  • 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 19:53:39. See the history of this page for a list of all contributions to it.