Math 2.0

Category theory; homological algebra Monoidal, enriched and higher categories Conjecture

Homotopy hypothesis

Builds on GroupoidFundamental groupoidHomotopy groupCW complexKan complexGeometric realizationSingular simplicial setQuillen equivalenceHigher category theory

Statement

Let nn be a natural number or n=∞n = \infty. A homotopy nn-type is a space XX whose homotopy groups πk(X,x)\pi_k(X, x) vanish for all k>nk > n and all base points xx. For n=∞n = \infty this is no condition. Any good definition of weak nn-groupoids should satisfy the following. Let Πn(X)\Pi_n(X) be the fundamental nn-groupoid of a space XX. Then X↦Πn(X)X \mapsto \Pi_n(X) induces an equivalence between the homotopy theory of homotopy nn-types and that of weak nn-groupoids. For n=∞n = \infty it says that weak ∞\infty-groupoids and spaces up to weak homotopy equivalence are the same homotopy theory.

Remarks

The hypothesis is a test that a definition of higher groupoid must pass, so its status depends on the chosen definition. For n=0n = 0 it says that sets correspond to spaces whose path components are weakly contractible, through X↦π0(X)X \mapsto \pi_0(X). For n=1n = 1 it is classical: the fundamental groupoid and the classifying space give an equivalence between groupoids up to equivalence and homotopy 11-types. If weak ∞\infty-groupoids are modelled by Kan complexes, the hypothesis holds. Quillen showed that geometric realization and the singular simplicial set functor form a Quillen equivalence. It relates simplicial sets with the Kan–Quillen model structure and topological spaces with the Quillen model structure. For other definitions of weak ∞\infty-groupoids, in particular algebraic ones, the question is a subject of research. Grothendieck stated the idea in his manuscript Pursuing Stacks (1983).

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

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CW complex, concept, no evidence levelCW complexconceptFundamental groupoid, concept, no evidence levelFundamental groupoidconceptGeometric realization, concept, no evidence levelGeometric realizationconceptGroupoid, concept, no evidence levelGroupoidconceptHigher category theory, concept, no evidence levelHigher category theoryconceptHomotopy group, concept, no evidence levelHomotopy groupconceptKan complex, concept, no evidence levelKan complexconceptQuillen equivalence, concept, no evidence levelQuillen equivalenceconceptSingular simplicial set, concept, no evidence levelSingular simplicial s…conceptHomotopy hypothesis, conjecture, no evidence levelHomotopy hypothesisconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. CW complex concept
  2. Fundamental groupoid concept
  3. Geometric realization concept
  4. Groupoid concept
  5. Higher category theory concept
  6. Homotopy group concept
  7. Kan complex concept
  8. Quillen equivalence concept
  9. Singular simplicial set concept
  10. Homotopy hypothesis conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Homotopy_hypothesis
About this page
Tag2E83D9ACDY
URIm2://2E83D9ACDY
Kindconjecture
WritingW0 stub
MSC18N65, 55U35, 18N20
StatusThis copy of the site has no build data. Run m2 status 2E83D9ACDY.
AboutGroupoid, Higher category theory
UsesGroupoid, Fundamental groupoid, Homotopy group, CW complex, Kan complex, Geometric realization, Singular simplicial set, Quillen equivalence, Higher category theory
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Changed2 November 2026

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