Homotopy hypothesis

In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states that the ∞-groupoids are spaces. If we model our ∞-groupoids as Kan complexes, then the homotopy types of the geometric realizations of these sets give models for every homotopy type. It is conjectured that there are many different "equivalent" models for ∞-groupoids all which can be realized as homotopy types.

See also

• Pursuing Stacks
• N-group (category theory)
• Quasi-category

References

• John Baez, The Homotopy Hypothesis
• Grothendieck, Alexander (2021). "Pursuing Stacks". arXiv:2111.01000 [math.CT].
• Lurie, Jacob (2009). Higher Topos Theory (AM-170). Princeton University Press. ISBN 9780691140490. JSTOR j.ctt7s47v.

External links

• homotopy hypothesis at the nLab
• What is the mistake in the proof of the Homotopy hypothesis by Kapranov and Voevodsky?
• Jacob Lurie's Home Page

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Category theory

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Key concepts Category Abelian Additive Concrete Pre-abelian Preadditive Bicategory Adjoint functors CCC Commutative diagram End Exponential Functor Kan extension Morphism Natural transformation Universal property Universal constructions Limits Terminal objects Products Equalizers Kernels Pullbacks Inverse limit Colimits Initial objects Coproducts Coequalizers Cokernels and quotients Pushout Direct limit Algebraic categories Sets Relations Magmas Groups Abelian groups Rings (Fields) Modules (Vector spaces) Constructions on categories Free category Functor category Kleisli category Opposite category Quotient category Product category Comma category Subcategory

Higher category theory

Key concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories Weak n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified concepts 2-group 2-ring En-ring (Traced)(Symmetric) monoidal category n-group n-monoid

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