Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the ∞-groupoids are spaces. One version of the hypothesis was claimed to be...

{Z} ,+)} , the additive group of integers. The homotopy hypothesis, a well-known conjecture in homotopy theory formulated by Alexander Grothendieck, states...

In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic...

homomorphism must map 0 to 0. Mathematics portal History of topos theory Homotopy hypothesis Intuitionistic type theory ∞-topos Quasitopos Geometric logic Generalized...

theorem. homotopy hypothesis The homotopy hypothesis states an ∞-groupoid is a space (less equivocally, an n-groupoid can be used as a homotopy n-type.)...

nt-status-of-grothendiecks-homotopy-hypothesis-and-whiteheads-algebraic-h https://ncatlab.org/nlab/show/algebraic+homotopy Handbook of Algebraic Topology...

morphism is an isomorphism. The homotopy hypothesis states that ∞-groupoids are equivalent to spaces up to homotopy.: 2–3  Alexander Grothendieck suggested...

In mathematical logic and computer science, homotopy type theory (HoTT) refers to various lines of development of intuitionistic type theory, based on...

category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as the...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

Introduction to Category Theory, ISBN 978-0521283045. Simpson, Carlos (2010). Homotopy theory of higher categories. arXiv:1001.4071. Bibcode:2010arXiv1001.4071S...

purposes of homotopy theory. Specifically, the category of simplicial sets carries a natural model structure, and the corresponding homotopy category is...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

fundamental group based at x0, denoted π1(X, x0). This is the group of homotopy classes of loops based at x0, with the group operation of concatenation...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

the fundamental groupoid of X): two loops (under equivalence relation of homotopy) may not have the same base point so they cannot multiply with each other...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

and cofibrations are examples of dual notions in algebraic topology and homotopy theory. In this context, the duality is often called Eckmann–Hilton duality...

instance of Grothendieck's homotopy hypothesis which says ∞-groupoids are spaces (or that they can model spaces from the homotopy theory point of view). The...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

{\displaystyle M[W^{-1}]\simeq {\text{Hot}}} where Hot denotes the homotopy category. Homotopy hypothesis ∞-groupoid Derivator N-group (category theory) Higher stack...

(for example, for a correct formulation of the Yoneda lemma). The homotopy hypothesis says that one can take an ∞-groupoid, concretely a Kan complex, as...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...

mathematics paper by Alexander Grothendieck K-theory Hilbert scheme Homotopy hypothesis Infinitesimal cohomology – cohomology theory for algebraic varieties...

concepts Categorification Enriched category Higher-dimensional algebra Homotopy hypothesis Model category Simplex category String diagram Topos n-categories...