mathematics, Grothendieck's homotopy hypothesis states (very roughly speaking) that the ∞-groupoids are spaces. One version of the hypothesis was claimed to be...

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...

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

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

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...

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...

In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences'...

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...

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...

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

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...

individual objects, consider homotopy groups of a product space, specifically the fundamental group of the torus. The homotopy groups of a product space...

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

equivalence between the category of sober spaces and spatial locales. The homotopy hypothesis relates Top with ∞Grpd, the category of ∞-groupoids. The conjecture...

rather evident sense. In the category of pointed spaces, fundamental in homotopy theory, the coproduct is the wedge sum (which amounts to joining a collection...

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

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...

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...

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...

sheaves of spaces. Mathematics portal Bousfield localization Homotopy hypothesis – Hypothesis that the ∞-groupoids are equivalent to the topological spaces...

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