In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized...

weak equivalence may refer to: Weak equivalence of categories Weak equivalence (homotopy theory) Weak equivalence (formal languages) Weak equivalence...

In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the ∞-groupoids are spaces. One...

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

Kan complexes and it furthermore models the homotopy theory of CW complexes up to weak homotopy equivalence, with the correspondence between simplicial...

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

Quasi-equivalence could refer to Weak equivalence (homotopy theory) Quasi-isometry Quasi-isomorphism The Caspar-Klug theory of viral capsids. This disambiguation...

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...

called simply the "homotopy category". One first defines a weak homotopy equivalence: a continuous map is called a weak homotopy equivalence if it induces...

admits a weak homotopy equivalence from a CW complex, this axiom reduces homology or cohomology theories on all spaces to the corresponding theory on CW...

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

F {\displaystyle F} and contractible total space, there is a weak homotopy equivalence F → Ω B . {\displaystyle F\to \Omega B.} : 408  For a fibration...

Harold (2011), An Introduction to Category Theory, ISBN 978-0521283045. Simpson, Carlos (2010). Homotopy theory of higher categories. arXiv:1001.4071....

}\to {\underline {\operatorname {Hom} }}(X,V)^{\simeq }} is a weak homotopy equivalence for each ∞-category V, where the superscript ≃ {\displaystyle...

n-th homotopy group of X with base point x. (For n = 0, π0(X) just means the set of path components of X.) A map f is a weak homotopy equivalence if the...

in the homotopy category of diagrams F ∈ Ho ( Top I ) {\displaystyle F\in {\text{Ho}}({\textbf {Top}}^{I})} , (where the homotopy equivalence of diagrams...

weak equivalences, which fulfill the properties of a model structure. Its fibrant objects are all ∞-categories and it furthermore models the homotopy...

Abstract homotopy theory and motivic homotopy theory are also outside the scope. Glossary of category theory covers (or will cover) concepts in theory of model...

homotopy equivalence. However these two spaces are not homotopy equivalent. So by the Whitehead theorem, the Warsaw circle does not have the homotopy...

topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological...

defined to be a weak equivalence if its geometric realization is a weak homotopy equivalence of spaces. A map of simplicial sets is defined to be a cofibration...

In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber) is part of a construction that associates a fibration...

simple homotopy equivalence is a finer invariant than homotopy equivalence by introducing an invariant called the torsion. The torsion of a homotopy equivalence...

was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. CW complexes have better categorical properties than simplicial complexes...

mathematics, A1 homotopy theory or motivic homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties...

{C}}[N^{hc}S(C)]}(x,y)|\to \operatorname {Map} _{C}(x,y)} is a weak homotopy equivalence for each pair of objects x , y {\displaystyle x,y} in C {\displaystyle...

Since simplicial sets have a good homotopy theory, one can ask questions about the meaning of the various homotopy groups πn(N(C)). One hopes that the...

captures all information about a topological space up to weak homotopy equivalence. Homotopy category of an ∞-category core of a category (while the fundamental...

In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built...

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