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...

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

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

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

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

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...

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

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...

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...

directed analogues of homotopy equivalence. For example, homotopy groups and fundamental n-groupoids of spaces generalize to homotopy monoids and fundamental...

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

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

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

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...

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

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

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

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...

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

associative and unital up to coherent equivalence or coherent isomorphism. A weak 0-category is just a set, and a weak 1-category is a ordinarily category...

particular homotopy theory, a continuous mapping between topological spaces i : A → X {\displaystyle i:A\to X} , is a cofibration if it has the homotopy extension...

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, more specifically in homotopy theory, a simplicial presheaf is a presheaf on a site (e.g., the category of topological spaces) taking...

that ∞-groupoids are equivalent to topological spaces modulo weak homotopy equivalence. Category of groups – Category whose objects are groups and whose...

representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed...

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