About: Homotopy colimit and limit

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In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category . The main idea is this: if we have a diagram considered as an object in the , (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone

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rdfs:label	Homotopy colimit and limit (en)
rdfs:comment	In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category . The main idea is this: if we have a diagram considered as an object in the , (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone (en)
foaf:depiction
dct:subject	Homotopy theory Category theory Homotopical algebra
Wikipage page ID	40774899 (xsd:integer)
Wikipage revision ID	1104064041 (xsd:integer)
Link from a Wikipage to another Wikipage	Algebraic topology Homeomorphic Homotopy equivalence Derivator Derived category Mathematics Nerve (category theory) Contractible space Homotopy cofiber Opposite category Limit (category theory) Functor Mapping cylinder Join (topology) Cylinder (geometry) Homotopy theory Category theory Homotopical algebra Cofibration Homotopical algebra Homotopy fiber Category (mathematics) Completion (ring theory) Pushout (category theory) Mapping telescope Weak equivalence (homotopy theory) Simplicial set Right adjoint Slice category Mapping path space Model categories dbr:Arxiv:1810.06632 dbr:Cohomology_of_categories dbr:Completion_of_a_spectrum dbr:Homotopy_category_of_diagrams dbr:Spectral_sequence_of_homotopy_colimits
Link from a Wikipage to an external page	https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/abs/homotopy-colimits-in-the-category-of-small-categories/32ACB453B263410CE1643E682D7DE335 https://pages.uoregon.edu/ddugger/hocolim.pdf https://www.math.ucla.edu/~mikehill/Research/surv-douglas2-201.pdf https://mathoverflow.net/q/135462
sameAs	Homotopy colimit and limit Homotopy colimit and limit
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thumbnail	wiki-commons:Special:FilePath/Homotopy_colimit_A-X-Y.png?width=300
has abstract	In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category . The main idea is this: if we have a diagram considered as an object in the , (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone which are objects in the homotopy category , where is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category since the latter homotopy functor category has functors which picks out an object in and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, which give techniques for constructing homotopy limits and colimits in terms of other homotopy categories, such as derived categories. Another perspective formalizing these kinds of constructions are derivatorspg 193 which are a new framework for homotopical algebra. (en)
prov:wasDerivedFrom	wikipedia-en:Homotopy_colimit_and_limit?oldid=1104064041&ns=0
page length (characters) of wiki page	12946 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Homotopy_colimit_and_limit
is Link from a Wikipage to another Wikipage of	Homotopy colimit Hocolimit Homotopy limit Homotopy limits Homotopy pullback Homotopy pushout
is Wikipage redirect of	Homotopy colimit Hocolimit Homotopy limit Homotopy limits Homotopy pullback Homotopy pushout
is foaf:primaryTopic of	wikipedia-en:Homotopy_colimit_and_limit

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