Cocycle category: Difference between revisions

Content deleted Content added

Inline

Revision as of 21:26, 18 December 2018

In category theory, a branch of mathematics, the cocycle category of objects X, Y in a model category is a category in which the objects are pairs of maps $X{\overset {f}{\leftarrow }}Z{\overset {g}{\rightarrow }}Y$ and the morphisms are obvious commutative diagrams between them.^[1] It is denoted by $H(X,Y)$ . (It may also be defined using the language of 2-category.)

One has: if the model category is right proper and is such that weak equivalences are closed under finite products,

\pi _{0}H(X,Y)\to [X,Y],\quad (f,g)\mapsto g\circ f^{-1}

is bijective.

References

^ Jardine, J. F. (2009). Algebraic Topology Abel Symposia Volume 4. Berlin Heidelberg: Springer. pp. 185–218. ISBN 978-3-642-01200-6.

Jardine, J.F. (2007). "Simplicial presheaves" (PDF).

[Jardine2009-1] Jardine, J. F. (2009). Algebraic Topology Abel Symposia Volume 4. Berlin Heidelberg: Springer. pp. 185–218. ISBN 978-3-642-01200-6.

[1]

Revision as of 11:14, 28 August 2017 edit KolbertBot (talk \| contribs) Bots 1,166,042 edits m Bot: HTTP→HTTPS ← Previous edit		Revision as of 21:26, 18 December 2018 edit undo Bradleyagin (talk \| contribs) 251 edits No edit summary Next edit →
Line 1:		Line 1:
	In [[category theory]], a branch of [[mathematics]], the '''~~cocyle~~ category''' of objects ''X'', ''Y'' in a [[model category]] is a [[category (mathematics)\|category]] in which the objects are pairs of maps <math>X \overset{f}\leftarrow Z \overset{g}\rightarrow Y</math> and the [[morphism]]s are obvious [[commutative diagram]]s between them.<ref name=Jardine2009>{{cite book\|last=Jardine\|first=J. F.\|title=Algebraic Topology Abel Symposia Volume 4\|year=2009\|publisher=Springer\|location=Berlin Heidelberg\|isbn=978-3-642-01200-6\|pages=185–218\|url=https://link.springer.com/chapter/10.1007/978-3-642-01200-6_8}}</ref> It is denoted by <math>H(X, Y)</math>. (It may also be defined using the language of [[2-category]].)		In [[category theory]], a branch of [[mathematics]], the '''cocycle category''' of objects ''X'', ''Y'' in a [[model category]] is a [[category (mathematics)\|category]] in which the objects are pairs of maps <math>X \overset{f}\leftarrow Z \overset{g}\rightarrow Y</math> and the [[morphism]]s are obvious [[commutative diagram]]s between them.<ref name=Jardine2009>{{cite book\|last=Jardine\|first=J. F.\|title=Algebraic Topology Abel Symposia Volume 4\|year=2009\|publisher=Springer\|location=Berlin Heidelberg\|isbn=978-3-642-01200-6\|pages=185–218\|url=https://link.springer.com/chapter/10.1007/978-3-642-01200-6_8}}</ref> It is denoted by <math>H(X, Y)</math>. (It may also be defined using the language of [[2-category]].)

	One has: if the model category is right proper and is such that [[equivalence of categories\|weak equivalences]] are closed under finite products,		One has: if the model category is right proper and is such that [[equivalence of categories\|weak equivalences]] are closed under finite products,