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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=http://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 [[weak equivalence]]s are closed under [[finite]] products,
:<math>\pi_0 H(X, Y) \to [X, Y], \quad (f, g) \mapsto g \circ f^{-1}</math>
is [[bijective]].

== References ==
{{reflist}}
* {{cite web | first=J.F. | last=Jardine | year=2007 | url=http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf | title=Simplicial presheaves }}

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[[Category:Algebraic topology]]