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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle X \overset{f}\leftarrow Z \overset{g}\rightarrow Y} and the morphisms are obvious commutative diagrams between them.^[1] It is denoted by ${\displaystyle H(X,Y)}$. (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,

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

is bijective.