Cocycle category

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 ${\displaystyle X{\overset {f}{\leftarrow }}Z{\overset {g}{\rightarrow }}Y}$ 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-categories.)

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

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

is bijective.