Cocycle category

Short description: Category-theoretic construction

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 $X\stackrel{f}{\leftarrow }Z\stackrel{g}{\to }Y$ and the morphisms are obvious commutative diagrams between them. 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,

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

is bijective.

References

• Jardine, J.F. (2007). "Simplicial presheaves". http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf.