Cocycle category

In category theory, a branch of mathematics, the cocyle 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

1. 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).