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 $X{\overset {f}{\leftarrow }}Z{\overset {g}{\rightarrow }}Y$ 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

Jardine, J. F. (2009). "Cocycle Categories". Algebraic Topology Abel Symposia Volume 4. Berlin Heidelberg: Springer. pp. 185–218. doi:10.1007/978-3-642-01200-6_8. ISBN 978-3-642-01200-6.

Jardine, J.F. (2007). "Simplicial presheaves" (PDF). Archived from the original (PDF) on 2013-10-17. Retrieved 2013-10-16.

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[Jardine2009-1] Jardine, J. F. (2009). "Cocycle Categories". Algebraic Topology Abel Symposia Volume 4. Berlin Heidelberg: Springer. pp. 185–218. doi:10.1007/978-3-642-01200-6_8. ISBN 978-3-642-01200-6.