Cocycle category explained

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} → 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], (f, g) \mapsto g \circ f ^{-1} is bijective.

References

• Web site: J.F. . Jardine . 2007 . Simplicial presheaves . 2013-10-16 . https://web.archive.org/web/20131017085805/http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf . 2013-10-17 . dead .

Notes and References

1. Book: Jardine, J. F.. Algebraic Topology Abel Symposia Volume 4. 2009. Springer. Berlin Heidelberg. 978-3-642-01200-6. 185–218. 10.1007/978-3-642-01200-6_8. Cocycle Categories.