Cocycle category
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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 [math]\displaystyle{ X \overset{f}\leftarrow Z \overset{g}\rightarrow Y }[/math] and the morphisms are obvious commutative diagrams between them.[1] It is denoted by [math]\displaystyle{ H(X, Y) }[/math]. (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,
- [math]\displaystyle{ \pi_0 H(X, Y) \to [X, Y], \quad (f, g) \mapsto g \circ f^{-1} }[/math]
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". http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf.
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