category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, the cocycle category of objects ''X'', ''Y'' in a

model category In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstrac ...

is a category in which the objects are pairs of maps

X \overset\leftarrow Z \overset\rightarrow Y

and the

morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms a ...

s are obvious

commutative diagram 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the s ...

s between them. It is denoted by

H(X, Y)

. (It may also be defined using the language of

2-category In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of catego ...

.) 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

, Y The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline ...

\quad (f, g) \mapsto g \circ f^ is bijective.

References

* {{cite web , first=J.F. , last=Jardine , year=2007 , url=http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf , title=Simplicial presheaves , access-date=2013-10-16 , archive-url=https://web.archive.org/web/20131017085805/http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf , archive-date=2013-10-17 , url-status=dead Algebraic topology