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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'. These abstract from a conventional homotopy category, of topological spaces or of chain complexes (derived category theory). This concept was introduced by .
In recent decades, the language of model categories has been used in some parts of algebraic ''K''-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.
==Motivation==
Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets.
Another model category is the category of chain complexes of ''R''-modules for a commutative ring ''R''. Homotopy theory in this context is homological algebra. Homology can then be viewed as a type of homotopy, allowing generalizations of homology to other objects, such as groups and ''R''-algebras, one of the first major applications of the theory. Because of the above example regarding homology, the study of closed model categories is sometimes thought of as homotopical algebra.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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