Proper model structure: Difference between revisions

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{{Short description|Special kind of model structure}}

In [[higher category theory]] in [[mathematics]], a '''proper model structure''' is a [[model structure]] in which additionally weak equivalences are preserved under [[Pullback (category theory)|pullback]] (fiber product) along fibrations, called ''right proper'', and [[Pushout (category theory)|pushouts]] (cofiber product) along cofibrations, called ''left proper''. It is helpful to construct weak equivalences and hence to find isomorphic objects in the [[homotopy theory]] of the [[Model category|model structure]].

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== Literature ==

* {{cite journal |last=Rezk |first=Charles |author-link=Charles Rezk |year=2000 |title=Every homotopy theory of simplicial algebras admits a proper model |url= |journal=[[Topology and Its Applications]] |series= |volume=119 |issue= |pages=65-9465–94 |arxivdoi=math10.1016/0003065S0166-8641(01)00057-8 |doiarxiv=10.48550/arXiv.math/0003065 }}

* {{cite book |last=Hirschhorn |first=Philip |url=https://webhomes.maths.ed.ac.uk/~v1ranick/papers/hirschhornloc.pdf |title=Model Categories and Their Localizations |date=2002 |publisher=[[Mathematical Surveys and Monographs]] |isbn=978-0-8218-4917-0 |___location= |language=en |authorlink=}}

* {{cite web |last=Joyal |first=André |author-link=André Joyal |date=2008 |title=The Theory of Quasi-Categories and its Applications |url=https://ncatlab.org/nlab/files/JoyalTheoryOfQuasiCategories.pdf |language=en}}