Proper 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 (fiber product) along fibrations, called right proper, and 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 structure.

Definition

For every model category, one has:Hirschhorn 2002, Proposition 13.1.2

Pushouts of weak equivalences between cofibrant objects along cofibrations are again weak equivalences.
Pullbacks of weak equivalences between fibrant objects along fibrations are again weak equivalences.

A model category is then called:Rezk 2000, 2.1. Definition of properness

left proper, if pushouts of weak equivalences along cofibrations are again weak equivalences.
right proper, if pullbacks of weak equivalences along fibrations are again weak equivalences.
proper, if it is both left proper and right proper.

Properties

A model category, in which all objects are cofibrant, is left proper.Rezk 2000, Remark 2.8.
A model category, in which all objects are fibrant, is right proper.

For a model category $\mathcal{M}$ and a morphism $f\colon X\rightarrow Y$ in it, there is a functor $f^*\colon
Y\backslash\mathcal{M}\rightarrow X\backslash\mathcal{M}$ by precomposition and a functor $f_*\colon
\mathcal{M}/X\rightarrow\mathcal{M}/Y$ by postcomposition. Furthermore, pushout defines a functor $Y+_X-\colon
X\backslash\mathcal{M}\rightarrow Y\backslash\mathcal{M}$ and pullback defines a functor $X\times_Y-\colon
\mathcal{M}/Y\rightarrow\mathcal{M}/X$ . One has:Rezk 2000, Proposition 2.7.

$\mathcal{M}$ is left proper if and only if for every weak equivalence $f\colon X\rightarrow Y$ , the adjunction $f_*\colon$

\mathcal{M}/X\rightarrow\mathcal{M}/Y\colon X\times_Y- forms a Quillen adjunction.

$\mathcal{M}$ is right proper if and only if for every weak equivalence $f\colon X\rightarrow Y$ , the adjunction $Y+_X-\colon$

X\backslash\mathcal{M}\rightarrow Y\backslash\mathcal{M}\colon f^* forms a Quillen adjunction.

Examples

The Joyal model structure is left proper,Lurie 2009, Higher Topos Theory, Proposition A.2.3.2. but not right proper.Lurie 2009, Higher Topos Theory, Remark 1.3.4.3. Left properness follows from all objects being cofibrant.
The Kan–Quillen model structure is proper.Joyal 2008, Theorem 6.1. on p. 293Cisinki 2019, Corollary 3.1.28. Left properness follows from all objects being cofibrant.

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=65–94 |doi=10.1016/S0166-8641(01)00057-8 |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}}
{{cite book |last1=Lurie |first1=Jacob |author-link=Jacob Lurie |title=Higher Topos Theory |title-link=Higher Topos Theory |publisher=Princeton University Press |year=2009 |isbn=978-0-691-14049-0 |series=Annals of Mathematics Studies |volume=170 |mr=2522659 |arxiv=math.CT/0608040}}
{{cite book |last=Cisinski |first=Denis-Charles |author-link=Denis-Charles Cisinski |url=https://cisinski.app.uni-regensburg.de/CatLR.pdf |title=Higher Categories and Homotopical Algebra |date=2019-06-30 |publisher=Cambridge University Press |isbn=978-1108473200 |location= |language=en |authorlink=}}

References

External links

proper model category at the nLab

Category:Higher category theory