Injective and projective model structure

In higher category theory in mathematics, injective and projective model structures are special model structures on functor categories into a model category. Both model structures do not have to exist, but there are conditions guaranteeing their existence. An important application is for the study of limits and colimits, which are functors from a functor category and can therefore be made into Quillen adjunctions.

Definition

Let $\mathcal{I}$ be a small category and $\mathcal{C}$ be a model category. For two functors $F,G\colon
\mathcal{I}\rightarrow\mathcal{C}$ , a natural transformation $\eta\colon
F\Rightarrow G$ is composed of morphisms $\eta_X\colon
FX\rightarrow GX$ in $\operatorname{Ar}\mathcal{C}$ for all objects $X$ in $\operatorname{Ob}\mathcal{I}$ . For those it hence be studied if they are fibrations, cofibrations and weak equivalences, which might lead to a model structure on the functor category $\operatorname{Fun}(\mathcal{I},\mathcal{C})$ .

Injective cofibrations and injective weak equivalences are the natural transformations, which componentswise only consist of cofibrations and weak equivalences respectively. Injective fibrations are those natural transformations which have the right lifting property with respect to all injective trivial cofibrations.Lurie 2009, Definition A.3.3.1.
Projective fibrations and projective weak equivalences are the natural transformations, which componentswise only consist of fibrations and weak equivalences respectively. Projective cofibrations are those natural transformations which have the left lifting property with respect to all projective trivial fibrations.Lurie 2009, Definition A.3.3.1.Cisinski 2019, 2.3.10.

For a model structure, the injective trivial cofibrations also have to have the right lifting property with respect to all injective fibrations and the projective trivial fibrations also have to have the left lifting property with respect to all projective cofibrations. Since both doesn't have to be the case, the injective and projective model structure doesn't have to exist.

The functor category $\operatorname{Fun}(\mathcal{I},\mathcal{C})$ with the initial and projective model structure is denoted $\operatorname{Fun}(\mathcal{I},\mathcal{C})_\mathrm{inj}$ and $\operatorname{Fun}(\mathcal{I},\mathcal{C})_\mathrm{proj}$ respectively.

Properties

If $\mathcal{I}$ ist the category assigned to a small well-ordered set with initial element and if $\mathcal{C}$ has all small colimits, then the projective model structure on $\operatorname{Fun}(\mathcal{I},\mathcal{C})$ exists.Cisinki 2019, Proposition 2.3.13.

Quillen adjunctions

Let $\mathcal{C}$ be a combinatorical model category. Let $F\colon
\mathcal{I}\rightarrow\mathcal{J}$ be a functor between small categories, then there is a functor $F^*\colon
\mathbf{Fun}(\mathcal{J},\mathcal{C})\rightarrow\mathbf{Fun}(\mathcal{I},\mathcal{C})$ by precomposition. Since $\mathcal{C}$ has all small limits and small colimits, this functor has a left adjoint $F_!\colon
\mathbf{Fun}(\mathcal{I},\mathcal{C})\rightarrow\mathbf{Fun}(\mathcal{J},\mathcal{C}),
F_!(G)=\operatorname{Lan}_F(G)$ with $F_!\dashv F^*$ known as left Kan extension as well as a right adjoint $F_*\colon
\mathbf{Fun}(\mathcal{I},\mathcal{C})\rightarrow\mathbf{Fun}(\mathcal{J},\mathcal{C}),
F_*(G)=\operatorname{Ran}_F(G)$ with $F^*\dashv F_!$ known as right Kan extension. While the former adjunction is a Quillen adjunction between the projective model structures, the latter is a Quillen adjunctions between the injective model structures.Lurie 2009, Proposition A.2.8.7.

Literature

{{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

model structure on functors at the nLab

Category:Higher category theory

Category:Simplicial sets