nLab
projectively cofibrant diagram

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Model category theory

model category

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Contents

Definition

Let 𝒞\mathcal{C} be a (cofibrantly generated) model category and let 𝒟\mathcal{D} be any category, regarded as a diagram-shape in the following.

Write [𝒟,𝒞] proj[\mathcal{D}, \mathcal{C}]_{proj} for the projective model structure on the functor category of functors from 𝒟\mathcal{D} to 𝒞\mathcal{C}, hence of 𝒟\mathcal{D}-diagrams in 𝒞\mathcal{C}.

Definition

A functor/diagram X:𝒟𝒞X : \mathcal{D} \to \mathcal{C} is a projectively cofibrant diagram in 𝒞\mathcal{C} if it is a cofibrant object in the projective model structure [𝒟,𝒞] proj[\mathcal{D}, \mathcal{C}]_{proj}.

This means that a diagram X:𝒟𝒞X :\mathcal{D}\to \mathcal{C} is projectively cofibrant precisely if the inclusion X\emptyset \to X of the initial diagram has the left lifting property with respect to natural transformations of diagrams

(ApB):𝒞𝒟 (A \stackrel{p}{\to} B) : \mathcal{C} \to \mathcal{D}

which are projective acyclic fibrations, hence which are such that for each c𝒞c \in \mathcal{C} the component η c:A(c)B(c)\eta_c : A(c) \to B(c) is an acyclic fibration in 𝒞\mathcal{C}.

This means that FF is projectively cofibrant precisely if for every diagram of natural transformations

A p X B \array{ &&A \\ & &\downarrow^{\mathrlap{p}} \\ X &\to& B }

with pp as above, there exists a lift σ\sigma in

A σ p X B[𝒟,𝒞]. \array{ &&A \\ & {}^{\mathllap{\sigma}}\nearrow &\downarrow^{\mathrlap{p}} \\ X &\to& B } \;\;\;\;\;\; \in [\mathcal{D}, \mathcal{C}] \,.

making the triangle commute.

=–

Properties

The main point of projectively cofibrant diagrams is that the ordinary colimit over them is a presentation of the homotopy colimit:

because the (colimit \dashv constant diagram)-adjunction

(limconst):𝒞constlim[𝒞,𝒟] (\underset{\longrightarrow}{\lim} \dashv const) : \mathcal{C} \stackrel{\overset{\underset{\longrightarrow}{\lim}}{\leftarrow}}{\underset{const}{\to}} [\mathcal{C}, \mathcal{D}]

is a Quillen adjunction (because constconst is by the very definition of the projective model structure a right Quillen functor), the homotopy colimit, being the left derived functor 𝕃lim\mathbb{L}\underset{\longrightarrow}{\lim} of the colimit, is computed as the ordinary colimit evaluated on a cofibrant resolution QXQ X of a diagram X:𝒟𝒞X : \mathcal{D} \to \mathcal{C}:

(𝕃lim)(X)lim)(QX). (\mathbb{L} \underset{\longrightarrow}{\lim})(X) \simeq \underset{\longrightarrow}{\lim})(Q X) \,.

Examples

For specific diagram shapes

Example

A span diagram X 1X 0X 1X_1 \leftarrow X_0 \to X_1 is projectively cofibrant precisely if the two morphisms are cofibrations in 𝒟\mathcal{D} and X 0X_0, hence all three objects, are cofibrant.

The colimit over such a diagram is the homotopy pushout of the span.

Example

A cotower diagram

X 0X 1X 2 X_0 \to X_1 \to X_2 \to \cdots

is projectively cofibrant precisely if every morphism is a cofibration and if the first object X 0X_0, and hence all objects, are cofibrant in 𝒟\mathcal{D}.

The colimit over such a diagram is a homotopy sequential colimit.

Example

A parallel morphisms diagram

X 0gfX 1 X_0 \stackrel{\overset{f}{\to}}{\underset{g}{\to}} X_1

is projectively cofibrant precisely if X 0X_0 is cofibrant, and if the morphism (f,g):X 0X 0X 1(f,g) : X_0 \coprod X_0 \to X_1 is a cofibration.

This implies that also ff and gg are cofibrations and hence that X 1X_1 is cofibrant.

The colimit over such a diagram is a homotopy coequalizer.

For specific ambient model categories

Let 𝒞=\mathcal{C} = sSet Quillen{}_{Quillen} be the standard model structure on simplicial sets. Then [𝒟,𝒞] proj[\mathcal{D}, \mathcal{C}]_{proj} is the projective model structure on simplicial presheaves.

For the following see at model structure on simplicial presheaves the section Cofibrant objects for more details (due to Dan Dugger).

Proposition

A sufficient condition for a diagram X:𝒟sSetX : \mathcal{D} \to sSet to be projectively cofibrant is:

  1. XX is degreewise a coproducts of representables

    X n= iU i n{U i n𝒞[𝒞,Set]} X_n = \coprod_{i} U^n_i \;\;\;\; \{U^n_i \in \mathcal{C} \hookrightarrow [\mathcal{C}, Set]\}
  2. the degenerate cells in each degree form a separate coproduct summand;

    X n=NonDegenerateDegenerate. X_n = NonDegenerate \coprod Degenerate \,.
Example

A split hypercover is of this form.

Proposition

For X:𝒟sSetX : \mathcal{D} \to sSet any simplicial presheaf, a cofibrant resolution is given by

(QX) n: U 0U nX nU 0, (Q X)_n : \coprod_{U_0 \to \cdots \to U_n \to X_n} U_0 \,,

where the coproduct runs over all sequences of morphisms between representables U iU_i, as indicated.

References

See the references at homotopy colimit and generally at model category.

Related discussion is at

Related discussion is for instance also in

where cofibrant cotowers are mentioned as example 2.3.15.

Revised on January 26, 2014 08:08:08 by Anonymous Coward (84.238.145.173)