nLab projectively cofibrant diagram



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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}.


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}.


We unwind the condition in def. .

First of all it says of course that a diagram X:𝒟𝒞X \colon \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 in turn 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.


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) \,.


For specific diagram shapes


A span diagram X 1X 0X 2X_1 \leftarrow X_0 \to X_2 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.


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.


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, see also Richard Garner’s answer on MathOverflow).


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

  1. XX is degreewise a coproduct of retracts 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. Each simplicial level, as a presheaf of sets, can be presented as the coproduct of two presheaves of sets, one of which is the presheaf of degenerate simplices:

    X n=NonDegenerate nDegenerate n. X_n = NonDegenerate_n \coprod Degenerate_n \,.

A split hypercover is of this form.


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.


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.

Last revised on April 4, 2020 at 23:20:02. See the history of this page for a list of all contributions to it.