nLab
projectively cofibrant diagram

Definition

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

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

Definition

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

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

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

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

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

$$\begin{array}{ccc}& & A\\ & & {\downarrow }^p\\ X& \to & B\end{array}$$\array{ &&A \\ & &\downarrow^{\mathrlap{p}} \\ X &\to& B }

with $p$ as above, there exists a lift $\sigma$ in

$$\begin{array}{ccc}& & A\\ & {}^{\sigma }\nearrow & {\downarrow }^p\\ X& \to & B\end{array}\in [𝒟,𝒞].$$\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 $⊣$ constant diagram)-adjunction

$$(\underset{⟶}{\lim }⊣\mathrm{const}):𝒞\stackrel{\stackrel{\underset{⟶}{\lim }}{\leftarrow }}{\underset{\mathrm{const}}{\to }}[𝒞,𝒟]$$(\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 $\mathrm{const}$ is by the very definition of the projective model structure a right Quillen functor), the homotopy colimit, being the left derived functor $𝕃\underset{⟶}{\lim }$ of the colimit, is computed as the ordinary colimit evaluated on a cofibrant resolution $QX$ of a diagram $X:𝒟\to 𝒞$:

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

Examples

For specific diagram shapes

Example

A span diagram $X_1\leftarrow X_0\to X_1$ is projectively cofibrant precisely if the two morphisms are cofibrations in $𝒟$ and $X_0$, hence all three objects, are cofibrant.

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

Example

A cotower diagram

$$X_0\to X_1\to X_2\to \cdots$$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 $X_0$, and hence all objects, are cofibrant in $𝒟$.

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

Example

A parallel morphisms diagram

$$X_0\stackrel{\stackrel{f}{\to }}{\underset{g}{\to }}{X}_1$$X_0 \stackrel{\overset{f}{\to}}{\underset{g}{\to}} X_1

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

This implies that also $f$ and $g$ are cofibrations and hence that $X_1$ is cofibrant.

The colimit over such a diagram is a homotopy coequalizer.

For specific ambient model categories

Let $𝒞=$ sSet${}_{\mathrm{Quillen}}$ be the standard model structure on simplicial sets. Then $[𝒟,𝒞{]}_{\mathrm{proj}}$ 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 $X:𝒟\to \mathrm{sSet}$ to be projectively cofibrant is:

1. $X$ is degreewise a coproducts of representables

   $$X_n=\coprod _i{U}_i^n\{U_i^n\in 𝒞\hookrightarrow [𝒞,\mathrm{Set}]\}$$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;

   $$X_n=\mathrm{NonDegenerate}\coprod \mathrm{Degenerate}.$$X_n = NonDegenerate \coprod Degenerate \,.

Example

A split hypercover is of this form.

Proposition

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

$$(QX{)}_n:\coprod _{U_0\to \cdots \to U_n\to X_n}{U}_0,$$(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 $U_i$, as indicated.

References

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

Related discussion is at

• MathOverflow When are “diagrams of cofibrations” projectively cofibrant

Related discussion is for instance also in

• Urs Schreiber, differential cohomology in a cohesive topos

where cofibrant cotowers are mentioned as example 2.3.15.