nLab projectively cofibrant diagram

Definition

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

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

Definition

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

Remark

We unwind the condition in def. .

First of all it says of course that a diagram $X \colon \mathcal{D}\to \mathcal{C}$ is projectively cofibrant precisely if the inclusion $\emptyset \to X$ of the initial diagram has the left lifting property with respect to natural transformations of diagrams

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

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

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

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

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

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

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

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

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

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

Examples

For specific diagram shapes

Example

A span diagram $X_1 \leftarrow X_0 \to X_2$ is projectively cofibrant precisely if the two morphisms are cofibrations in $\mathcal{D}$ 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

$$X_0 \to X_1 \to X_2 \to \cdots$$

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

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

Example

A parallel morphisms diagram

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

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

Proposition

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

1. $X$ is degreewise a coproduct of retracts of representables

   $$X_n = \coprod_{i} U^n_i \;\;\;\; \{U^n_i \in \mathcal{C} \hookrightarrow [\mathcal{C}, Set]\}$$

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:

   $$X_n = NonDegenerate_n \coprod Degenerate_n \,.$$

Example

A split hypercover is of this form.

Proposition

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

$$(Q X)_n : \coprod_{U_0 \to \cdots \to U_n \to X_n} U_0 \,,$$

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.