Contents

model category

for ∞-groupoids

# Contents

## 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 $X_1 \leftarrow X_0 \to X_1$ 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).

###### Proposition

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

1. $X$ is degreewise a coproducts of representables

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

$X_n = NonDegenerate \coprod Degenerate \,.$
###### 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.

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.