model category

for ∞-groupoids

# Contents

## Definition

###### Proposition

For $C$ a model category and $X \in C$ an object, the over category $C/X$ as well as the undercategory $X/C$ inherit themselves structures of model categories whose fibrations, cofibrations and weak equivalences are precisely the morphism that become fibrations, cofibrations and weak equivalences in $C$ under the forgetful functor $C/X \to C$ or $X/C \to C$.

## Properties

###### Proposition

If $C$ is

then so are $C/X$ and $X/C$.

The proofs are in (OvMod).

###### Proposition

If $C$ is a combinatorial model category, then so is $C/X$.

###### Proof

By basic properties of locally presentable categories they are stable under slicing. Hence with $C$ locally presentable also $C/X$ is, and by prop. 2 with $C$ cofibrantly generated also $C/X$ is.

###### Proposition

If $C$ is a simplicial model category and $X \in C$ is fibrant, then the overcategory $C/X$ with the above slice model structure is a presentation of the over-(∞,1)-category $C^\circ / X$: we have an equivalence of (∞,1)-categories

$(C/X)^\circ \simeq C^\circ / X \,.$
###### Proof

It is clear that we have an essentially surjective (∞,1)-functor $C^\circ/X \to (C/X)^\circ$. What has to be shown is that this is a full and faithful (∞,1)-functor in that it is an equivalence on all hom-∞-groupoids $C^\circ/X(a,b) \simeq (C/X)^\circ(a,b)$.

To see this, notice that the hom-space in an over-(∞,1)-category $C^\circ/X$ between objects $a : A \to X$ and $b : B \to X$ is given (as discussed there) by the (∞,1)-pullback

$\array{ C^\circ/X(A \stackrel{a}{\to} X, B \stackrel{b}{\to} X) &\to& C^\circ(A,B) \\ \downarrow && \downarrow^{\mathrlap{b_*}} \\ {*} &\stackrel{a}{\to}& C^\circ(A,X) }$

in ∞Grpd.

Let $A \in C$ be a cofibrant representative and $b : B \to X$ be a fibration representative in $C$ (which always exists) of the objects of these names in $C^\circ$, respectively. In terms of these we have a cofibration

$\array{ \emptyset &&\hookrightarrow&& A \\ & \searrow && \swarrow_{\mathrlap{a}} \\ && X }$

in $C/X$, exhibiting $a$ as a cofibrant object of $C/X$; and a fibration

$\array{ B &&\stackrel{b}{\to}&& X \\ & {}_{\mathllap{b}}\searrow && \swarrow_{\mathrlap{Id}} \\ && X }$

in $C/X$, exhibiting $b$ as a fibrant object in $C/X$.

Moreover, the diagram in sSet given by

$\array{ C/X(a, b) &\to& C(A,B) \\ \downarrow && \downarrow^{\mathrlap{b_*}} \\ {*} &\stackrel{a}{\to}& C(A,X) }$

is

1. a pullback diagram in sSet (by the definition of morphism in an ordinary overcategory);

2. a homotopy pullback in the model structure on simplicial sets, because by the axioms on the sSet${}_{Quillen}$ enriched model category $C$ and the above (co)fibrancy assumptions, all objects are Kan complexes and the right vertical morphism is a Kan fibration.

3. has in the top left the correct derived hom-space in $C/X$ (since $a$ is cofibrant and $b$ fibrant).

This means that this correct hom-space $C/X(a,b) \simeq (C/X)^\circ(a,b) \in sSet$ is indeed a model for $C^\circ/X(a,b) \in \infty Grpd$.

## References

• Hirschhorn, Overcategories and undercategories of model categories (pdf)

Revised on February 6, 2013 18:05:27 by Urs Schreiber (82.113.106.234)