nLab
model structure on an over category

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Definition

Proposition

For CC a model category and XCX \in C an object, the over category C/XC/X as well as the undercategory X/CX/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 CC under the forgetful functor C/XCC/X \to C or X/CCX/C \to C.

Properties

Proposition

If CC is

then so are C/XC/X and X/CX/C.

The proofs are in (OvMod).

Proposition

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

Proof

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

Proposition

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

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

It is clear that we have an essentially surjective (∞,1)-functor C /X(C/X) 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 /X(a,b)(C/X) (a,b)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 /XC^\circ/X between objects a:AXa : A \to X and b:BXb : B \to X is given (as discussed there) by the (∞,1)-pullback

C /X(AaX,BbX) C (A,B) b * * a C (A,X) \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 ACA \in C be a cofibrant representative and b:BXb : B \to X be a fibration representative in CC (which always exists) of the objects of these names in C C^\circ, respectively. In terms of these we have a cofibration

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

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

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

in C/XC/X, exhibiting bb as a fibrant object in C/XC/X.

Moreover, the diagram in sSet given by

C/X(a,b) C(A,B) b * * a C(A,X) \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{}_{Quillen} enriched model category CC 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/XC/X (since aa is cofibrant and bb fibrant).

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