nLab
model structure on an over category

Contents

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 rational \infty-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

Cofibrant generation, properness, combinatoriality

Proposition

If 𝒞\mathcal{C} is

then so are 𝒞 /X\mathcal{C}_{/X} and 𝒞 X/\mathcal{C}^{X/}.

More in detail, if I,JMor(𝒞)I,J \subset Mor(\mathcal{C}) are the classes of generating cofibrations and of generating acylic cofibrations of 𝒞\mathcal{C}, respectively, then

  • the generating (acyclic) cofibrations of 𝒞 X/\mathcal{C}^{X/} are the image under X()X \sqcup(-) of those of 𝒞\mathcal{C}.

This is spelled out in (Hirschhorn 05).

Proposition

If 𝒞\mathcal{C} is a combinatorial model category, then so is 𝒞 /X\mathcal{C}_{/X}.

Proof

By basic properties of locally presentable categories they are stable under slicing. Hence with 𝒞\mathcal{C} locally presentable also 𝒞 /X\mathcal{C}_{/X} is, and by prop. with 𝒞\mathcal{C} cofibrantly generated also 𝒞 /X\mathcal{C}_{/X} is.

Proposition

If 𝒞\mathcal{C} is an cartesian enriched model category?, then so is 𝒞 /X\mathcal{C}_{/X}.

Proof

By basic properties of cartesian enriched categories? they are stable under slicing, where tensoring is computed in 𝒞\mathcal{C}. Hence with 𝒞\mathcal{C} enriched also 𝒞 /X\mathcal{C}_{/X} is. The pushout product axiom now follows from the fact that in overcategories pushouts can be computed in the underlying category 𝒞\mathcal{C}. The unit axiom? follows from the unit axiom of 𝒞\mathcal{C} using the fact that tensorings are computed in 𝒞\mathcal{C}.

As presentations for over (∞,1)-categories

When restricted to fibrant objects, the operation of forming the model structure on an overcategory presents the operation of forming the over (∞,1)-category of an (∞,1)-category.

More explicitly, for any model category 𝒞\mathcal{C}, let

(1)γ:𝒞L W𝒞 \gamma \colon \mathcal{C} \longrightarrow L_W \mathcal{C}

denote the localization of an (infinity,1)-category|localization]] (as an (∞,1)-category) inverting the weak equivalences (as e.g. given by simplicial localization; see also the model structure on relative categories). Then:

Proposition

If 𝒞\mathcal{C} is a model category and X𝒞X \in \mathcal{C} is fibrant, then γ\gamma (1) induces an (∞,1)-functor 𝒞/XL W(𝒞)/γ(X)\mathcal{C}/X \to L_W(\mathcal{C})/\gamma(X), which in turn induces an equivalence of (∞,1)-categories

L W(𝒞/X)L W(𝒞)/γ(X). L_W(\mathcal{C}/X) \overset{\simeq}{\longrightarrow} L_W(\mathcal{C})/\gamma(X) \,.

This main result is corollary 7.6.13 of Cisinski 20. Model categories are (∞,1)-categories with weak equivalences and fibrations as defined in Cisinski Def. 7.4.12.

We spell out a proof for the special case that 𝒞\mathcal{C} carries the extra structure of a simplicial model category (this proof was written in 2011 when no comparable statement seemed to be available in the literature):

Proposition

If 𝒞\mathcal{C} is a simplicial model category and X𝒞X \in \mathcal{C} is fibrant, then the overcategory 𝒞/X\mathcal{C}/X with the above slice model structure is a presentation of the over-(∞,1)-category L W𝒞/γ(X)L_W \mathcal{C} / \gamma(X): we have an equivalence of (∞,1)-categories

L W(𝒞/X)(L W𝒞)/γ(X). L_W(\mathcal{C}/X) \simeq (L_W\mathcal{C}) / \gamma(X) \,.
Proof

We write equivalently () L W()(-)^\circ \coloneqq L_W(-).

It is clear that we have an essentially surjective (∞,1)-functor 𝒞 /X(𝒞/X) \mathcal{C}^\circ/X \to (\mathcal{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)\mathcal{C}^\circ/X(a,b) \simeq (\mathcal{C}/X)^\circ(a,b).

To see this, notice that the hom-space in an over-(∞,1)-category 𝒞 /X\mathcal{C}^\circ/X between objects a:AXa \colon A \to X and b:BXb \colon 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{ \mathcal{C}^\circ/X(A \stackrel{a}{\to} X, B \stackrel{b}{\to} X) &\to& \mathcal{C}^\circ(A,B) \\ \big\downarrow && \big\downarrow^{\mathrlap{b_*}} \\ {*} &\stackrel{a}{\to}& \mathcal{C}^\circ(A,X) }

in ∞Grpd.

Let ACA \in C be a cofibrant representative and b:BXb \colon 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/X\mathcal{C}/X, exhibiting aa as a cofibrant object of 𝒞/X\mathcal{C}/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/X\mathcal{C}/X, exhibiting bb as a fibrant object in 𝒞/X\mathcal{C}/X.

Moreover, the diagram in sSet given by

𝒞/X(a,b) 𝒞(A,B) b * * a 𝒞(A,X) \array{ \mathcal{C}/X(a, b) &\longrightarrow& \mathcal{C}(A,B) \\ \big\downarrow && \big\downarrow^{\mathrlap{b_*}} \\ {*} &\stackrel{a}{\to}& \mathcal{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 pullback power axiom 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)sSet\mathcal{C}/X(a,b) \simeq (\mathcal{C}/X)^\circ(a,b) \in sSet is indeed a model for 𝒞 /X(a,b)Grpd\mathcal{C}^\circ/X(a,b) \in \infty Grpd.

Quillen adjunctions between slice categories

Proposition

Given an adjunction LRL\dashv R with L:ABL\colon A\to B and R:BAR\colon B\to A, the following compositions define two Quillen ajdunctions between associated slice categories. If XAX\in A, then

L:A/XB/LX:RL:A/X\leftrightarrows B/L X:R

is an adjunction, where is the composition R:B/LXA/RLXA/XR\colon B/L X\to A/R L X\to A/X, the second arrow is the base change functor along the unit XRLXX\to R L X. If YBY\in B, then

L:A/RYB/Y:RL:A/R Y\leftrightarrows B/Y:R

is an adjunction, where L:A/RYB/LRYB/YL\colon A/R Y\to B/L R Y\to B/Y. The first adjunction is a Quillen equivalence if XX is cofibrant and LXL X is fibrant. The second adjunction is a Quillen equivalence if YY is fibrant.

Proof

These adjunctions are Quillen adjunctions because their left (respectively right) adjoints are left (respectively right) Quillen functors: in the model structures on slice categories (co)fibrations and weak equivalences are created by the forgetful functor to AA or BB, see Hirschhorn’s note (Hirschhorn 05). An object in A/XA/X given by an arrow ZXZ\to X is cofibrant if and only if ZZ is cofibrant and fibrant if and only if ZXZ\to X is a fibration. Quillen’s criterion for Quillen equivalences now yields the statements about equivalences.

Examples

References

Last revised on April 19, 2021 at 03:00:45. See the history of this page for a list of all contributions to it.