nLab
model structure on an over category

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

Cofibrant generation, properness, combinatoriality

Proposition

If $\mathcal{C}$ is

• a cofibrantly generated model category

• or a proper model category

• or a cellular model category

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

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

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

This is spelled out in (Hirschhorn 05).

Proposition

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

Proof

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

Proposition

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

Proof

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

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 $\mathcal{C}$, let

(1)$$\gamma \colon \mathcal{C} \longrightarrow L_W \mathcal{C}$$

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

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

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

$$L_W(\mathcal{C}/X) \simeq (L_W\mathcal{C}) / \gamma(X) \,.$$

Proof

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

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

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

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

in ∞Grpd.

Let $A \in C$ be a cofibrant representative and $b \colon B \to X$ 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 $\mathcal{C}/X$, exhibiting $a$ as a cofibrant object of $\mathcal{C}/X$; and a fibration

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

in $\mathcal{C}/X$, exhibiting $b$ as a fibrant object in $\mathcal{C}/X$.

Moreover, the diagram in sSet given by

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

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${}_{Quillen}$ 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 $\mathcal{C}/X(a,b) \simeq (\mathcal{C}/X)^\circ(a,b) \in sSet$ is indeed a model for $\mathcal{C}^\circ/X(a,b) \in \infty Grpd$.

Quillen adjunctions between slice categories

Proposition

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

$$L:A/X\leftrightarrows B/L X:R$$

is an adjunction, where is the composition $R\colon B/L X\to A/R L X\to A/X$, the second arrow is the base change functor along the unit $X\to R L X$. If $Y\in B$, then

$$L:A/R Y\leftrightarrows B/Y:R$$

is an adjunction, where $L\colon A/R Y\to B/L R Y\to B/Y$. The first adjunction is a Quillen equivalence if $X$ is cofibrant and $L X$ is fibrant. The second adjunction is a Quillen equivalence if $Y$ 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 $A$ or $B$, see Hirschhorn’s note (Hirschhorn 05). An object in $A/X$ given by an arrow $Z\to X$ is cofibrant if and only if $Z$ is cofibrant and fibrant if and only if $Z\to X$ is a fibration. Quillen’s criterion for Quillen equivalences now yields the statements about equivalences.

Examples

• the classical model structure on pointed topological spaces is the model structure on the undercategory under the point of the classical model structure on topological spaces.

Related concepts

• over-category

  • slice category

  • under category

  • over topos

• over (∞,1)-category,

  • model structure on an over-category

  • over-(∞,1)-topos

• opposite model structure

References

• Philip Hirschhorn, Theorem 7.6.5 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)

• Philip Hirschhorn, Overcategories and undercategories of model categories, 2005 (pdf, arXiv:1507.01624)

• Denis-Charles Cisinski, Higher category theory and homotopical algebra, 2020 (pdf)