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

Proposition

If $C$ is

• a cofibrantly generated model category

• or a proper model category

• or a cellular model category

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

$$(C/X{)}^{\circ }\simeq C^{\circ }/X.$$(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

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

$$\begin{array}{ccccc}\varnothing & & \hookrightarrow & & A\\ & \searrow & & {\swarrow }_a\\ & & X\end{array}$$\array{ \emptyset &&\hookrightarrow&& A \\ & \searrow && \swarrow_{\mathrlap{a}} \\ && X }

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

$$\begin{array}{ccccc}B& & \stackrel{b}{\to }& & X\\ & {}_b\searrow & & {\swarrow }_{\mathrm{Id}}\\ & & X\end{array}$$\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

$$\begin{array}{ccc}C/X(a,b)& \to & C(A,B)\\ \downarrow & & {\downarrow }^{b_{*}}\\ *& \stackrel{a}{\to }& C(A,X)\end{array}$$\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${}_{\mathrm{Quillen}}$ 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 $C/X(a,b)\simeq (C/X{)}^{\circ }(a,b)\in \mathrm{sSet}$ is indeed a model for $C^{\circ }/X(a,b)\in \mathrm{\infty }\mathrm{Grpd}$.

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

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