on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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 $X/C \to C$.
If $\mathcal{C}$ is
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
This is spelled out in (Hirschhorn 05).
If $\mathcal{C}$ is a combinatorial model category, then so is $\mathcal{C}_{/X}$.
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.
If $\mathcal{C}$ is an cartesian enriched model category?, then so is $\mathcal{C}_{/X}$.
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 $\mathcal{C}$.
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
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:
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
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):
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
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
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
in $\mathcal{C}/X$, exhibiting $a$ as a cofibrant object of $\mathcal{C}/X$; and a fibration
in $\mathcal{C}/X$, exhibiting $b$ as a fibrant object in $\mathcal{C}/X$.
Moreover, the diagram in sSet given by
is
a pullback diagram in sSet (by the definition of morphism in an ordinary overcategory);
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;
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 $\mathcal{C}^\circ/X(a,b) \in \infty Grpd$.
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
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
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.
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.
model structure on an over-category
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)
Last revised on April 19, 2021 at 03:00:45. See the history of this page for a list of all contributions to it.