nLab slice model structure

Definition

Proposition

(slice model structure)
For $\mathcal{C}$ a model category and $X \in \mathcal{C}$ any object, the slice category $\mathcal{C}_{/X}$ as well as the coslice category $\mathcal{C}^{X/}$ inherit themselves structures of model categories, whose fibrations, cofibrations and weak equivalences are precisely the morphism whose images under the forgetful functors $\mathcal{C}_{/X} \to \mathcal{C}$ or $\mathcal{C}^{X/} \to \mathcal{C}$ are fibrations, cofibrations or weak equivalences, respectively, in $\mathcal{C}$.

(Hirschhorn 2002, Thm. 7.6.5, May & Ponto 2012, Th. 15.3.6)

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 $\mathcal{C}$.

(Hirschhorn (2021); May & Ponto (2012, Th. 15.3.6)).

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 enriched model category over a cartesian closed model category, then so is its enriched slice category $\mathcal{C}_{/X}$.

Proof

By basic properties of enriched categories over cartesian closed categories they are stable under slicing, where tensoring is computed in $\mathcal{C}$ (see at enriched slice category). Hence with $\mathcal{C}$ enriched also $\mathcal{C}_{/X}$ is. The pushout product axiom now follows from the fact that in overcategories pushouts are reflected in the underlying category $\mathcal{C}$ (by this Prop.). 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 (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

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

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

Base change Quillen adjunction

Proposition

(left base change Quillen adjunction)
For $\mathcal{C}$ a model category and $c_1 \xrightarrow{f} c_2$ any morphism in $\mathcal{C}$, the left base change adjunction $(f_! \dashv f^\ast)$ along $f$ (where $f_!$ is postcomposition with and $f^\ast$ is pullback along $f$) is a Quillen adjunction between the slice model structures (from Prop. ):

$$\mathcal{C}_{/c_1} \underoverset {\underset{f^\ast}{\longleftarrow}} {\overset{f_!}{\longrightarrow}} {\;\;\;\;\bot_{\mathrlap{Qu}}\;\;\;\;} \mathcal{C}_{/c_2} \,.$$

Proof

Since the left adjoint $f_!$ is the postcomposition operation, it manifestly preserves the classes of underlying morphisms, hence in particular preserves the classes of (acyclic) cofibrations in the slice model structure (by Prop. ), hence is a left Quillen functor.

Proposition

(left base change Quillen equivalence)

Let $\mathcal{C}$ be a model category, and $\phi \colon S \overset{ \in \mathrm{W} }{\longrightarrow} T$ be a weak equivalence in $\mathcal{C}$.

Then the left base change Quillen adjunction along $\phi$ (Prop. ) is a Quillen equivalence

$$\mathcal{C}_{/T} \underoverset {\underset{\phi^*}{\longrightarrow}} {\overset{\phi_!}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C}_{/S}$$

if and only if $\phi$ has this property:

$(\ast)$ The pullback (base change) of $\phi$ along any fibration is still a weak equivalence.

Notice that the property $(\ast)$ of $\phi$ is implied as soon as either:

• $\mathcal{C}$ is right proper, or

• $\phi$ is an acyclic fibration, or

• both $S$ and $T$ are fibrant objects

(for the first this follows by definition; for the second by the fact that $\phi^\ast$ is a right Quillen functor by Prop. ; for the third by this Prop. on recognizing homotopy pullbacks).

Proof

Using the characterization of Quillen equivalences by derived adjuncts (here), the base change adjunction is a Quillen equivalence iff for

• any cofibrant object $X \to S$ in the slice over $S$ (i.e. $X$ is cofibrant in $\mathcal{C}$)

• and a fibrant object $p \colon Y \to T$ in the slice over $T$ (i.e. $p$ is a fibration in $\mathcal{C}$),

we have that

(1) $X \to \phi^*(Y) = S \times_T Y$ is a weak equivalence

iff

(2) $\phi_!(X) \to Y$ is a weak equivalence.

But the latter morphism is the top composite in the following commuting diagram:

$$\array{ X &\longrightarrow& S \times_T Y &\overset{p^\ast \phi}{\longrightarrow}& Y \\ &\searrow& \big\downarrow &{}^{_{(pb)}}& \big\downarrow {}^{\mathrlap{p \in Fib}} \\ && S &\underset{\phi \in \mathrm{W} }{\longrightarrow}& T }$$

Hence the two-out-of-three-property says that (1) is equivalent to (2) if $p^\ast \phi$ is a weak equivalence.

Conversely, taking $X \to \phi^\ast(X)$ to be a weak equivalence (hence a cofibrant resolution of $\phi^\ast(X)$), two-out-of-three implies that if $(\phi_! \dashv \phi^\ast)$ is a Quillen equivalence, then $p^\ast \phi$ is a weak equivalence.

In particular:

Proposition

The following are equivalent:

1. $\mathcal{C}$ is a right proper model category.

2. If $f \colon c_1 \to c_2$ is any weak equivalence in $\mathcal{C}$, then the left base change Quillen adjunction $(f_! \dashv f^\ast)$ (from Prop. ) is a Quillen equivalence.

This is due to Rezk 02, Prop. 2.5.

Sliced Quillen adjunctions

Proposition

(slice Quillen adjunctions)
Given a Quillen adjunction

$$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\;\;\bot_{\mathrlap{Qu}}\;\;\;\;} \mathcal{C} \,,$$

then:

1. for any object $b \in \mathcal{C}$ the sliced adjunction over $b$ is a Quillen adjunction between the corresponding slice model categories (Prop. ):

   (2)$$\mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot_{\mathrlap{Qu}}} \mathcal{C}_{/b} \mathrlap{\,,}$$

2. for any object $b \in \mathcal{D}$ the sliced adjunction over $b$ is a Quillen adjunction between the corresponding slice model categories (Prop. ):

   (3)$$\mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot_{\mathrlap{Qu}}} \mathcal{C}_{/R(b)} \mathrlap{\,,}$$

(e.g. Li 2016, Prop. 2.5 (2))

Proof

Consider the first case: By the nature of the sliced adjunction, its left adjoint $L_{/b}$ acts as $L$ on underlying morphisms. But since $L$ is assumed to be a left Quillen functor and since the (acyclic) cofibrations of the slice model structure are those of underlying morphisms (Prop. ), $L_{/b}$ preserves them and is hence itself a left Quillen functor.

The second case is directly analogous: Here it is evident that $R_{/b}$ is a right Quillen functor, since it acts via $R$ on underlying morphisms, and $R$ is right Quillen by assumption.

Proposition

(sliced Quillen equivalences)
Consider a Quillen equivalence

$$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\;\;\simeq_{\mathrlap{Qu}}\;\;\;\;} \mathcal{C} \,.$$

Then:

1. for a cofibrant object $b \in \mathcal{C}$ such that $L(b)$ is a fibrant object, the sliced Quillen adjunction (2) from Prop. is itself a Quillen equivalence:

$$\mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\simeq_{\mathrlap{Qu}}} \mathcal{C}_{/b}$$

1. for a fibrant object $b \in \mathcal{D}$, the sliced Quillen adjunction (3) from Prop. is itself a Quillen equivalence:

$$\mathcal{D}_{/b} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\simeq_{\mathrlap{Qu}}} \mathcal{C}_{/R(b)}$$

(e.g. Li 16, Prop. 3.1)

Proof

It is sufficient to check that the derived adjunction unit and derived adjunction counit are weak equivalences (…)

Examples

Pointed objects

Example

(model categories of pointed objects)
For every model category $\mathcal{C}$, its category of pointed objects, hence the category under the terminal object $\mathcal{C}^{\ast/}$ carries the under-category model structure: the canonical model structure on pointed objects.

For instance, the classical model structure on pointed topological spaces is the model structure on the undercategory under the point (the category of pointed objects) of the classical model structure on topological spaces.

Proposition

(induced Quillen adjunction on model categories of pointed objects)
Given a Quillen adjunction between model categories

$$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \mathcal{C} \,,$$

there is induced a Quillen adjunction between the corresponding model categories of pointed objects

$$\mathcal{D}^{\ast\!/} \underoverset {\underset{R^{\ast\!/}}{\longrightarrow}} {\overset{L^{\ast\!/}}{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \mathcal{C}^{\ast\!/} \,,$$

where

• the right adjoint acts directly as $R$ on the triangular commuting diagrams in $\mathcal{C}$ that define the morphisms in $\mathcal{C}^{\ast\!/}$;

• the left adjoint is the composite of the corresponding direct application of $L$ followed by pushout along the adjunction counit $L(\ast) \simeq L \circ R(\ast) \xrightarrow{ \;\epsilon_\ast \; } \ast$ (using that $R(\ast) \simeq \ast$ since right adjoints preserve limits and hence terminal objects):

  $$L^{\ast\!/} \;\colon\; \mathcal{C}^{\ast\!/} \xrightarrow{ \;\; L \;\; } \mathcal{D}^{L(\ast)\!/} \;\simeq\; \mathcal{D}^{L\circ R(\ast)\!/} \xrightarrow{ \;\; (-) \sqcup \epsilon_\ast \;\; } \mathcal{D}^{\ast\!/} \,.$$

Proof

It is fairly straightforward to check this directly (e.g. Hovey 1999, Prop. 1.3.5), but it is also a special case of Prop. — to make this explicit, notice that passing to opposite categories with their opposite model structures turns the original Quillen adjunction into the opposite Quillen adjunction:

$$\mathcal{C}^{op} \underoverset {\underset{L^{op}}{\longrightarrow}} {\overset{R^{op}}{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \mathcal{D}^{op} \,.$$

Now the passage to pointed objects corresponds to slicing (instead of co-slicing), since

(4)$$\mathcal{C}^{\ast\!/} \;\; \simeq \big( \mathcal{C}^{op}_{/\ast} \big)^{op} \,, \;\;\;\;\;\;\; \text{similarly} \;\;\; \mathcal{D}^{\ast\!/} \;\; \simeq \big( \mathcal{D}^{op}_{/R(\ast)} \big)^{op} \,,$$

whence item (1) in Prop. says that there is a Quillen adjunction of the form

$$\mathcal{C}^{op}_{/R(\ast)} \underoverset {\underset{L^{op}_{/\ast}}{\longrightarrow}} {\overset{R^{op}_{/\ast}}{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \mathcal{D}^{op}_{/\ast} \,,$$

hence with opposite Quillen adjunction of the required form

$$\mathcal{D}^{\ast\!/} \simeq \big(\mathcal{D}^{op}_{/R(\ast)}\big)^{op} \underoverset {\underset{ R^{\ast\!/} \,\coloneqq\, \big( R^{op}_{/\ast} \big)^{op} }{\longrightarrow}} {\overset{ L^{\ast\!/} \,\coloneqq\, \big( L^{op}_{/\ast} \big)^{op} }{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \big(\mathcal{C}^{op}_{/\ast}\big)^{op} \simeq \mathcal{C}^{\ast\!/} \,,$$

with $R^{op}$ acting directly as $R$ on underlying diagrams, and with $L^{op}$ acting as the composite of $L$ following by pullback – in $\mathcal{C}^{op}$ – along the adjunction unit of $(R^{op} \dashv L^{op})$. Since the component morphism of the unit of the opposite adjunction $(R^{op} \dashv L^{op})$ is that of the adjunction unit of $(L \dashv R)$, and since pullback in an opposite category is pushout in the original category, this implies the claim.

Slices over simplicial sets

Example

(Borel model structure)
For $G \,\in\, Grp(sSet)$ a simplicial group and $\overline{W}G \,\in\, sSet$ its simplicial classifying space, the slice model structure (Prop. ) over $\overline{W}G$ of the classical model structure on simplicial sets is Quillen equivalent to the Borel model structure of $G$-actions (see this Prop.):

$$G Act\big(sSet_{Qu}\big)_{proj} \underoverset {\underset{ \big((-) \times W G\big)/G }{\longrightarrow}} {\overset{ (-) \times_{\overline{W}G} W G }{\longleftarrow}} {\bot} \big(sSet_{Qu}\big)_{/\overline{W}G}$$

For more on this see also at $\infty$-action.

Lemma

For any simplicial set $B \,\in$ sSet and any pair of Kan fibrations $X \underoverset{\in Fib}{p}{\longrightarrow} B$ and $X' \underoverset{\in Fib}{p'}{\longrightarrow} B$, a morphism

$$\array{ X && \xrightarrow{\;\; f \;\;} && X \\ & {}_{\mathllap{p}}\searrow && \swarrow{}_{\mathrlap{p'}} \\ && B } \;\;\;\;\;\; \in (sSet_{Qu})_{/B}$$

is a simplicial weak homotopy equivalence (hence a weak equivalence in the slice model structure, from Prop. , over $B$ of the classical model structure on simplicial sets) if and only if so are all its restrictions (all its base changes by pullback) to all (homotopy) fibers $X_b$

$$b^\ast(f) \;\colon\; X_b \xrightarrow{\;\; f_b\;\;} X'_b \,,$$

for all points $b \,\in\, B_0$.

Proof

In one direction, assume that $f$ is a weak equivalence. By Prop. the pullback operation $b^\ast$ is a right Quillen functor. Therefore Ken Brown's lemma (here) implies that it preserves weak equivalences between fibrant objects. Since $p$ and $p'$ are fibrant by assumption, this means that $b^\ast(f)$ is a weak equivalence.

In the other direction, assume that $b^\ast(f)$ is a weak equivalence for all $b \,\in\, B_0$. Then for any $x \,\in\, X_0$ let $b \,\coloneqq\, p(x) \,\in\, B_0$ and consider the resulting morphism of homotopy fiber-diagrams:

and, in turn, the induced morphim of long exact sequences of homotopy groups, which has the following segments, for all $n \,\in\, \mathbb{N}_+$ (where $x' \,\coloneqq\, f(x)$):

Now the (non-abelian) five lemma implies that $\pi_n(f,x)$ is an isomorphism, for all $n \in \mathbb{N}_+$ and all $x \in X$.

It only remains to see that $\pi_0(f)$ is an isomorphism. This follows by the same argument after replacing $B$ by the connected components $\tilde B \xhookrightarrow{\;} B = B \sqcup (B \setminus \tilde B)$ which are, under $\pi_0$, in the image of $p$. This yields a morphism of exact sequences of the above form by replacing the rightmost item by the singleton; and the conclusion follows.

Related concepts

• parametrized homotopy theory

• 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, J. Homotopy Relat. Struct. 16 (2021) 753–768 [arXiv:1507.01624, doi:10.1007/s40062-021-00294-4]

• Peter May, Kate Ponto, Thm. 15.3.6 in: More concise algebraic topology, University of Chicago Press (2012) (ISBN:9780226511795, pdf)

• Zhi-Wei Li, A note on the model (co-)slice categories, Chinese Annals of Mathematics, Series B volume 37, pages 95–102 (2016) (arXiv:1402.2033, doi:10.1007/s11401-015-0955-z)

See also:

• Charles Rezk, Every homotopy theory of simplicial algebras admits a proper model, Topology and its Applications, Volume 119, Issue 1, 2002, Pages 65-94, (doi:10.1016/S0166-8641(01)00057-8, arXiv:math/0003065)

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