nLab
slice model structure

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

(slice model structure)
For 𝒞\mathcal{C} a model category and X𝒞X \in \mathcal{C} any object, the slice category 𝒞 /X\mathcal{C}_{/X} as well as the coslice category 𝒞 X/\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 𝒞 /X𝒞\mathcal{C}_{/X} \to \mathcal{C} or 𝒞 X/𝒞\mathcal{C}^{X/} \to \mathcal{C} are fibrations, cofibrations or weak equivalences, respectively, in 𝒞\mathcal{C}.

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

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}.

(Hirschhorn 05, May & Ponto 2012, Th. 15.3.6).

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

Base change Quillen adjunction

Proposition

(left base change Quillen adjunction)
For 𝒞\mathcal{C} a model category and c 1fc 2c_1 \xrightarrow{f} c_2 any morphism in 𝒞\mathcal{C}, the left base change adjunction (f 1f *)(f_1 \dashv f^\ast) along ff (where f !f_! is postcomposition with and f *f^\ast is pullback along ff) is a Quillen adjunction:

𝒞 /c 1 Quf *f !𝒞 /c 2. \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 1f_1 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 𝒞\mathcal{C} be a model category, and ϕ:SWT\phi \colon S \overset{ \in \mathrm{W} }{\longrightarrow} T be a weak equivalence in 𝒞\mathcal{C}.

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

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

if and only if ϕ\phi has this property:

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

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

(for the first this follows by definition; for the second by the fact that ϕ *\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 XSX \to S in the slice over SS (i.e. XX is cofibrant in 𝒞\mathcal{C})

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

we have that

(1) Xϕ *(Y)=S× TYX \to \phi^*(Y) = S \times_T Y is a weak equivalence

iff

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

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

X S× TY p *ϕ Y (pb) pFib S ϕW T \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 *ϕp^\ast \phi is a weak equivalence.

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

In particular:

Proposition

The following are equivalent:

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

  2. If f:c 1c 2f \colon c_1 \to c_2 is any weak equivalence in 𝒞\mathcal{C}, then the left base change Quillen adjunction (f !f *)(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

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

then:

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

    (2)𝒟 /L(c) QuR /bL /b𝒞 /b, \mathcal{D}_{/L(c)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\bot_{\mathrlap{Qu}}} \mathcal{C}_{/b} \mathrlap{\,,}
  2. for any object b𝒟b \in \mathcal{D} the sliced adjunction over bb is a Quillen adjunction between the corresponding slice model categories (Prop. ):

    (3)𝒟 /c QuR /bL /b𝒞 /R(b), \mathcal{D}_{/c} \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 /bL_{/b} acts as LL on underlying morphisms. But since LL 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 /bL_{/b} preserves them and is hence itself a left Quillen functor.

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

Proposition

(sliced Quillen equivalences)
Consider a Quillen equivalence

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

Then:

  1. for a cofibrant object b𝒞b \in \mathcal{C} such that L(b)L(b) is a fibrant object, the sliced Quillen adjunction (2) from Prop. is itself a Quillen equivalence:
𝒟 /L(b) QuR /bL /b𝒞 /b \mathcal{D}_{/L(b)} \underoverset {\underset{\;\;\;\;R_{/b}\;\;\;\;}{\longrightarrow}} {\overset{\;\;\;\;L_{/b}\;\;\;\;}{\longleftarrow}} {\simeq_{\mathrlap{Qu}}} \mathcal{C}_{/b}
  1. for a fibrant object b𝒟b \in \mathcal{D}, the sliced Quillen adjunction (3) from Prop. is itself a Quillen equivalence:
𝒟 /b QuR /bL /b𝒞 /R(b) \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

Proposition

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

𝒟 QuRL𝒞, \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

𝒟 */ QuR */L */𝒞 */, \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 RR on the triangular commuting diagrams in 𝒞\mathcal{C} that define the morphisms in 𝒞 */\mathcal{C}^{\ast\!/};

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

    L */:𝒞 */L𝒟 L(*)/𝒟 LR(*)/()ϵ *𝒟 */. 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:

𝒞 op QuL opR op𝒟 op. \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)𝒞 */(𝒞 /* op) op,similarly𝒟 */(𝒟 /R(*) op) op, \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

𝒞 /R(*) op QuL /* opR /* op𝒟 /* op, \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

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

References

See also:

Last revised on August 20, 2021 at 06:28:10. See the history of this page for a list of all contributions to it.