nLab geometry of physics -- basic notions of higher topos theory

Basic notions of higher topos theory

We have discussed basic notions of topos theory above and of homotopy theory (above). The combination of the two is higher topos theory which we discuss here.

We had explained how toposes may be thought of as categories of generalized spaces and how homotopy theory is about relaxing the concept of equality to that of gauge transformation/homotopy and higher gauge transformation/higher homotopy. Accordingly, higher toposes may be thought of as higher categories of generalized spaces whose probe are defined only up to gauge transformation/homotopy. Examples of such include orbifolds and Lie groupoids.

(…)

Locally presentable $\infty$-Categories

The analog of the notion of locally presentable categories (Def. ) for model categories (Def. ) are combinatorial model categories (Def. ) below. In addition to the ordinary condition of presentability of the underlying category, these are required to be cofibrant generation (Def. below) in that all cofibrations are retracts of relative cell complex-inclusions.

That this is indeed the correct model category-analog of locally presentable categories is the statement of Dugger's theorem (Def. below).

Hence as we pass to the localization of the very large category of combinatorial model categories at the Quillen equivalences, we obtain a homotopy-theoretic refinement of the very large category PrCat of locally presentable categories: Ho(CombModCat) (Def. ). An object in Ho(CombModCat) we also refer to as a locally presentable (∞,1)-category, and a morphism in Ho(CombModCat) we also refer to as the equivalence class of an (∞,1)-colimit-preserving (∞,1)-functor.

$\,$

Definition

(cofibrantly generated model category)

A model category $\mathcal{C}$ (def. ) is called cofibrantly generated if there exists two small subsets

$$I, J \subset Mor(\mathcal{C})$$

of its class of morphisms, such that

1. $I$ and $J$ have small domains according to def. ,

2. the (acyclic) cofibrations of $\mathcal{C}$ are precisely the retracts, of $I$-relative cell complexes ($J$-relative cell complexes), def. .

Proposition

For $\mathcal{C}$ a cofibrantly generated model category, def. , with generating (acylic) cofibrations $I$ ($J$), then its classes $W, Fib, Cof$ of weak equivalences, fibrations and cofibrations are equivalently expressed as injective or projective morphisms (def. ) this way:

1. $Cof = (I Inj) Proj$

2. $W \cap Fib = I Inj$;

3. $W \cap Cof = (J Inj) Proj$;

4. $Fib = J Inj$;

Proof

It is clear from the definition that $I \subset (I Inj) Proj$, so that the closure property of prop. gives an inclusion

$$Cof \subset (I Inj) Proj \,.$$

For the converse inclusion, let $f \in (I Inj) Proj$. By the small object argument, prop. , there is a factorization $f\colon \overset{\in I Cell}{\longrightarrow}\overset{I Inj}{\longrightarrow}$. Hence by assumption and by the retract argument lemma , $f$ is a retract of an $I$-relative cell complex, hence is in $Cof$.

This proves the first statement. Together with the closure properties of prop. , this implies the second claim.

The proof of the third and fourth item is directly analogous, just with $J$ replaced for $I$.

Definition

(combinatorial model category)

A combinatorial model category is a model category (Def. ) which is

1. locally presentable (Def. )

2. cofibrantly generated model category (Def. )

Example

(classical model structure on simplicial sets is combinatorial model category)

The classical model structure on simplicial sets (Def. ) $sSet_{Qu}$ is a combinatorial model category (Def. ).

Example

(category of simplicial presheaves)

Let $\mathcal{C}$ be a small (Def. ) sSet-enriched category (Def. with Example ) and consider the enriched presheaf category (Example )

$$sPSh(\mathcal{C}) \;\coloneqq\; [\mathcal{C}^{op}, sSet]$$

This is called the category of simplicial presheaves on $\mathcal{C}$.

By Prop. this is equivalent (Def. ) to the category of simplicial objects in the category of presheaves over $\mathcal{C}$ (Example ):

(1)$$[\mathcal{C}^{op}, sSet] \;\simeq\; [\Delta^{op}, \mathcal{C}^{op}, Set]$$

This implies for instance that if

$$\mathcal{D} \overset{F}{\longrightarrow} \mathcal{D}$$

a functor, the induced adjoint triple (Remark ) of sSet-enriched functor Kan extensions (Prop. )

$$[\mathcal{C}^{op}, sSet] \; \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{Lan_F}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{F^\ast}{\longleftarrow} \\ \underoverset{\phantom{AA}\phantom{\bot}\phantom{AA}}{Ran_F}{\longrightarrow} } \; [\mathcal{D}^{op}, sSet]$$

is given simplicial-degreewise by the corresponding Set-enriched Kan extensions.

Proposition

(model categories of simplicial presheaves)

Let $\mathcal{C}$ be a small (Def. ) sSet-enriched category (Def. with Example ). Then the category of simplicial presheaves $[\mathcal{C}^{op}, sSet]$ (Example ) carries the following two structures of a model category (Def. )

1. the projective model structure on simplicial presheaves

   $$[\mathcal{C}^{op}, sSet_{Qu}]_{proj}$$

   has as weak equivalences and fibrations those natural transformations $\eta$ whose component on every object $c \in \mathcal{C}$ is a weak equivalences or fibration, respectively, in the classical model structure on simplicial sets (Def. );

2. the injective model structure on simplicial presheaves

   $$[\mathcal{C}^{op}, sSet_{Qu}]_{inj}$$

   has as weak equivalences and cofibrations those natural transformations $\eta$ whose component on every object $c \in \mathcal{C}$ is a weak equivalences or cofibration, respectively, in the classical model structure on simplicial sets (Def. );

Moreover, the identity functors constitute a Quillen equivalence (Def. ) between these two model structures

(2)$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\simeq_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$$

Remark

The Quillen adjunction (2) in Prop. implies in particular that

1. every projective cofibration is in particular an objectwise cofibration;

2. every injective fibration is in particular an objectwise fibration;

Proposition

(some projectively cofibrant simplicial presheaves)

Let $\mathcal{C}$ be a small (Def. ). Then a sufficient condition for a simplicial presheaf over $\mathcal{C}$ (Def. )

$$\mathbf{X} \;\in\; [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$$

to be a cofibrant object with respect to the projective model structure on simplicial presheaves (Prop. ) is that

1. $\mathbf{X}$ is degreewise a coproduct of representable presheaves

   $$\mathbf{X}_k \;\simeq\; \underset{i_k}{\coprod} y(X_{i_k})$$

2. the degeneracy maps are inclusions of direct summands.

In particular every representable presheaf, regarded as a simplicially constant simplicial presheaf, is projectively cofibrant.

(Dugger 00, section 9, lemma 2.7)

The following concept of left Bousfield localization is the analog for model categories of the concept of reflection onto local objects (Def. ):

Definition

(left Bousfield localization)

A left Bousfield localization $\mathcal{C}_{loc}$ of a model category $\mathcal{C}$ (Def. ) is another model category structure on the same underlying category with the same cofibrations,

$$Cof_{loc} = Cof$$

but more weak equivalences

$$W_{loc} \supset W \,.$$

We say that this is localization at $W_{loc}$.

Notice that:

Proposition

(left Bousfield localization is Quillen reflection)

Given a left Bousfield localization $\mathcal{C}_{loc}$ of $\mathcal{C}$ as in def. , then the identity functor exhibits a Quillen reflection (Def. )

$$\mathcal{C}_{loc} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{C} \,.$$

In particular, by Prop. , the induced adjunction of derived functors (Prop. ) exhibits a reflective subcategory inclusion of homotopy categories (Def. )

$$Ho(\mathcal{C}_{loc}) \underoverset {\underset{\phantom{AA}\mathbb{R} id \phantom{AA}}{\hookrightarrow}} {\overset{\mathbb{L}id}{\longleftarrow}} {\bot} Ho(\mathcal{C}) \,.$$

Proof

We claim that

1. $Fib_{loc} \subset Fib$;

2. $W_{loc} \cap Fib_{loc} = W \cap Fib$;

Using the properties of the weak factorization systems (def.) of (acyclic cofibrations, fibrations) and (cofibrations, acyclic fibrations) for both model structures we get

$$\begin{aligned} Fib_{loc} &= (Cof_{loc} \cap W_{loc})Inj \\ &\subset (Cof_{loc} \cap W)Inj \\ & = Fib \end{aligned}$$

and

$$\begin{aligned} Fib_{loc} \cap W_{loc} & = Cof_{loc} Inj \\ & = Cof \, Inj \\ & = Fib \cap W \end{aligned} \,.$$

Next to see that the identity functor constitutes a Quillen adjunction (Def. ): By construction, $id \colon \mathcal{C} \to \mathcal{C}_{loc}$ preserves cofibrations and acyclic cofibrations, hence is a left Quillen functor.

To see that the derived adjunction counit (Def. ) is a weak equivalence:

Since we have an adjoint pair of identity functors, the ordinary adjunction counit is the identity morphisms and hence the derived adjunction counit on a fibrant object $c$ is just a cofibrant resolution-morphism

$$Q(c) \underoverset{ \in W_{\mathcal{D}} \cap Fib_{\mathcal{D}} }{p_c}{\longrightarrow} c$$

but regarded in the model structure $\mathcal{D}_{loc}$. Hence it is sufficient to see that acyclic fibrations in $\mathcal{D}$ remain weak equivalences in the left Bousfield localized model structure. In fact they even remain acyclic fibrations, bu the first point above.

We may also easily check directly the equivalent statement (via Prop. ) that the induced adjunction of derived functors on homotopy categories is a reflective subcategory-inclusion:

Since $Cof_{loc} = Cof$ the notion of left homotopy in $\mathcal{C}_{loc}$ is the same as that in $\mathcal{C}$, and hence the inclusion of the subcategory of local cofibrant-fibrant objects into the homotopy category of the original cofibrant-fibrant objects is clearly a full subcategory inclusion. Since $Fib_{loc} \subset Fib$ by the first statement above, on these cofibrant-fibrant objects the right derived functor of the identity is just the identity and hence does exhibit this inclusion. The left adjoint to this inclusion is given by $\mathbb{L}id$, by the general properties of Quillen adjunctions (Prop. )).

Example

(left Bousfield localization is Quillen reflection)

Let

Proof

We consider the case of left Bousfield localizations, the other case is formally dual.

A left Bousfield localization is a Quillen adjunction by identity functors (this Remark)

$$\mathcal{D}_{loc} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {{}_{\phantom{Qu}} \bot_{Qu}} \mathcal{D}$$

This means that the ordinary adjunction counit is the identity morphisms and hence that the derived adjunction counit on a fibrant object $c$ is just a cofibrant resolution-morphism

$$Q(c) \underoverset{ \in W_{\mathcal{D}} \cap Fib_{\mathcal{D}} }{p_c}{\longrightarrow} c$$

but regarded in the model structure $\mathcal{D}_{loc}$. Hence it is sufficient to see that acyclic fibrations in $\mathcal{D}$ remain weak equivalences in the left Bousfield localized model structure. In fact they even remain acyclic fibrations, by this Remark.

The following proposition says that Definition of combinatorial model categories is indeed the suitable analog of Def. of locally presentable categories:

Proposition

(Dugger's theorem)

Let $\mathcal{C}$ be a combinatorial model category (Def. ). Then there exists

1. a small category $\mathcal{S}$;

2. a small set $S \subset Mor_{[\mathcal{S}^{op}, sSet]}$ in its category of simplicial presheaves (Example );

1 a Quillen equivalence (Def. )

$$[\mathcal{S}^{op}, sSet_{Qu}]_{proj,S} \underoverset {\underset{\phantom{AAAA}}{\longrightarrow}} {\overset{}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C}$$

between $\mathcal{C}$ and the left Bousfield localization (Def. ) of the projective model structure on simplicial presheaves over $\mathcal{C}$ (Prop. ) at the set $S$.

Definition

(homotopy category of presentable (∞,1)-categories)

Write $CombModCat$ for the very large category whose objects are combinatorial model categories (Def. ) and whose morphisms are left Quillen functors between them (Def. ).

We write

Ho(CombModCat)$\coloneqq CombModCat\big[ QuillenEquivs^{-1}\big]$

for its localization (Def. ) at the Quillen equivalences (Def. ).

We say:

• an object in Ho(CombModCat) is a locally presentable (∞,1)-category,

• a morphism in Ho(CombModCat) is the equivalence class of an(∞,1)-colimit-preserving (∞,1)-functor;

• an isomorphism in Ho(CombModCat) is an equivalence of (∞,1)-categories.

The following example is the genralization of the category of sets (Def. ) as we pass to homotopy theory:

Example

(∞Grpd)

The image of the classical model structure on simplicial sets $sSet_{Qu}$ (Def. ), which is combinatorial model category by example , under the localization to Ho(CombModCat) (Def. ), we call the presentable (∞,1)-category of ∞-groupoids:

$$\array{ CombModCat &\overset{\phantom{AA}\gamma\phantom{AA}}{\longrightarrow}& Ho(CombModCat) \\ sSet_{Qu} &\mapsto& \infty Grpd }$$

In order to get good control over left Bousfield localization (Def. ) and hence over presentable (∞,1)-categories (Def. ) we need the analog of Prop. , saying that reflective localization are reflections onto their full subcategories of local objects. For this, in turn, we need a good handle on the hom-infinity-groupoids:

Definition

(simplicial model category)

An sSet${}_{Quillen}$-enriched model category or simplicial model category, for short is a category $\mathcal{C}$ (Def. ) equipped with

1. the structure of an sSet-enriched category (Def. via Example ), which is also tensored and cotensored over sSet (Def. )

   (with sSet (Def. ), equipped with its canonical structure of a cosmos from Prop. , Example ),

2. the structure of a model category (Def. )

such that these two structures are compatible in the following way:

• for every cofibration $X \overset{f}{\to} Y$ and every fibration $A \overset{g}{\to} B$ in $\mathcal{C}$, the induced pullback powering-morphism of hom-simplicial sets

  (3)$$\mathcal{C}(Y,A) \longrightarrow \mathcal{C}(X,A) \underset{\mathcal{C}(X,B)}{\times} \mathcal{C}(Y,B)$$

  is a Kan fibration (Def. ), and is a weak homotopy equivalence (Def. ) as soon as one of the two morphisms is a weak equivalence in $\mathcal{C}$.

Proposition

(in simplicial model category enriched hom-functor out of cofibrant into fibrant is homotopical functor)

Let $\mathcal{C}$ be a simplicial model category (Def. ).

If $Y \in \mathcal{C}$ is a cofibrant object, then the enriched hom-functor (Example ) out of $X$

$$\mathcal{C}(Y,-) \;\colon\; \mathcal{C} \longrightarrow sSet_{Qu}$$

preserves fibrations and acyclic fibrations.

If $A \in \mathcal{C}$ is a fibrant object, then the enriched hom-functor (Example ) into $X$

$$\mathcal{C}(-,A) \;\colon\; \mathcal{C}^{op} \longrightarrow sSet_{Qu}$$

sends cofibrations and acyclic cofibrations in $\mathcal{C}$ to fibrations and acyclic fibrations, respectively, in the classical model structure on simplicial sets.

Proof

In the first case, consider the comparison morphism (3) for $X =\emptyset$ the initial object, in the second case consider it for $B = \ast$ the terminal object (Def. )

Since $\mathcal{C}$ is a tensored and cotensored category, Prop. says that

$$\mathcal{C}(\emptyset, -) \;\simeq\; \ast \phantom{AA} \text{and} \phantom{AA} \mathcal{C}(-,\ast) \;\ast\; \;\;\; \in sSet \,.$$

This means that in the first case the comparison morphism

$$\mathcal{C}(Y,A) \longrightarrow \mathcal{C}(X,A) \underset{\mathcal{C}(X,B)}{\times} \mathcal{C}(Y,B)$$

(3) becomes equal to the top morphism in the following diagram

$$\array{ \mathcal{C}(Y,A) &\overset{\mathcal{C}(Y,g)}{\longrightarrow}& \mathcal{C}(Y,B) \\ \Big\downarrow && \Big\downarrow \\ \ast &\underset{\phantom{AAA}}{\longrightarrow}& \ast }$$

while in the second case it becomes equal to the left morphism in

$$\array{ \mathcal{C}(Y,A) &\overset{\phantom{\mathcal{C}(Y,g)}}{\longrightarrow}& \ast \\ {}^{\mathllap{ \mathcal{C}(f,A) }}\Big\downarrow && \Big\downarrow \\ \mathcal{C}(X,A) &\underset{\phantom{AAA}}{\longrightarrow}& \ast }$$

Hence the claim follows by the defining condition on the comparison morphism in a simplicial model category.

Definition

(derived hom-functor)

Let $\mathcal{C}$ be a simplicial model category (Def. ).

By Prop. and by Ken Brown's lemma (Prop. ), the enriched hom-functor (Example ) has a right derived functor (Def. ) when its first argument is cofibrant and its second argument is fibrant. The combination is called the derived hom-functor

$$\mathbb{R}hom \;\colon\; Ho(\mathcal{C})^{op} \times Ho(\mathcal{C}) \longrightarrow Ho(sSet_{Quillen})$$

In view of the Quillen equivalence $sSet_{Qu} \simeq_{Qu} Top_{Qu}$ (Theorem ), the simplicial sets (Kan complexes) $\mathbb{R}hom(X,A)$ are also called the derived hom-spaces.

In the presence of functorial cofibrant resolution $Q$ and fibrant resolution $P$ (Def. ) this is given by the ordinary enriched hom-functor $\mathcal{C}(-,-)$ as

$$\mathbb{R}hom(X,Y) \;\simeq\; \mathcal{C}(Q X, P Y) \,.$$

Proposition

(recognition of simplicial Quillen adjunctions)

Let $\mathcal{C}$ and $\mathcal{D}$ be two simplicial model categories (Def. ) such that $\mathcal{D}$ is also a left proper model category (Def. ). Then for an sSet-enriched adjunction (Def. ) of the form

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} \mathcal{D}$$

to be Quillen adjunction (Def. , hence a simplicial Quillen adjunction) it is sufficient that the following two conditions hold:

1. $L$ preserves cofibrations,

2. $R$ preserves fibrant objects

(i.e. this already implies that $R$ preserves all fibrations).

(Lurie HTT, cor. A.3.7.2)

Proposition

(model structure on simplicial presheaves is left proper combinatorial simplicial model category)

Let $\mathcal{C}$ be a small (Def. ) sSet-enriched category (Def. with Example ). Then the injective and projective model structure on simplicial presheaves over $\mathcal{C}$ (Prop. )

$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \;, \phantom{A} [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \;\;\; \in CombModCat$$

are

1. proper model categories (Def. ),

2. simplicial model categories (Def. ),

3. combinatorial model categories (Def. ).

The following is the model category-analog of the concept of local objects from Def. :

Definition

(local objects and local morphisms in a model category)

Let $\mathcal{C}$ be a simplicial model category (Def. ) and let $S \subset Mor_{\mathcal{C}}$ be a sub-class of its class of morphisms. Then

1. an object $A \in \mathcal{C}$ is called a (derived-)local object if for every $X \overset{s}{\to} Y \; \in S$ the value of the derived hom-functor (Def. ) out of $s$ into $X$ is a weak equivalence (i.e. an isomorphism in the classical homotopy category $Ho(sSet)$)

   $$\mathbb{R}Hom(s,A) \;\colon\; \mathbb{R}Hom(Y,A) \overset{\simeq}{\longrightarrow} \mathbb{R}Hom(X,A)$$

2. a morphism $X \overset{f}{\to} Y$ in $\mathcal{C}$ is called a (derived-)local morphism if for every local object $A$ we have

   $$\mathbb{R}Hom(f,A) \;\colon\; \mathbb{R}Hom(Y,A) \overset{\simeq}{\longrightarrow} \mathbb{R}Hom(X,A)$$

The following is the model category-analog of the characterization from Prop. of reflective localizations as reflections onto local objects:

Proposition

(existence of left Bousfield localization for left proper simplicial combinatorial model categories)

Let $\mathcal{C}$ be a combinatorial model category (Def. ) which is left proper (Def. ) and simplicial (Def. ), and let $S \subset Mor_{\mathcal{C}}$ be a small set of its morphisms.

Then the left Bousfield localization (Def. ) of $\mathcal{C}$ at $S$, namely at the class of $S$-local morphisms (Def. ) exist, to be denoted $L_S \mathcal{C}$, and it has the following properties:

1. $L_S \mathcal{C}$ is itself a left proper simplicial combinatorial model category;

2. the fibrant objects of $L_S \mathcal{C}$ are precisely those fibrant objects of $\mathcal{C}$ which in addition are $S$-local objects (Def. );

3. the homotopy category (Def. ) of $L_S \mathcal{C}$ is the full subcategory of that of $\mathcal{C}$ on ( the images under localization of) the $S$-local objects.

   $$Ho(L_S \mathcal{C}) \hookrightarrow Ho(\mathcal{C})$$

The following class of examples of left Bousfield localizations generalizes those of Def. from 1-categories to locally presentable (∞,1)-categories:

Definition

(homotopy localization of combinatorial model categories)

Let $\mathcal{C}$ be a combinatorial model category (Def. ) which, by Dugger's theorem (Prop. ) is Quillen equivalent to a left Bousfield localization of a model category of simplicial presheaves over some small simplicial category $\mathcal{S}$

$$\mathcal{C} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\bot_{Qu}} [\mathcal{S}^{op}, sSet_{Qu}]_{proj} \; \in CombModCat \;\text{i.e.}\; \mathcal{C} \underoverset {\underset{}{\hookrightarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longleftarrow}} {\bot} PSh_\infty(\mathcal{S}) \; \in Ho(CombModCat)$$

Let moreover

$$\mathbb{A} \in [\mathcal{S}^{op}, sSet_{Qu}]$$

be any object. Then the homotopy localization of $\mathcal{C}$ at $\mathbb{A}$ is the further left Bousfield localization (Def. ) at the morphisms of the form

$$X \times \mathbb{A} \overset{p_1}{\longrightarrow} X$$

for all $X \in \mathcal{S}$:

$$[\mathcal{S}^{op}, sSet_{Qu}]_{proj, \mathbb{A}} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\bot_{Qu}} [\mathcal{S}^{op}, sSet_{Qu}]_{proj} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\simeq_{Qu}} \mathcal{C} \;\;\;\; \in CombModCat \,.$$

The image of this homotopy localization in Ho(CombModCat) (Def. ) we denote by

$$\mathcal{C}_{\mathbb{A}} \underoverset {\underset{\iota}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} \mathcal{C} \;\;\; \in Ho(CombModCat) \,.$$

$\infty$-Modalities

The following is an homotopy theoretic analog of adjoint triples (Remark ):

Definition

(Quillen adjoint triple)

Let $\mathcal{C}_1, \mathcal{C}_2, \mathcal{D}$ be model categories (Def. ), where $\mathcal{C}_1$ and $\mathcal{C}_2$ share the same underlying category $\mathcal{C}$, and such that the identity functor on $\mathcal{C}$ constitutes a Quillen equivalence (Def. )

(4)$$\mathcal{C}_2 \underoverset {\underset{ \phantom{AA}id\phantom{AA} }{\longrightarrow}} {\overset{ \phantom{AA}id\phantom{AA} }{\longleftarrow}} {{}_{\phantom{Qu}}\simeq_{Qu}} \mathcal{C}_1$$

Then a Quillen adjoint triple

$$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

$$\mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

is a pair of Quillen adjunctions (Def. ), as shown, together with a 2-morphism in the double category of model categories (Def. )

(5)$$\array{ \mathcal{D} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{C}_1 \\ {}^{\mathllap{C}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{C}_2 &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{C}_2 }$$

whose derived natural transformation $Ho(id)$ (Def. ) is invertible (a natural isomorphism).

If two Quillen adjoint triples overlap

$$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$$

$$\mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$$

$$\mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2$$

we speak of a Quillen adjoint quadruple, and so forth.

Proposition

(Quillen adjoint triple induces adjoint triple of derived functors on homotopy categories)

Given a Quillen adjoint triple (Def. ), the induced derived functors (Def. ) on the homotopy categories (Def. ) form an ordinary adjoint triple (Remark ):

$$\mathcal{C}_{1/2} \array{ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{Qu}}{C}{\longleftarrow} \\ \overset{\phantom{AA}R\phantom{AA}}{\longrightarrow} \\ } \mathcal{D} \phantom{AAAA} \overset{Ho(-)}{\mapsto} \phantom{AAAA} Ho(\mathcal{C}) \array{ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}L}{\longrightarrow} \\ \underoverset{{}_{\phantom{Qu}}\bot_{\phantom{Qu}}}{\mathbb{L}C \simeq \mathbb{R}C}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}R\phantom{AA}}{\longrightarrow} \\ } Ho(\mathcal{D})$$

Proof

This follows immediately from the fact that passing to homotopy categories of model categories is a double pseudofunctor from the double category of model categories to the double category of squares in Cat (Prop. ).

Example

(Quillen adjoint triple from left and right Quillen functor)

Given an adjoint triple (Remark )

$$\mathcal{C} \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{L}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} \\ \underoverset{\phantom{AA}\phantom{\bot}\phantom{AA}}{R}{\longrightarrow} } \mathcal{D}$$

such that $C$ is both a left Quillen functor as well as a right Quillen functor (Def. ) for given model category-structures on the categories $\mathcal{C}$ and $\mathcal{D}$. Then this is a Quillen adjoint triple (Def. ) of the form

$$\mathcal{C} \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

$$\mathcal{C} \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

Proof

The condition of a Quillen equivalence (4) is trivially satisfied (by Prop. ). Similarly the required 2-morphism (5)

$$\array{ \mathcal{C} &\overset{\phantom{A}C\phantom{A}}{\longrightarrow}& \mathcal{D} \\ {}^{\mathllap{C}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ \mathcal{D} &\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}& \mathcal{D} }$$

exists trivially. To check that its derived natural transformation (Def. ) is a natural isomorphism we need to check (by Prop. ) that for every fibrant and cofibrant object $d \in \mathcal{D}$ the composite

$$Q C(d) \overset{ p_{C(d)} }{\longrightarrow} C(d) \overset{ j_{C(d)} }{\longrightarrow} P C(C)$$

is a weak equivalence. But this is trivially the case, by definition of fibrant resolution/cofibrant resolution (Def. ; in fact, since $C$ is assumed to be both left and right Quillen, also $C(d)$ is a fibrant and cofibrant objects and hence we may even take both $p_{C(d)}$ as well as $j_{C(d)}$ to be the identity morphism).

The following is the analog in homotopy theory of the adjoint triple of the adjoint triple colimit/constant functor/limit (Def. ):

Example

(Quillen adjoint triple of homotopy limits/colimits of simplicial sets)

Let $\mathcal{C}$ be a small category (Def. ), and write $[\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj}$ for the projective/injective model structure on simplicial presheaves over $\mathcal{C}$ (Prop. ), which participate in a Quillen equivalence of the form

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{\phantom{AA}id\phantom{AA}}{\longleftarrow}} {\simeq_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$$

Moreover, the constant diagram-assigning functor

$$[\mathcal{C}^{op}, sSet] \overset{const}{\longleftarrow} sSet$$

is clearly a left Quillen functor for the injective model structure, and a right Quillen functor for the projective model structure.

Together this means that in the double category of model categories (Def. ) we have a 2-morphism of the form

$$\array{ sSet_{Qu} &\overset{const}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{const}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{id}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{inj} }$$

Moreover, the derived natural transformation $Ho(id)$ (Prop. ) of this square is invertible, if for every Kan complex $X$

$$Q const X \overset{}{\longrightarrow} const X \longrightarrow P const X$$

is a weak homotopy equivalence (by Prop. ), which here is trivially the case.

Therefore we have a Quillen adjoint triple (Def. ) of the form

$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{ \underset{\longrightarrow}{\lim} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu} } sSet_{Qu}$$

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{const}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu} } sSet_{Qu}$$

The induced adjoint triple of derived functors on the homotopy categories (via Prop. ) is the homotopy colimit/homotopy limit adjoint triple

$$Ho([\mathcal{C}^{op}, sSet]) \; \array{ \overset{\phantom{AA}\mathbb{L}\underset{\longrightarrow}{\lim}\phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA}const\phantom{AA}}{\longleftarrow} \\ \overset{\phantom{AA}\mathbb{R}\underset{\longleftarrow}{\lim}\phantom{AA}}{\longrightarrow} } \; Ho(sSet)$$

More generally:

Example

(Quillen adjoint triple of homotopy Kan extension of simplicial presheaves)

Let $\mathcal{C}$ and $\mathcal{D}$ be small categories (Def. ), and let

$$\mathcal{C} \overset{\phantom{AA}F\phantom{AA}}{\longrightarrow} \mathcal{D}$$

be a functor between them. By Kan extension (Prop. ) enriched over sSet (Example ) this induces an adjoint triple between categories of simplicial presheaves (Def. ):

$$[\mathcal{C}^{op}, sSet] \array{ \underoverset{\bot}{ \phantom{AA}F_!\phantom{AA} }{\longrightarrow} \\ \underoverset{\bot}{ \phantom{AA}F^\ast\phantom{AA} }{\longleftarrow} \\ \overset{ \phantom{AA}F_\ast\phantom{AA} }{\longrightarrow} } [\mathcal{D}^{op}, sSet]$$

where

$$F^\ast \mathbf{X} \;\coloneqq\; \mathbf{X}(F(-))$$

is the operation of precomposition with $F$. This means that $F^\ast$ preserves all objectwise cofibrations/fibrations/weak equivalences in the model structure on simplicial presheaves (Prop. ). Hence it is

1. a right Quillen functor (Def. ) $[\mathcal{D}^{op}, sSet]_{proj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$;

2. a left Quillen functor (Def. ) $[\mathcal{D}^{op}, sSet]_{inj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{inj}$;

and since

$$[\mathcal{D}^{op}, sSet]_{inj} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{AA}\bot\phantom{AA}} [\mathcal{D}^{op}, sSet]_{proj}$$

is also a Quillen adjunction (Def. ), these imply that $F^\ast$ is also

• a right Quillen functor $[\mathcal{D}^{op}, sSet]_{inj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$.

• a left Quillen functor $[\mathcal{D}^{op}, sSet]_{proj} \overset{F^\ast}{\to} [\mathcal{C}^{op}, sSet_{Qu}]_{inj}$.

In summary this means that we have 2-morphisms in the double category of model categories (Def. ) of the following form:

$$\array{ [\mathcal{D}^{op}, sSet_{Qu}]_{proj} &\overset{\phantom{AA}F^\ast\phantom{AA}}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{F^\ast}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{id}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{inj} } \phantom{AAA} \text{and} \phantom{AAA} \array{ [\mathcal{D}^{op}, sSet_{Qu}]_{inj} &\overset{\phantom{AA}F^\ast\phantom{AA}}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \\ {}^{\mathllap{F^\ast}} \Big\downarrow &\swArrow_{\mathrlap{id}}& \Big\downarrow{}^{ \mathrlap{ id } } \\ [\mathcal{C}^{op}, sSet_{Qu}]_{inj} &\underset{id}{\longrightarrow}& [\mathcal{C}^{op}, sSet_{Qu}]_{proj} }$$

To check that the corresponding derived natural transformations $Ho(id)$ are natural isomorphisms, we need to check (by Prop. ) that the composites

$$Q_{inj} F^\ast \mathbf{X} \overset{ p_{F^\ast \mathbf{X}} }{\longrightarrow} F^\ast \mathbf{X} \overset{ j_{F^\ast \mathbf{X}} }{\longrightarrow} P_{proj} F^\ast \mathbf{X}$$

are invertible in the homotopy category $Ho([\mathcal{C}^{op}, sSet_{Qu}]_{inj/proj})$ (Def. ), for all fibrant-cofibrant simplicial presheaves $\mathbf{X}$ in $[\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj}$. But this is immediate, since the two factors are weak equivalences, by definition of fibrant/cofibrant resolution (Def. ).

Hence we have a Quillen adjoint triple (Def. ) of the form

(6)$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} \array{ \underoverset{\bot}{\phantom{AA}F_!\phantom{AA}}{\longrightarrow} \\ \underoverset{\bot}{\phantom{AA}F^\ast\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}F_\ast\phantom{AA}}{\longrightarrow} } [\mathcal{D}^{op}, sSet_{Qu}]_{proj} \phantom{AAA} \text{and} \phantom{AAA} [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} \array{ \underoverset{\bot}{\phantom{AA}F_!\phantom{AA}}{\longrightarrow} \\ \underoverset{\bot}{\phantom{AA}F^\ast\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}F_\ast\phantom{AA}}{\longrightarrow} } [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$$

The corresponding derived adjoint triple on homotopy categories (Prop. ) is that of homotopy Kan extension:

$$Ho([\mathcal{C}^{op}, sSet]) \array{ \underoverset{\bot \phantom{\simeq A_a}}{ \phantom{A}\mathbb{L}F_! \phantom{\simeq A_a}\phantom{A} }{\longrightarrow} \\ \underoverset{\phantom{\simeq A_a} \bot}{ \phantom{A}\mathbb{R}F^\ast \simeq \mathbb{L}F^\ast\phantom{A} }{\longleftarrow} \\ \overset{ \phantom{A} \phantom{A_a \simeq} \mathbb{R}F_\ast\phantom{A} }{\longrightarrow} } Ho([\mathcal{D}^{op}, sSet])$$

Example

(Quillen adjoint quadruple of homotopy Kan extension of simplicial presheaves along adjoint pair)

Let $\mathcal{C}$ and $\mathcal{D}$ be small categories (Def. ), and let

$$\mathcal{C} \underoverset {\underset{\phantom{AA}R\phantom{AA}}{\longleftarrow}} {\overset{\phantom{AA}L\phantom{AA}}{\longrightarrow}} {\bot} \mathcal{D}$$

be a pair of adjoint functors (Def. ). By Kan extension this induces an adjoint quadruple (Prop. ) between categories of simplicial presheaves (Def. )

$$[\mathcal{C}^{op}, sSet] \array{ \underoverset{\bot \phantom{\simeq A_a}}{ L_! \phantom{\simeq A_a} }{\longrightarrow} \\ \underoverset{\bot \phantom{\simeq} \bot }{ L^\ast \simeq R_! }{\longleftarrow} \\ \underoverset{\phantom{A_a \simeq}\bot}{ L_\ast \simeq R^\ast }{\longrightarrow} \\ \overset{ \phantom{A_a \simeq } R_\ast }{\longrightarrow} } [\mathcal{D}^{op}, sSet]$$

By Example the top three as well as the bottom three of these form Quillen adjoint triples (Def. ) for model structures on simplicial presheaves (Prop. ) in two ways (6). If for the top three we choose the first version, and for the bottom three the second version from (6), then these combine to a Quillen adjoint quadruple of the form

$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{= A_a}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$$

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{L^\ast = R_!}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$$

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{A=} R_\ast}{\longleftarrow}} {\overset{L_\ast = R^\ast}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$$

Example

(Quillen adjoint quintuple of homotopy Kan extension of simplicial presheaves along adjoint triple)

Let $\mathcal{C}$ and $\mathcal{D}$ be small categories (Def. ) and let

$$\mathcal{C} \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{L}{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{C}{\longleftarrow} } \mathcal{D}$$

be a triple of adjoint functors (Remark ). By Kan extension (Prop. ) enriched over sSet (Def. ) this induces an adjoint quintuple between categories of simplicial presheaves

(7)$$[\mathcal{C}^{op}, sSet] \array{ \underoverset{\bot \phantom{\simeq A_a \simeq A_a}}{ L_! \phantom{\simeq A_a \simeq A_a} }{\longrightarrow} \\ \underoverset{\bot \phantom{\simeq} \bot \phantom{\simeq A_a} }{ L^\ast \simeq C_! \phantom{\simeq A_a} }{\longleftarrow} \\ \underoverset{\phantom{A_a \simeq}\bot \phantom{\simeq} \bot}{ L_\ast \simeq C^\ast \simeq R_! }{\longrightarrow} \\ \underoverset{\phantom{A_a \simeq A_a } \bot}{ \phantom{A_a \simeq } C_\ast \simeq R^\ast }{\longrightarrow} \\ \underoverset{\phantom{A_a \simeq A_a } \phantom{\bot}}{ \phantom{A_a \simeq C_\ast \simeq} R_\ast }{\longrightarrow} } [\mathcal{D}^{op}, sSet]$$

By Example the top four functors in (7) form a Quillen adjoint quadruple (Def. ) on model structures on simplicial presheaves (Prop. ) ending in a right Quillen functor

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \overset{C_\ast \simeq R^\ast}{\longrightarrow} [\mathcal{C}^{op}, sSet_{Qu}]_{inj} \,.$$

But $R^\ast$ here is also a left Quillen functor (as in Example ), and hence this continues by one more Quillen adjoint triple via Example to a Quillen adjoint quintuple of the form

$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$$

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$$

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$$

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$$

Alternatively, we may regard the bottom four functors in (7) as a Quillen adjoint quadruple via example , whose top functor is then the left Quillen functor

$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \overset{ L^\ast }{\longleftarrow} [\mathcal{D}^{op}, sSet_{Qu}]_{proj} \,.$$

But this is also a right Quillen functor (as in Example ) and hence we may continue by one more Quillen adjoint triple upwards (via Example ) to obtain a Quillen adjoint quintuple, now of the form

$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{L_! \phantom{\simeq A_a \simeq A_a}}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$$

$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj} \underoverset {{\longrightarrow}} {\overset{L^\ast \simeq C_! \phantom{\simeq A_a}}{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$$

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{}{\longleftarrow}} {\overset{L_\ast \simeq C^\ast \simeq R_!}{\longrightarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{proj}$$

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \phantom{ A_a \simeq A_a \simeq} R_\ast }{\longrightarrow}} {\overset{ \phantom{A_a \simeq} C_\ast \simeq R^\ast }{\longleftarrow}} {\phantom{\phantom{A}{}_{Qu}}\bot_{Qu}\phantom{A}} [\mathcal{D}^{op}, sSet_{Qu}]_{inj}$$

We now discuss how to extract derived adjoint modalities from systems of Quillen adjoint triples. First we consider some preliminary lemmas.

Lemma

(derived adjunction units of Quillen adjoint triple)

Consider a Quillen adjoint triple (Def. )

$$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

$$\mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

such that the two model structures $\mathcal{C}_1$ and $\mathcal{C}_2$ on the category $\mathcal{C}$ share the same class of weak equivalences.

Then:

1. the derived adjunction unit of $(L \dashv C)$ in $\mathcal{C}_1$ (Def. ) differs only by a weak equivalence from the plain adjunction unit (Def. ).

2. the derived adjunction counit of $(C \dashv R)$ (Def. ) differs only by a weak equivalence form the plain adjunction counit (Def. ).

Proof

By Def. , the derived adjunction unit is on cofibrant objects $c \in \mathcal{C}_1$ given by

$$c \overset{\eta_c}{\longrightarrow} C L (c) \overset{ C(j_{L(c)}) }{\longrightarrow} C P L (c)$$

Here the fibrant resolution-morphism $j_{P(c)}$ is an acyclic cofibration in $\mathcal{D}$. Since $C$ is also a left Quillen functor $\mathcal{D} \overset{C}{\to} \mathcal{C}_2$, the comparison morphism $C(j_{L(c)})$ is an acyclic cofibration in $\mathcal{C}_2$, hence in particular a weak equivalence in $\mathcal{C}_2$ and therefore, by assumption, also in $\mathcal{C}_1$.

The derived adjunction counit of the second adjunction is

$$C Q R (c) \overset{ C(p_{R(c)}) }{\longrightarrow} C R (c) \overset{ \epsilon_c }{\longrightarrow} c$$

Here the cofibrant resolution-morphisms $p_{R(c)}$ is an acyclic fibration in $\mathcal{D}$. Since $C$ is also a right Quillen functor $\mathcal{D} \overset{C}{\to} \mathcal{C}_1$, the comparison morphism $C(p_{R(c)})$ is an acyclic fibration in $\mathcal{C}_1$, hence in particular a weak equivalence there, hence, by assumption, also a weak equivalence in $\mathcal{C}_2$.

Lemma

(fully faithful functors in Quillen adjoint triple)

Consider a Quillen adjoint triple (Def. )

$$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L}{\longrightarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

$$\mathcal{C}_2 \underoverset {\underset{R}{\longrightarrow}} {\overset{C}{\longleftarrow}} {{}_{\phantom{Qu}}\bot_{Qu}} \mathcal{D}$$

If $L$ and $R$ are fully faithful functors (necessarily jointly, by Prop. ), then so are their derived functors $\mathbb{L}L$ and $\mathbb{R}R$ (Prop. ).

Proof

We discuss that $R$ being fully faithful implies that $\mathbb{R}R$ is fully faithful. Since also the derived functors form an adjoint triple (by Prop. ), this will imply the claim also for $L$ and $\mathbb{L}L$, by Prop. .

By Lemma the derived adjunction counit of $C \dashv R$ is, up to weak equivalence, the ordinary adjunction counit. But the latter is an isomorphism, since $R$ is fully faithful (by this Prop.). In summary this means that the derived adjunction unit of $(C \dashv R)$ is a weak equivalence, hence that its image in the homotopy category is an isomorphism. But the latter is the ordinary adjunction unit of $\mathbb{L}C \dashv \mathbb{R}R$ (by this Prop.), and hence the claim follows again by that Prop..

Lemma

(fully faithful functors in Quillen adjoint quadruple)

Given a Quillen adjoint quadruple (Def. )

$$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$$

$$\mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$$

$$\mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2$$

if any of the four functors is fully faithful functor, then so is its derived functor.

Proof

Observing that each of the four functors is either the leftmost or the rightmost adjoint in the top or the bottom adjoint triple within the adjoint quadruple, the claim follows by Lemma .

In summary:

Proposition

(derived adjoint modalities from fully faithful Quillen adjoint quadruples)

Given a Quillen adjoint quadruple (Def. )

$$\mathcal{C}_1 \underoverset {{\longleftarrow}} {\overset{L \phantom{= A}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$$

$$\mathcal{C}_2 \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_1$$

$$\mathcal{C}_2 \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} \mathcal{D}_2$$

then the corresponding derived functors form an adjoint quadruple

$$Ho(\mathcal{C}) \array{ \underoverset{\phantom{AA}\bot\phantom{AA}}{ \mathbb{L}L }{\longrightarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{ \mathbb{L}C \simeq \mathbb{R}C \simeq \mathbb{L}L' }{\longleftarrow} \\ \underoverset{\phantom{AA}\bot\phantom{AA}}{ \mathbb{R}R \simeq \mathbb{L}C' \simeq \mathbb{R}C' }{\longrightarrow} \\ \underoverset{\phantom{AA}\phantom{\bot}\phantom{AA}}{ \mathbb{R}R' }{\longleftarrow} } Ho(\mathcal{D})$$

Moreover, if one of the functors in the Quillen adjoint quadruple is a fully faithful functor, then so is the corresponding derived functor.

Hence if the original adjoint quadruple induces an adjoint modality on $\mathcal{C}$ (Def. )

$$\bigcirc \dashv \Box \dashv \lozenge$$

or on $\mathcal{D}$

$$\Box \dashv \bigcirc \dashv \triangle$$

then so do the corresponding derived functors on the homotopy categories, respectively.

Proof

The existence of the derived adjoint quadruple followy by Prop. and by uniqueness of adjoints (this Prop.).

The statement about fully faithful functors is Lemma . The reformulation in terms of adjoint modalities is by this Prop.

$\,$

$\infty$-Toposes

The characterization of sheaf toposes as the left exact reflective localizations of presheaf toposes (Prop. ) now has an immediate generalization from the realm of locally presentable categories to that of combinatorial model categories and their corresponding locally presentable (∞,1)-categories (Def. ): This yields concept of model toposes and (∞,1)-toposes (Def. below).

$\,$

Definition

(model topos and (∞,1)-topos)

A combinatorial model category (Def. ) is a model topos if it has a presentation via Dugger's theorem (Prop. )

(8)$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj,S} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \;\; \in CombModCat$$

such that the left derived functor $\mathbb{L}id$ preserves finite homotopy limits.

We denote the image of such a combinatorial model category under the localization functor $\gamma$ in Ho(CombModCat) (Def. ) by

$$Sh_\infty(\mathcal{C}) \;\coloneqq\; \gamma([\mathcal{C}^{op}, sSet_{Qu}]_{proj,S}) \;\in\; Ho(CombModCat)$$

and call this an (∞,1)-topos over a site $\mathcal{C}$. Moreover, we denote the image of the defining Quillen adjunction (8) in Ho(CombModCat) by

$$Sh_\infty(\mathcal{C}) \underoverset {\underset{\phantom{AAAA}}{\hookrightarrow}} {\overset{lex}{\longleftarrow}} {\bot} PSh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat) \,.$$

The following construction generalizes the Cech groupoid (Example ) as groupoids are generalized to Kan complexes (Def. ):

Example

(Cech nerve)

Let $\mathcal{C}$ be a site (Def. ). Then for every object $X \in \mathcal{C}$ and every covering $\{U_i \overset{\iota_i}{\to}X\}$ there is a simplicial presheaf (Example )

$$C(\{U_i\}) \;\in\; [\mathcal{C}^{op}, sSet]$$

which in degree $k$ is given by the disjoint union of the $k$-fold fiber products of presheaves over $y(X)$ of the patches $y(U_i) \in [\mathcal{C}^{op}, Set]$ of the cover, regarded as presheaves under the Yoneda embedding (Prop. )

$$C(\{U_i\})_k \;\coloneqq\; \underset{i_1, \cdots, i_k}{\coprod} y(U_{i_1}) \times_{y(X)} y(U_{i_2}) \times_{y(X)} \cdots \times_{y(X)} y(U_{i_k}) \,.$$

The face maps are the evident projection morphisms, and the degeneracy maps the evident diagonal morphisms.

This is called the Cech nerve of the given cover.

By the definition of fiber products there is a canonical morphism of simplicial presheaves from the Cech nerve to $y(X)$

(9)$$C(\{U_i\}) \overset{p_{\{U_i\}}}{\longrightarrow} y(X)$$

We call this the Cech nerve projection.

More generally, for

$$\mathbf{Y} \overset{f}{\longrightarrow} \mathbf{X} \;\;\; \in [\mathcal{C}^{op}, Set]$$

any morphism of presheaves, there is the correspnding Cech nerve simplicial presheaf

$$C(f) \in [\mathcal{C}^{op}, sSet]$$

which in degree $k$ is the $k$-fold fiber product of $f$ with itself:

$$C(f)_k \;\coloneqq\; \underset{ k \; \text{factors} }{ \underbrace{ \mathbf{Y} \times_{\mathbf{X}} \cdots \times_{\mathbf{X}} \mathbf{Y} }} \,.$$

The following is the generalization of Prop. , saying that Cech nerves are codescent-objects for (∞,1)-sheaves:

Proposition

(topological localization)

Let $\mathcal{C}$ be a site (Def. ) and let

$$S \;\coloneqq\; \big\{ C(\{U_i\}) \overset{p_{\{U_i\}}}{\longrightarrow} y(X) \;\vert\; \{U_i \overset{\iota_i}{\to} X\}_i \; \text{covering} \big\} \subset Mor_{[\mathcal{C}^{op}, sSet]}$$

be the set of projections (9) out of the Cech nerves (Example ) for coverings of all objects in the site, as a subset of the class of morphisms of simplicial presheaves over $\mathcal{C}$ (Example ).

Then the left Bousfield localization (Def. ) of the projective or injective model structure on simplicial presheaves (Prop. ), to be denoted

$$[\mathcal{C}^{op}, sSet_{Qu}]_{{proj/inj} \atop {loc}} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj}$$

and to be called the (projective or injective) local model structure on simplicial presheaves, is left exact, in that it exhibits a model topos according to Def. , hence in that its image in Ho(CombModCat) is an (∞,1)-topos

$$Sh_\infty(\mathcal{C}) \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{lex}{\longleftarrow}} {\bot} PSh_\infty(\mathcal{C}) \,.$$

Proposition

(Quillen equivalence between projective and injective topological localization)

Let $\mathcal{C}$ be a site (Def. ) and let

$$S \;\coloneqq\; \big\{ C(\{U_i\}) \overset{p_{\{U_i\}}}{\longrightarrow} y(X) \;\vert\; \{U_i \overset{\iota_i}{\to} X\}_i \; \text{covering} \big\} \subset Mor_{[\mathcal{C}^{op}, sSet]}$$

be the set of projections (9) out of the Cech nerves (Example ) for coverings of all objects in the site, as a subset of the class of morphisms of simplicial presheaves over $\mathcal{C}$ (Example ).

If each Cech nerve $C(\{U_i\})$ is already a cofibrant object in the projective model structure on simplicial presheaves (prop. ) then the identity functors constitute a Quillen equivalence (Def. ) between the corresponding topological localizations (Def. ) of the projective and the injective model structure on simplicial presheaves:

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj \atop loc} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc}$$

Proof

First to see that we have a Quillen adjunction (Def. ): By Prop. this is the case before left Bousfield localization. By the nature of left Bousfield localization, and since the model structures are left proper simplicial model categories (by Prop. ), by Prop. it is sufficient to check that the right Quillen functor preserves fibrant objects. By Prop. this means to check that it preserves $S$-local objects. But since $C(\{U_i\})$ is assumed to be projectively cofibrant, and since injectively fibrant objects are already projectively fibrant, the condition on an injectively local object according to Def. is exactly the same as for a projectively local object.

Now to see that this Quillen adjunction is a Quillen equivalence, it is sufficient to check that the corresponding left/right derived functors induce an equivalence of categories on homotopy categories. By Prop. this is the case before left Bousfield localization. By Prop. it is thus sufficient to check that derived functors (before localization) preserves $S$-local objects. By Prop. for this it is sufficient that the Quillen functors themselves preserve local objects. For the right Quillen functor we have just seen this in the previous paragaraph, for the left Quillen functor it follows analogously.

Example

(homotopy localization at $\mathbb{A}^1$ over the site of $\mathbb{A}^n$s)

Let $\mathcal{C}$ be any site (Def. ), and write $[\mathcal{C}^{op}, sSet_{Qu}]_{proj, loc}$ for its local projective model category of simplicial presheaves (Prop. ).

Assume that $\mathcal{C}$ contains an object $\mathbb{A} \in \mathcal{C}$, such that every other object is a finite product $\mathbb{A}^n \coloneqq \underset{n \; \text{factors}}{\underbrace{\mathbb{A} \times \cdots \times \mathbb{A}}}$, for some $n \in \mathbb{N}$. (In other words, assume that $\mathcal{C}$ is also the syntactic category of Lawvere theory.)

Consider the $\mathbb{A}^1$-homotopy localization (Def. ) of the (∞,1)-sheaf (∞,1)-topos over $\mathcal{C}$ (Prop. )

$$Sh_\infty(\mathcal{C})_{\mathbb{A}} \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} Sh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat)$$

hence the left Bousfield localization of model categories

$$[\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc,\mathbb{A}} \underoverset {\underset{\phantom{AA}id\phantom{AA}}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc} \;\; \in CombModCat$$

at the set of morphisms

$$S \;\coloneqq\; \big\{ \mathbb{A}^n \times \mathbb{A} \overset{p_1}{\longrightarrow} \mathbb{A}^n \big\}$$

(according to Prop. ).

Then this is equivalent (Def. ) to ∞Grpd (Def. ),

$$\infty Grpd \;\simeq\; Sh_\infty(\mathcal{C})_{\mathbb{A}} \underoverset {\underset{\phantom{AA}\iota\phantom{AA}}{\hookrightarrow}} {\overset{L_{\mathbb{A}}}{\longleftarrow}} {\bot} Sh_\infty(\mathcal{C}) \;\; \in Ho(CombModCat)$$

in that the (constant functor $\dashv$ limit)-adjunction (Def. )

(10)$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}} {\bot} sSet_{Qu} \;\;\;\; \in CombModCat$$

is a Quillen equivalence (Def. ).

Proof

First to see that (10) is a Quillen adjunction (Def. ): Since we have a simplicial Quillen adjunction before localization

$$[\mathcal{C}^{op}, sSet_{Qu}]_{inj} \underoverset {\underset{ \underset{\longleftarrow}{\lim} }{\longrightarrow}} {\overset{ \phantom{AA}const\phantom{AA} }{\longleftarrow}} {\bot} sSet_{Qu}$$

(by Example ) and since both model categories here are left proper simplicial model categories (by Prop. and Prop. ), and since left Bousfield localization does not change the class of cofibrations (by Def. ) it is sufficient to show that $\underset{\longleftarrow}{\lim}$ preserves fibrant objects (by Prop. ).

But by assumption $\mathcal{C}$ has a terminal object $\ast = \mathbb{A}^0$ (Def. ), which is hence the initial object of $\mathcal{C}^{op}$, so that the limit operation is given just by evaluation on that object:

$$\underset{\longleftarrow}{\lim} \mathbf{X} \;=\; \mathbf{X}(\mathbb{A}^0) \,.$$

Hence it is sufficient to see that an injectively fibrant simplicial presheaf $\mathbf{X}$ is objectwise a Kan complex. This is indeed the case, by Prop. .

To check that (10) is actually a Quillen equivalence (Def. ), we check that the derived adjunction unit and derived adjunction counit (Def. ) are weak equivalences:

For $X \in sSet$ any simplicial set (necessarily cofibrant), the derived adjunction unit is

$$X \overset{id_X}{\longrightarrow} const(X)(\mathbb{A}^0) \overset{ const(j_X)(\mathbb{A}^0) }{\longrightarrow} const(P X)(\mathbb{A}^0)$$

where $X \overset{j_X}{\longrightarrow} P X$ is a fibrant replacement (Def. ). But $const(-)(\mathbb{A}^0)$ is clearly the identity functor and the plain adjunction unit is the identity morphism, so that this composite is just $j_X$ itself, which is indeed a weak equivalence.

For the other case, let $\mathbf{X} \in [\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc, \mathbb{A}^1}$ be fibrant. This means (by Prop. ) that $\mathbf{X}$ is fibrant in the injective model structure on simplicial presheaves as well as in the local model structure, and is a derived-$\mathbb{A}^1$-local object (Def. ), in that the derived hom-functor out of any $\mathbb{A}^n \times \mathbb{A}^1 \overset{p_1}{\longrightarrow} \mathbb{A}^n$ into $\mathbf{X}$ is a weak homotopy equivalence:

$$\mathbb{R}Hom( p_1 ) \;\colon\; \mathbb{R}Hom( \mathbb{A}^n , \mathbf{X}) \overset{\in W}{\longrightarrow} \mathbb{R}Hom( \mathbb{A}^n \times \mathbb{A}^1 , \mathbf{X})$$

But since $\mathbf{X}$ is fibrant, this derived hom is equivalent to the ordinary hom-functor (Lemma ), and hence with the Yoneda lemma (Prop. ) we have that

$$\mathbf{X}(p_1) \;\colon\; \mathbf{X}(\mathbb{A}^n) \overset{\in W}{\longrightarrow} \mathbf{X}(\mathbb{A}^{n+1})$$

is a weak equivalence, for all $n \in \mathbb{N}$. By induction on $n$ this means that in fact

$$\mathbf{X}(\mathbb{A}^0) \overset{\in W}{\longrightarrow} \mathbf{X}(\mathbb{A}^n)$$

is a weak equivalence for all $n \in \mathbb{N}$. But these are just the components of the adjunction counit

$$const (\mathbf{X}(\mathbb{A}^0)) \underoverset{\in W}{\epsilon}{\longrightarrow} \mathbf{X}$$

which is hence also a weak equivalence. Hence for the derived adjunction counit

$$const (Q \mathbf{X})(\mathbb{A}^0) \overset{const(p_{\mathbf{X}}(\mathbb{A}^0))}{\longrightarrow} const (\mathbf{X}(\mathbb{A}^0)) \underoverset{\in W}{\epsilon}{\longrightarrow} \mathbf{X}$$

to be a weak equivalence, it is now sufficient to see that the value of a cofibrant replacement $p_{\mathbf{X}}$ on $\mathbb{A}^0$ is a weak equivalence. But by definition of the weak equivalences of simplicial presheaves these are objectwise weak equivalences.

Proposition

(Cech nerve-projection of local epimorphism is local weak equivalence)

Let $\mathcal{C}$ be a site (Def. ) and let

$$\mathbf{Y} \overset{\phantom{A}f\phantom{A}}{\longrightarrow} \mathbf{X} \;\;\; \in [\mathcal{C}^{op}, Set]$$

be a local epimorphism (Def. ) in its category of presheaves. Then the corresponding Cech nerve-projection (Def. )

$$C(f) \overset{}{\longrightarrow} \mathbf{X} \;\;\; \in [\mathcal{C}^{op}, sSet_{Qu}]_{proj,loc}$$

is a weak equivalence in the local projective model structure on simplicial presheaves (Prop. ).

(Dugger-Hollander-Saksen 02, corollary A.3)

$\,$

Gros $\infty$-Toposes

We have established above enough higher category theory/homotopy theory that it is now fairly straightforward to generalize the discussion of gros toposes to model toposes/(∞,1)-toposes.

$\,$

Cohesive $\infty$-Toposes

The following is a refinement to homotopy theory of the notion of cohesive topos (Def. ):

Definition

(cohesive model topos)

An (∞,1)-topos $\mathbf{H}$ (Def. ) is called a cohesive (∞,1)-topos if it is presented by a model topos $[\mathcal{C}^{op}, sSet_{Qu}]_{loc}$ (Def. ) which admits a Quillen adjoint quadruple (Def. ) to the classical model category of simplicial sets (Def. ) of the form

$$\array{ [\mathcal{C}^{op}, sSet_{Qu}]_{proj/inj} \array{ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AA}\Pi\phantom{AA}}{\longrightarrow} \\ \underoverset{{}_{\phantom{A}}\bot_{Qu} }{\phantom{A}Disc\phantom{A}}{\hookleftarrow} \\ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AA}\Gamma\phantom{AA}}{\longrightarrow} \\ \overset{\phantom{A}coDisc\phantom{A}}{\hookleftarrow} \\ } sSet_{Qu} }$$

such that

1. $(Disc \dashv \Gamma)$ is a Quillen coreflection (Def. );

2. $(\Gamma \dashv coDisc)$ is a Quillen reflection (Def. );

3. $\Pi$ preserves finite products.

The following is the analog of Example :

Example

(Quillen adjoint quadruple on simplicial presheaves over site with finite products)

Let $\mathcal{C}$ be a small category (Def. ) with finite products (hence with a terminal object $\ast \in \mathcal{C}$ and for any two objects $X,Y \in \mathcal{C}$ their Cartesian product $X \times Y \in \mathcal{C}$). By Example the terminal object is witnessed by an adjunction

(11)$$\ast \underoverset {\underset{}{\hookrightarrow}} {\overset{\phantom{AAAA}}{\longleftarrow}} {\bot} \mathcal{C}$$

Consider the category of simplicial presheaves $[\mathcal{C}^{op}, sSet]$ (Example ) with its projective and injective model structure on simplicial presheaves (Prop. ).

Then Kan extension (Prop. ) enriched over sSet (Example ) along the adjoint pair (11) yields a simplicial Quillen adjoint quadruple (Def. )

(12)$$[\mathcal{C}^{op}, Set_{Qu}]_{proj/inj} \array{ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \underoverset{{}_{\phantom{A}}\bot_{Qu}}{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } sSet_{Qu}$$

such that:

1. the functor $\Gamma$ sends a simplicial presheaf $\mathbf{Y}$ to its simplicial set of global sections, which here is its value on the terminal object:

   (13)$$\begin{aligned} \Gamma \mathbf{Y} & = \underset{\underset{\mathcal{C}}{\longleftarrow}}{\lim} \mathbf{Y} \\ & \simeq \mathbf{Y}(\ast) \end{aligned}$$

2. $(Disc \dashv \Gamma)$ is a Quillen coreflection (Def. )

3. $(\Gamma \dashv coDisc)$ is a Quillen reflection (Def. );

4. $\Pi$ preserves finite products:

Hence the category of simplicial presheaves over a small category with finite products is a cohesive (∞,1)-topos (Def. ).

Proof

The Quillen adjoint quadruple follows as the special case of Example applied to the adjoint pair

$$\ast \underoverset {\underset{}{\hookrightarrow}} {\overset{}{\longleftarrow}} {\bot} \mathcal{C}$$

given by inclusion of the terminal object (Example ).

Since the plain adjoint quadruple has $(\Pi \dashv \Disc)$ a reflective subcategory inclusion and $(Disc \dashv \Gamma)$ a coreflective subcategory inclusion (Example ) the Quillen (co-)reflection follows by Prop.

The following is a refinement to homotopy theory of the notion of cohesive site (Def. ):

Definition

(∞-cohesive site)

We call a site $\mathcal{C}$ (Def. ) ∞-cohesive if the following conditions are satisfied:

1. The category $\mathcal{C}$ has finite products;

2. For every covering family $\{U_i \to X\}_i$ in the given coverage on $\mathcal{C}$, the induced Cech nerve simplicial presheaf (Example ) $C(\{U_i\}) \in [\mathcal{C}^{op}, sSet]$ satisfies the following conditions

   1. $C(\{U_i\})$ is a cofibrant object in the projective model structure on simplicial presheaves $[\mathcal{C}^{op}, sSet_{Qu}]_{proj}$ (Prop. )

   2. The simplicial set obtained as the degreewise colimit over the Cech nerve is weakly homotopy equivalent to the point

      $$\underset{ \underset{ \mathcal{C}^{op} }{\longrightarrow} }{\lim} C(\{U_i\}) \simeq \ast$$

   3. The simplicial set obtained at the degreewise limit over the Cech nerve is weakly homotopy equivalent to the underlying set of points of $X$:

      $$\underset{\underset{\mathcal{C}^{op}}{\longleftarrow}}{C(\{U_i\})} \simeq Hom_{\mathcal{C}}(\ast, X) \,.$$

The following is the analog of Prop. :

Proposition

(model topos over ∞-cohesive site is cohesive model topos)

Let $\mathcal{C}$ be an ∞-cohesive site (Def. ). Then the (∞,1)-topos (Def. ) over it, obtained by topological localization (Prop. ) is a cohesive (∞,1)-topos (Def. ).

Proof

By Example we have the required Quillen adjoint quadruple on the projective model structure on simplicial presheaves, i.e. before left Bousfield localization at the Cech nerve projections

$$[\mathcal{C}^{op}, Set_{Qu}] \array{ \overset{\phantom{AAA} \Pi \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } sSet_{Qu}$$

Hence it remains to see that these Quillen adjunctions pass to the local model structures $[\mathcal{C}^{op}, Set_{Qu}]_{proj/inj, loc}$ from Prop. , and that $Disc$ and $coDisc$ then still participate in Quillen (co-)reflections.

By Prop. and Prop. all model structures involved are left proper simplicial model categories, and hence we may appeal to Prop. for recognition of the required Quillen adjunctions. Since, moreover, left Bousfield localization does not change the class of cofibrations (Def. ), this means that we are reduced to checking that all right Quillen functors in the above global Quillen adjoint quadruple preserve fibrant objects with respect to the local model structure.

For the Quillen adjunctions

$$(\Pi \dashv Disc), (\Gamma \dashv coDisc) \;\colon\; [\mathcal{C}^{op}, sSet_{Qu}]_{proj} \leftrightarrow sSet_{Qu}$$

this means to check that for every Kan complex $S \in sSet$ the simplicial presheaves $Disc(S)$ and $coDisc(S)$ are derived-local objects (Def. , Prop. ) with respect to the Cech nerve projections. Since $Disc$ and $coDisc$ are right Quillen functors with respect to the global model projective model structure, $Disc(S)$ and $coDisc(S)$ are globally projectively fibrant simplicial presheaves. Since, moreover, $C(\{U_i\})$ is projectively cofibrant by assumption, and since the representables $X \in \mathcal{C}$ are projectively cofibrant by Prop. , the value of the derived hom-functor reduces to that of the ordinary enriched hom-functor (Def. ), and hence the condition is that

$$\array{ [\mathcal{C}^{op}, sSet]( X, Disc(S) ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( C(\{U_i\}), Disc(S) ) \\ [\mathcal{C}^{op}, sSet]( X, coDisc(S) ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( C(\{U_i\}), coDisc(S) ) }$$

are weak equivalences. But now by the ordinary adjunction hom-isomorphism (?), these are identified with

$$\array{ [\mathcal{C}^{op}, sSet]( \underset{\longrightarrow}{\lim} X, S ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( \underset{\longrightarrow}{\lim}C(\{U_i\}), S ) \\ [\mathcal{C}^{op}, sSet]( \underset{\longleftarrow}{\lim}X, S ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( \underset{\longleftarrow}{\lim}C(\{U_i\}), S ) }$$

Since the colimit of a representable is the singleton (Lemma ) and since the limit over the opposite of a category with terming object is evaluation at that object, this in turn is equivalent to

$$\array{ [\mathcal{C}^{op}, sSet]( \ast, S ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( \underset{\longrightarrow}{\lim}C(\{U_i\}), S ) \\ [\mathcal{C}^{op}, sSet]( Hom_{\mathcal{C}}(\ast, X), S ) &\overset{\in W}{\longrightarrow}& [\mathcal{C}^{op}, sSet]( \underset{\longleftarrow}{\lim}C(\{U_i\}), S ) }$$

Here we recognize the internal hom in simplicial sets from the weak equivalences of the definition of an ∞-cohesive site (Def. ), which necessarily go between cofibrant simplicial sets, into a fibrant simplicial set $S$. Hence this is the derived hom-functor (Def. ) in the classical model structure on simplicial sets. Since the latter is a simplicial model category (Def. ) by Prop. , these morphisms are indeed weak equivalences of simplicial sets.

This establishes that $(\Pi \dashv Disc)$ and $(\Gamma \dashv \coDisc)$ descent to Quillen adjunctions on the local model structure. Finally, it is immediate that $\Gamma$ preserves fibrant objects, and hence also $(Disc \dashv \Gamma)$ passes to the local model structure.

The following is the analog in homotopy theory of the cohesive adjoint modalities from Def. :

Definition

(adjoint triple of derived adjoint modal operators on homotopy category of cohesive model topos)

Given a cohesive model topos (Def. ), its adjoint quadruple (Remark ) of derived functor between homotopy categorues (via Prop. )

(14)$$\Pi \dashv Disc \dashv \Gamma \dashv coDisc \;\;\colon\;\; Ho([\mathcal{C}^{op}, sSet_{Qu}]_{loc}) \array{ \overset{\phantom{AAA} \Pi_0 \phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AA} Disc \phantom{AA} }{\hookleftarrow} \\ \overset{\phantom{AAA} \Gamma \phantom{AAA} }{\longrightarrow} \\ \overset{\phantom{AA} coDisc \phantom{AA} }{\hookleftarrow} } Ho(sSet)$$

induce, by composition of functors, an adjoint triple (Remark ) of adjoint modalities (via Prop. ):

$$ʃ \dashv \flat \dashv \sharp \;\;\colon\;\; Ho([\mathcal{C}^{op}, sSet_{Qu}]_{loc}) \array{ \overset{ ʃ \;\coloneqq\; Disc \circ \Pi_0 }{\hookleftarrow} \\ \overset{\flat \;\coloneqq\; Disc \circ \Gamma }{\longrightarrow} \\ \overset{ \sharp \;\coloneqq\; coDisc\circ \Gamma }{\hookleftarrow} } Ho([\mathcal{C}^{op}, sSet_{Qu}]_{loc}) \,.$$

Since $Disc$ and $coDisc$ are fully faithful functors by assumption, these are (co-)modal operators (Def. ), (by Prop. and Prop. ).

We pronounce these as follows:

$\phantom{A}$ shape modality $\phantom{A}$	$\phantom{A}$ flat modality $\phantom{A}$	$\phantom{A}$ sharp modality $\phantom{A}$
$\phantom{A}$ $ʃ \;\coloneqq\; Disc \circ \Pi_0$ $\phantom{A}$	$\phantom{A}$ $\flat \;\coloneqq\; Disc \circ \Gamma$ $\phantom{A}$	$\phantom{A}$ $\sharp \;\coloneqq\; coDisc \circ \Gamma$ $\phantom{A}$

and we refer to the corresponding modal objects (Def. ) as follows:

• a flat-comodal object

  $$\flat X \underoverset{\simeq}{\phantom{A}\epsilon^\flat_X \phantom{A}}{\longrightarrow} X$$

  is called a discrete object;

• a sharp-modal object

  $$X \underoverset{\simeq}{\phantom{A}\eta^\sharp_X\phantom{A}}{\longrightarrow} \sharp X$$

  is called a codiscrete object;

• a sharp-submodal object

  $$X \underoverset{mono}{\phantom{A}\eta^\sharp_X\phantom{A}}{\longrightarrow} \sharp X$$

  is a concrete object.

$\,$

Elastic $\infty$-Toposes

The following is a refinement to homotopy theory of the notion of elastic topos (Def. ):

Definition

(elastic model topos

Given a cohesive model topos $[\mathcal{C}_{red}^{op}, sSet_{Qu}]_{{proj/inj} \atop loc}$ (Def. ), a differentially cohesive or elastic model topos over it is another cohesive model topos $[\mathcal{C}_{red}^{op}, sSet_{Qu}]_{{proj/inj} \atop loc}$ equipped with a system of Quillen adjoint quadruples (Def. ) of the form

$$\array{ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \\ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{ \Pi_{inf} }{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \\ sSet_{Qu} \underoverset {{\longleftarrow}} {\overset{Disc_{red}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longleftarrow}} {\overset{ Disc_{inf} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {\underset{coDisc_{red}}{\longrightarrow}} {\overset{\Gamma_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {\phantom{\underset{id}{\longrightarrow}}} {\overset{id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu}} } \phantom{ [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} } \underoverset {\phantom{{\longleftarrow}}} {\overset{ \Gamma_{inf} }{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{[\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc}} }$$

such that

1. $(\iota_{inf} \dashv \Pi_{inf})$ is a Quillen coreflection (Def. );

2. $(\Pi_{inf} \dashv Disc_{inf})$ is a Quillen reflection (Def. ).

Definition

(∞-elastic site)

For $\mathcal{C}_{red}$ an ∞-cohesive site (Def. ), an infinitesimal neighbourhood site of $\mathcal{C}_{red}$ is a coreflective subcategory-inclusion into another ∞-cohesive site $\mathcal{C}$

$$\mathcal{C}_{red} \underoverset {\underset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow}} {\overset{\iota_{inf}}{\hookrightarrow}} {\bot} \mathcal{C}$$

such that

1. both $\iota_{inf}$ and $\Pi_{inf}$ send covers to covers;

2. the left Kan extension of $\iota_{inf}$ preserves fiber products $y(U_i) \times_{y(X)} y(u_j)$ of morphisms in a covering $\{U_i \overset{\iota_i}{\to} X\}$;

3. if $\{ U_i \overset{\iota_i}{\to} X \}$ is a covering family in $\mathcal{C}_{red}$, and $p(\widehat X) \longrightarrow X$ is any morphism in $\mathcal{C}_{red}$, then there is a covering familiy $\{ \widehat U_i \overset{\widehat\iota_j}{\to} \widehat X \}$ such that for all $i$ there is a $j$ and a commuting square of the form

   (15)$$\array{ \Pi_{inf}(\widehat U_j) &\longrightarrow& U_i \\ {}^{\mathllap{ \Pi_{inf}(\widehat\iota_j) }}\Big\downarrow && \Big\downarrow{}^{\mathrlap{ \iota_i }} \\ \Pi_{inf}(\widehat X) &\longrightarrow& X }$$

We also call this an ∞-elastic site, for short.

Proposition

(model topos over ∞-elastic site is elastic model topos)

Let

$$\mathcal{C}_{red} \underoverset {\underset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow}} {\overset{\iota_{inf}}{\hookrightarrow}} {\bot} \mathcal{C}$$

be an ∞-elastic site (Def. ). Then Kan extension (Prop. ) enriched over sSet (Example ) induces on the corresponding cohesive model toposes (Prop. ) the structure of an elastic model topos (Def. ).

Proof

By Example we have a Quillen adjoint quadruple for the global projective model structure on simplicial presheaves of the form

$$[\mathcal{C}^{op}_{red}, sSet_{Qu}]_{proj} \underoverset {{\longleftarrow}} {\overset{ \iota_{inf} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$$

$$[\mathcal{C}^{op}_{red}, sSet_{Qu}]_{inj} \underoverset {{\longrightarrow}} {\overset{C = L'}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj}$$

$$[\mathcal{C}^{op}_{red}, sSet_{Qu}]_{inj} \underoverset {\underset{\phantom{A=} R'}{\longleftarrow}} {\overset{R = C'}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj}$$

Here we denote left Kan extension along a functor by the same symbol as that functor, which is consistent by Prop. .

By Prop. all model categories appearing here are left proper simplicial model categories, and by Def. left Bousfield localization retains the class of cofibrations. Therefore Prop. says that to see that this is also a Quillen adjoint quadruple for the local model structure on simplicial presheaves (Prop. ) it is sufficient that, for each Quillen adjunction, the right adjoint preserves fibrant objects, hence Cech-local objects (Def. ).

For each right adjoint $R$ here this means to consider any covering $\{U_i \overset{}{\to} X\}$ (either in $\mathcal{C}_{red}$ or in $\mathcal{C}$) with induced Cech nerve $C(\{U_i\})$ (Example ) and to check that for a fibrant object $\mathbf{X}$ in the global projective/injective model structure on simplicial presheaves, that

$$[X, R\mathbf{X}] \longrightarrow [ C(\{U_i\}), R\mathbf{X} ]$$

is a weak equivalence. Notice that this is indeed already the image under the correct derived hom-functor, Def. , since both sites are assumed to be ∞-cohesive sites (Def. ), which means in particular that $C(\{U_i\})$ is projectively cofibrant, and hence also injectively cofibrant, by Prop. .

Now by the enriched adjunction-isomorphism (?) this means equivalently that

(16)$$[L X, \mathbf{X}] \longrightarrow [ L C(\{U_i\}), \mathbf{X} ]$$

is a weak equivalence. This we now check in each of the three cases:

For the case $(\iota_{inf} \dashv \Pi_{inf})$ we have that

$$\iota_{inf} C(\{U_i\}) \simeq C(\{\iota_{inf} U_i\})$$

by the assumption that $\iota_{inf}$ preserves fiber products of Yoneda embedding-images of morphisms in a covering. Moreover, by the assumption that $\iota_{inf}$ preserves covering-families, $C(\{\iota_{inf} U_i\})$ is itself the Cech nerve of a covering family, and hence (16) is a weak equivalence since $\mathbf{X}$ is assumed to be a local object.

The same argument directly applies also to $(\Pi_{inf} \dashv Disc_{inf})$, where now the respect of $\Pi_{inf}$ for fiber products follows already from the fact that this is a right adjoint (since right adjoints preserve limits, Prop. ).

In the same way, for $(Disc_{inf} \dashv \Gamma_{inf})$ we need to check that $[ C(\{Disc_{inf}U_i\}) \to Disc_{inf} X, \mathbf{X} ]$ is a weak equivalence. Now $Disc_{inf}$ is no longer a left Kan extension, hence $Disc_{inf}(U_i) \to Disc_{inf}(X)$ is no longer a morphism of representable presheaves. But the third assumption (15) on an $\infty$-elastic site manifestly means, under the adjunction isomorphism (?) for $(Pi_{inf} \dashv Disc_{inf})$ that $Disc_{inf}(U_i) \to Disc_{inf}(X)$ is a local epimorphism (Def. ). Therefore Prop. implies that

$$C(\{Disc_{inf} U_i\}) \to Disc_{inf} X$$

is a weak equivalence. With this, the fact (Prop. with Prop. ) that $[\mathcal{C}^{op}, sSet_{Qu}]_{inj, loc}$ is a simplicial model category (Def. ) implies that $[C(\{Disc_{inf} U_i\}) \to Disc_{inf} X, \mathbf{X}]$ is a weak equivalence.

The following is a refinement to homotopy theory of the adjoint modalities on an elastic topos from Def. :

Definition

(derived adjoint modalities on elastic model topos)

Given an elastic model topos (def. ), composition composition of the derived functors (Prop. ) yields via Prop. and Prop. , the following adjoint modalities (Def. ) on the homotopy category (Def. )

$$\Re \dashv \Im \dashv \& \;\;\colon\;\; Ho([\mathcal{C}^{op}, sSet]_{loc}) \array{ \overset{ \Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf} }{\longleftarrow} \\ \overset{\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf} }{\longrightarrow} \\ \overset{ \& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf} }{\longleftarrow} } Ho([\mathcal{C}^{op}, sSet]_{loc}) \,.$$

Since $\iota_{inf}$ and $Disc_{inf}$ are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. ^{op}, sSet]_{loc}) } and Prop. ).

We pronounce these as follows:

$\phantom{A}$ reduction modality $\phantom{A}$	$\phantom{A}$ infinitesimal shape modality $\phantom{A}$	$\phantom{A}$ infinitesimal flat modality $\phantom{A}$
$\phantom{A}$ $\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf}$ $\phantom{A}$	$\phantom{A}$ $\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf}$ $\phantom{A}$	$\phantom{A}$ $\& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf}$ $\phantom{A}$

and we refer to the corresponding modal objects (Def. ) as follows:

• a reduction-comodal object

  $$\Re X \underoverset{\simeq}{\epsilon^\Re_X}{\longrightarrow} X$$

  is called a reduced object;

• an infinitesimal shape-modal object

  $$X \underoverset{\simeq}{\phantom{A}\eta^\Im_X\phantom{A}}{\longrightarrow} \Im X$$

  is called a coreduced object.

Proposition

(progression of derived adjoint modalities on elastic model topos)

Let $[\mathcal{C}^{op}, sSet]_{{proj/inj} \atop loc}$ be an elastic model topos (Def. ) and consider the corresponding derived adjoint modalities which it inherits

1. for being a cohesive topos, from Def. ,

2. for being an elastic topos, from Def. :

$\phantom{A}$ shape modality $\phantom{A}$	$\phantom{A}$ flat modality $\phantom{A}$	$\phantom{A}$ sharp modality $\phantom{A}$
$\phantom{A}$ $ʃ \;\coloneqq\; Disc \circ \Pi$ $\phantom{A}$	$\phantom{A}$ $\flat \;\coloneqq\; Disc \circ \Gamma$ $\phantom{A}$	$\phantom{A}$ $\sharp \;\coloneqq\; coDisc \circ \Gamma$ $\phantom{A}$
$\phantom{A}$ reduction modality $\phantom{A}$	$\phantom{A}$ infinitesimal shape modality $\phantom{A}$	$\phantom{A}$ infinitesimal flat modality $\phantom{A}$
$\phantom{A}$ $\Re \;\coloneqq\; \iota_{inf} \circ \Pi_{inf}$ $\phantom{A}$	$\phantom{A}$ $\Im \;\coloneqq\; Disc_{inf} \circ \Pi_{inf}$ $\phantom{A}$	$\phantom{A}$ $\& \;\coloneqq\; Disc_{inf} \circ \Gamma_{inf}$ $\phantom{A}$

Then these arrange into the following progression, via the preorder on modalities from Def.

$$\array{ \Re &\dashv& \Im &\dashv& \& \\ && \vee && \vee \\ && ʃ &\dashv& \flat &\dashv& \sharp \\ && && \vee && \vee \\ && && \emptyset &\dashv& \ast }$$

where we display also the bottom adjoint modality $\emptyset \dashv \ast$ (Example ), for completeness.

Proof

This is just as in Prop. .

$\,$

Solid $\infty$-Toposes

The following is a refinement to homotopy theory of the notion of solid topos (Def. ):

Definition

(solid model topos)

Given an elastic model topos $[\mathcal{C}^{op}_{inf}, sSet_{Qu}]_{{proj/inj} \atop loc}$ (Def. ) a solid model topos over it is another elastic model topos $[\mathcal{C}^{op}, sSet_{Qu}]_{{proj/inj} \atop loc}$ and a system of Quillen adjoint quadruples (Def. ) as follows

$$\array{ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{even}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc} \\ \phantom{ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {\underset{id}{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{inf}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{\iota_{sup}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {{\longrightarrow}} {\overset{\Pi_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{ \Pi_{inf} }{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc} \underoverset {{\longleftarrow}} {\overset{id}{\longrightarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{\Pi_{sup}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {{\longleftarrow}} {\overset{Disc_{red}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longleftarrow}} {\overset{ Disc_{inf} }{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longrightarrow}} {\overset{\;\;id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu} } [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {{\longleftarrow}} {\overset{\;Disc_{sup}}{\longrightarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \\ sSet_{Qu} \underoverset {\underset{coDisc_{red}}{\longrightarrow}} {\overset{\Gamma_{red}}{\longleftarrow}} {\phantom{{}_{Qu}}\bot_{Qu}} [\mathcal{C}_{red}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} \underoverset {\phantom{\underset{id}{\longrightarrow}}} {\overset{id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu}} } \phantom{ [\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc} } \underoverset {\phantom{{\longleftarrow}}} {\overset{ \Gamma_{inf} }{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{[\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{inj\phantom{r} \atop loc}} \underoverset {\underset{}{\phantom{\longrightarrow}}} {\overset{id}{\phantom{\longleftarrow}}} {\phantom{\phantom{{}_{Qu}}\simeq_{Qu} }} \phantom{[\mathcal{C}_{inf}^{op}, sSet_{Qu}]_{proj \atop loc}} \underoverset {\phantom{{\longleftarrow}}} {\overset{\Gamma_{sup}}{\phantom{\longrightarrow}}} {\phantom{\phantom{{}_{Qu}}\bot_{Qu}}} \phantom{[\mathcal{C}^{op}, sSet_{Qu}]_{proj \atop loc}} }$$

such that

1. $(even \dashv \iota_{sup})$ is a Quillen reflection (def. );

2. $(\iota_{sup} \dashv \Pi_{sup})$ is a Quillen coreflection.

Definition

(∞-solid site)

For $\mathcal{C}_{red} \underoverset {\underset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow}} {\overset{\iota_{inf}}{\hookrightarrow}} {\bot} \mathcal{C}_{inf}$ an ∞-elastic site (Def. ) over an ∞-cohesive site (Def. ), a super-infinitesimal neighbourhood site is a reflective/coreflective subcategory-inclusion into another ∞-elastic site $\mathcal{C}_{red} \underoverset {\underset{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow}} {\overset{\iota_{inf}}{\hookrightarrow}} {\bot} \mathcal{C}$

$$\ast \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \phantom{\underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow}} \\ \underoverset{\bot}{\phantom{AA}\Pi\phantom{AA}}{\longleftarrow} \\ \underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow} } \mathcal{C}_{red} \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } \mathcal{C}_{inf} \array{ \underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}\iota_{sup}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{sup}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } \mathcal{C}_{sup}$$

such that

1. all of $even$, $\iota_{sup}$ and $\Pi_{inf}$ send covers to covers;

2. the left Kan extension of $even$ preserves fiber products $y(U_i) \times_{y(X)} y(u_j)$ of morphisms in a covering $\{U_i \overset{\iota_i}{\to} X\}$;

Proposition

(model topos over ∞-solid site is solid model topos)

Let

$$\ast \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \phantom{\underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow}} \\ \underoverset{\bot}{\phantom{AA}\Pi\phantom{AA}}{\longleftarrow} \\ \underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow} } \mathcal{C}_{red} \array{ \phantom{\underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow}} \\ \underoverset{\bot}{\phantom{AA}\iota_{inf}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{inf}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } \mathcal{C}_{inf} \array{ \underoverset{\bot}{\phantom{AA}even\phantom{AA}}{\longleftarrow} \\ \underoverset{\bot}{\phantom{AA}\iota_{sup}\phantom{AA}}{\hookrightarrow} \\ \underoverset{}{\phantom{AA}\Pi_{sup}\phantom{AA}}{\longleftarrow} \\ \phantom{\underoverset{}{\phantom{A}Disc\phantom{A}}{\hookrightarrow}} } \mathcal{C}_{sup}$$

be an ∞-solid site (Def. ). Then Kan extension (Prop. ) enriched over sSet (Example ) induces on the corresponding elastic model toposes (Prop. ) the structure of a solid model topos (Def. ).

The following is a refinement to homotopy theory of the modal operators on a solid topos from Def. :

Definition

(derived adjoint modalities on solid model topos)

Given a solid model topos $[\mathcal{C}^{op}, sSet_{Qu}]_{{proj/inj} \atop loc}$ (Def. ), composition of derived functors via Prop. and Prop. , the following adjoint modalities (Def. )

$$\rightrightarrows \;\dashv\; \rightsquigarrow \;\dashv\; Rh \;\;\colon\;\; \mathbf{H} \array{ \overset{ \rightrightarrows \;\coloneqq\; \iota_{sup} \circ even }{\longleftarrow} \\ \overset{\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup} }{\longrightarrow} \\ \overset{ Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup} }{\longleftarrow} } \mathbf{H} \,.$$

Since $\iota_{sup}$ and $Disc_{sup}$ are fully faithful functors by assumption, these are (co-)modal operators (Def. ) on the cohesive topos, by (Prop. and Prop. ).

We pronounce these as follows:

$\phantom{A}$ fermionic modality $\phantom{A}$	$\phantom{A}$ bosonic modality $\phantom{A}$	$\phantom{A}$ rheonomy modality $\phantom{A}$
$\phantom{A}$ $\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even$ $\phantom{A}$	$\phantom{A}$ $\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup}$ $\phantom{A}$	$\phantom{A}$ $Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup}$ $\phantom{A}$

and we refer to the corresponding modal objects (Def. ) as follows:

• a $\rightsquigarrow$-comodal object

  $$\overset{\rightsquigarrow}{X} \underoverset{\simeq}{\epsilon^\rightsquigarrow_X}{\longrightarrow} X$$

  is called a bosonic object;

• a $Rh$-modal object

  $$X \underoverset{\simeq}{ \eta^{Rh}_X}{\longrightarrow} Rh X$$

  is called a rheonomic object;

Proposition

(progression of adjoint modalities on solid topos)

Let $[\mathcal{C}^{op}, sSet_{Qu}]_{{proj/inj} \atop lco}$ be a solid model topos (Def. ) and consider the adjoint modalities which it inherits

1. for being a cohesive topos, from Def. ,

2. for being an elastic topos, from Def. ,

3. for being a solid topos, from Def. :

$\phantom{A}$ shape modality $\phantom{A}$	$\phantom{A}$ flat modality $\phantom{A}$	$\phantom{A}$ sharp modality $\phantom{A}$
$\phantom{A}$ $ʃ \;\coloneqq\; Disc \Pi$ $\phantom{A}$	$\phantom{A}$ $\flat \;\coloneqq\; Disc \circ \Gamma$ $\phantom{A}$	$\phantom{A}$ $\sharp \;\coloneqq\; coDisc \circ \Gamma$ $\phantom{A}$
$\phantom{A}$ reduction modality $\phantom{A}$	$\phantom{A}$ infinitesimal shape modality $\phantom{A}$	$\phantom{A}$ infinitesimal flat modality $\phantom{A}$
$\phantom{A}$ $\Re \;\coloneqq\; \iota_{sup} \iota_{inf} \circ \Pi_{inf}\Pi_{sup}$ $\phantom{A}$	$\phantom{A}$ $\Im \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Pi_{inf} \Pi_{sup}$ $\phantom{A}$	$\phantom{A}$ $\& \;\coloneqq\; Disc_{sup} Disc_{inf} \circ \Gamma_{inf}\Gamma_{sup}$ $\phantom{A}$
$\phantom{A}$ fermionic modality $\phantom{A}$	$\phantom{A}$ bosonic modality $\phantom{A}$	$\phantom{A}$ rheonomy modality $\phantom{A}$
$\phantom{A}$ $\rightrightarrows \;\coloneqq\; \iota_{sup} \circ even$ $\phantom{A}$	$\phantom{A}$ $\rightsquigarrow \;\coloneqq\; \iota_{sup} \circ \Pi_{sup}$ $\phantom{A}$	$\phantom{A}$ $Rh \;\coloneqq\; Disc_{sup} \circ \Pi_{sup}$ $\phantom{A}$

Then these arrange into the following progression, via the preorder on modalities from Def. :

$$\array{ id &\dashv& id \\ \vee && \vee \\ \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh \\ && \vee && \vee \\ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee && \vee \\ && && ʃ &\dashv& \flat &\dashv& \sharp \\ && && && \vee && \vee \\ && && && \emptyset &\dashv& \ast }$$

where we are displaying, for completeness, also the adjoint modalities at the bottom $\emptyset \dashv \ast$ and the top $id \dashv id$ (Example ).

Proof

This is just as in Prop. .

$\,$

(…)