nLab
reflective sub-(infinity,1)-category

Context

(,1)(\infty,1)-Category theory

Notions of subcategory

Contents

Idea

The notion of reflective (,1)(\infty,1)-subcategory is the generalization of the notion of reflective subcategory from category theory to (∞,1)-category theory.

Definition

Reflective sub-(,1)(\infty,1)-category

Definition

(local objects, local equivalences)

A full and faithful (∞,1)-functor

R:DC R : D \hookrightarrow C

exhibits DD as a reflective sub-(∞,1)-category (of CC) if it has a left adjoint (∞,1)-functor L:CDL : C \to D.

(LR):DRLC. (L \dashv R) : D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} C \,.

If LL moreover is a left exact functor in that it preserves finite (∞,1)-limits, then the embedding is called exact.

The (∞,1)-functor RR or its composite

Loc:=RL:CC Loc := R \circ L : C \to C

may be understood as exhibiting a localization of CC at those morphisms that LL sends to equivalences in DD. If LL preserves finite limits (is a left exact functor), then this is an exact localization

Local objects and local morphisms

One finds, as discussed below, that reflective subcategories may be entirely characterized by the class of morphisms that the localization functor Loc:CCLoc : C \to C sends to weak equivalences.

Definition

(local objects, local equivalences)

Let SMor(C)S \subset Mor(C) be a class of morphisms.

  • An object cCc \in C is called an SS-local object if for all morphisms f:xyf : x \to y in CC the induced morphism

    Hom C(f,c):Hom C(y,c)Hom C(x,c) Hom_C(f,c) : Hom_C(y,c) \to Hom_C(x,c)

    is an equivalence (of ∞-groupoids).

  • An morphism f:xyf : x \to y in CC is called an SS-local morphism or SS-local equivalence if for all SS-local objects cCc \in C we have that

    Hom C(f,c):Hom C(y,c)Hom C(x,c) Hom_C(f,c) : Hom_C(y,c) \to Hom_C(x,c)

    is an equivalence (of ∞-groupoids).

Notice that the class of SS-equivalences always contains SS itself. Hence passing from a collection SS to its class S¯\bar S of SS-equivalences is a kind of saturation procedure. This is formalized by the following definition, whose justification is given by the propositions below.

Definition

(strongly saturated class of morphisms)

For CC an (∞,1)-category with small (∞,1)-colimits, a class SC 1S \subset C_1 of morphisms in CC is said to be strongly saturated if its satisfies the following three conditions

  1. It is stable under pushouts;

  2. The full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors Func(Δ[1],C)Func(\Delta[1], C) on SS has all (∞,1)-colimits;

  3. it satisfies the 2-out-of-3 property.

Notice that this definition has some immediate consequences:

The identity Id Id_\emptyset on the initial object of CC, which is the initial object in Func(Δ[1],C)Func(\Delta[1],C) is in SS, since it is the colimit of the empty diagram. Moreover, every equivalence is a pushout of Id Id_{\emptyset} so

  • A strongly saturated class contains every equivalence.

Given any collection {S i} i\{S_i\}_i of strongly saturated classes of morphisms in CC, their intersection is clearly also strongly saturated. Therefore for every collection SS of morphisms, there is a smallest strongly saturated class S¯\bar S containing it. We say that SS generates the strongly saturated class S¯\bar S. If SS is a small set, then S¯\bar S is said to be of small generation.

The smallest strongly saturated class of morphism in CC is that containing only the equivalences of CC.

Of importance are the strongly saturated classes arising as follows.

Lemma

For CC and DD two (,1)(\infty,1)-categories that have small (∞,1)-colimits, and for F:CDF : C \to D an (∞,1)-functor that preserves small (,1)(\infty,1)-colimits, given a strongly saturated class of morphisms SS in DD, its preimage F 1(S)F^{-1}(S) is a strongly saturated class in CC.

In particular the class of morphisms in CC sent to equivalences by FF is strongly saturated.

Lemma

The class of S 0S_0-local equivalences for S 0S_0 any class of morphisms is strongly saturated.

Proof

For each object cCc \in C let j(c):CGrpd opj(c) : C \to \infty Grpd^{op} be the functor represented by cc. Let S cS_c be the class of morphisms sent by j(c)j(c) to weak equivalences in ∞Grpd. Since j(c)j(c) preserves small colimits, this is a strongly saturated class, by the above lemma. Now observe that SS is the intersection S= cS cS = \cap_c S_c where cc ranges over the S 0S_0-local objects.

In the following this language of local morphisms is used to characterize reflective (,1)(\infty,1)-subcategories.

Properties

Characterization of reflectors

The following proposition characterizes the reflectors of a reflective (,1)(\infty,1)-subcategory. (You can read this proposition as an evident statement on the characterization of adjoints, but maybe as a preparation for the proofs to come there is some value in looking at its concrete proof in this special case of an (,1)(\infty,1)-adjunction.)

Lemma

(universality of reflection counit)

Let CC be an (∞,1)-category and DCD \hookrightarrow C a full sub-(∞,1)-category. Then this inclusion has a left adjoint (∞,1)-functor precisely if

  • for every object cCc \in C there is a localization or reflection : a morphism f:cc¯f : c \to \bar c such that c¯DC\bar c \in D\hookrightarrow C and such that for all eDCe \in D \hookrightarrow C we have that

    Hom C(f,e):Hom C(c¯,e)Hom C(c,e) Hom_C(f,e) : Hom_C(\bar c,e) \to Hom_C(c,e)

    is an equivalence (of ∞-groupoids).

This appears as HTT, prop. 5.2.7.8.

Proof

We produce an evident cograph realization KK of the inclusion and check that it being also a coCartesian fibration, hence exhibiting RR as a right adjoint, is equivalent to the second statement.

Let KC×Δ[1]K \subset C \times \Delta[1] be the full subcategory on those objects (c,i)(c,i) for which cDc \in D if i=1i = 1. Let p:KΔ[1]p : K \to \Delta[1] be the induced projection. One checks that this is the correspondence which is associated to the inclusion functor DCD \hookrightarrow C.

Therefore by the properties of adjoint (∞,1)-functors, we have that the inclusion functor has a left adjoint precisely if pp is not only a Cartesian fibration but also a coCartesian fibration.

To see that this is the case precisely if every cc has a reflection f:cdf : c \to d, recall the characterization of coCartesian morphisms f˜:(c,0)(d,1)\tilde f : (c,0) \to (d,1) as those making the squares

Hom K((d,1),(e,i)) Hom K(f˜,(e,i)) Hom K((c,0),(e,i)) Hom Δ[1](1,i) Hom Δ[1](0,i) \array{ Hom_K((d,1),(e,i)) &\stackrel{Hom_K(\tilde f,(e,i))}{\to} & Hom_K((c,0),(e,i)) \\ \downarrow && \downarrow \\ Hom_{\Delta[1]}(1, i) &\stackrel{}{\to}& Hom_{\Delta[1]}(0, i) }

being homotopy pullback squares, for all (e,i)K(e,i) \in K. Now in Δ[1]\Delta[1] all hom-objects are either empty or are points, so that the bottom morphism becomes the identity on the point if i=1i = 1. Since for i=0i = 0 everything becomes entirely trivial we consider the case that i=1i =1 and hence eDe \in D.

In that case the homotopy-pullback property is equivalent to the top morphism being an equivalence, hence to

Hom C(d,e)Hom C(f,e)Hom C(c,a) Hom_C(d,e) \stackrel{Hom_C(f,e)}{\to} Hom_C(c,a)

being an equivalence. This way the reflectors are identified precisely with the coCartesian morphisms in KΔ[1]K \to \Delta[1] that exhibit the left adjoint (∞,1)-functor to the inclusion functor.

The following proposition asserts that localizations are entirely determined by the corresponding local objects.

Proposition

Let

DRLC D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} C

be a localization of the (,1)(\infty,1)-category CC and let

Loc:CLDRC Loc : C \stackrel{L}{\to} D \overset{R}{\hookrightarrow} C

be the corresponding localization (∞,1)-monad. Write SMor(C)S \subset Mor(C) for the collection of morphisms that LocLoc sends to equivalences.

Then

  • an object cCc \in C is an SS-local object precisely if it is in the essential image of LocLoc (equivalent to an object of the form LocxLoc x);

  • every SS-local morphism is already in SS.

This is HTT, prop 5.5.4.2.

Proof

The reasoning is entirely analogous to the 1-categorical case (see for instance localization, reflective subcategory and geometric embedding).

First notice that because DCD \hookrightarrow C is a full and faithful (∞,1)-functor we have that the counit LRId DL R \stackrel{\simeq}{\to} Id_D is an equivalence. From this it follows that precomposition with the unit i z:zLoczi_z : z \to Loc z of morphisms in the image of LocLoc is a weak equivalence: for all z,xCz,x \in C we have

Hom C(i z,Locz):Hom C(Locz,Locx)Hom C(z,Locx). Hom_C(i_z, Loc z) : Hom_C(Loc z, Loc x) \stackrel{\simeq}{\to} Hom_C(z, Loc x) \,.

If zz is itself in the image of LocLoc, then this means that precomposition with the unit zLoczz \to Loc z is an isomorphism on hom-sets in the homotopy category of LocCLoc C, hence by the Yoneda lemma is itself an isomorphism in the homotopy category, hence i z:zLoczi_z : z \to Loc z is a weak equivalence if zz is itself in the image of LocLoc.

Applying this statement to the naturality square for the natural transformation IdLocId \to Loc on i si_s

s i s Locs i s Loci s Locs i Locs LocLocs \array{ s &\stackrel{i_s}{\to} & Loc s \\ \downarrow^{\mathrlap{i_s}} && \downarrow^{\mathrlap{Loc i_s}} \\ Loc s &\stackrel{i_{Loc s}}{\to}& Loc Loc s }

we find that Loci si LocsLoc i_s \simeq i_{Loc s}, hence that Loci sLoc i_s is a weak equivalence, and hence that i si_s is in SS, for all sCs \in C.

Now to show that for all xXx \in X the object LocxLoc x is SS-local, let f:yzf : y \to z be in SMor(C)S \subset Mor(C) and consider the induced square

Hom C(Locz,Locx) Hom C(Locy,Locx) Hom C(z,Locx) Hom C(y,Locx). \array{ Hom_C(Loc z, Loc x) &\stackrel{\simeq}{\to}& Hom_C(Loc y, Loc x) \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} \\ Hom_C(z, Loc x) &\to& Hom_C(y, Loc x) } \,.

Here the vertical morphisms are equivalences by the above remark, and the top morphism is an equivalence by the assumption that ff in in SS. It follows that the bottom morphism is an equivalence. This says that LocxLoc x is SS-local, for all xCx \in C.

Conversely, to show that for sCs \in C an SS-local object, we have that ss is in the essential image of LocLoc use that since i s:sLocsi_s : s \to Loc s is in SS, we have an equivalence Hom C(i s,s):Hom C(Locs,s)Hom C(s,s)Hom_C(i_s, s) : Hom_C(Loc s, s) \stackrel{\simeq}{\to} Hom_C(s,s). The pre-image of the identity under this equivalence is hence a left-inverse LocssLoc s \to s of sLocss \to Loc s. But this means that LocssLoc s \to s is itself in SS (since the morphisms in SS evidently satisfy 2-out-of-three), hence by applying the same argument again, we find that the left inverse LocssLoc s \to s has itself a left inverse. That implies that it is actually an inverse of sLocss \to Loc s, hence that this is an equivalence. So this shows that the SS-local ss is indeed in the essential iamge of LocLoc.

Finally, to show that every SS-local morphism is already in SS, let f:xyf : x \to y be such an SS-local morphism and consider the square

x f y i x i y Locx Locf Locy. \array{ x &\stackrel{f}{\to}& y \\ \downarrow^{\mathrlap{i_x}} && \downarrow^{\mathrlap{i_y}} \\ Loc x &\stackrel{Loc f}{\to}& Loc y } \,.

By the above we know now that the vertical morphisms here are also SS-local. It follows that the image of Locf:LocxLocyLoc f : Loc x \to Loc y on the homotopy category of Ho(LocC)Ho(Loc C) corepresents an isomorphism, hence by the Yoneda lemma that LoffLof f is a weak equivalence. Hence ff is indeed in SS.

Reflective localization at a set of morphisms

Above is discussed that every reflective subcategory is the localization at the collection local morphisms, those which the left adjoint functor inverts. One can turn this around and define or construct reflective (,1)(\infty,1)-subcategories by specifying collections of local morphisms.

Proposition

(localization proposition)

Let CC be a presentable (∞,1)-category and S 0S_0 be a small set of morphisms of CC.

Then the full sub-(∞,1)-category

R:DC R : D \hookrightarrow C

on S 0S_0-local objects is a reflective (,1)(\infty,1)-subcategory.

If L:CDL : C \to D denotes the left adjoint (∞,1)-functor of the inclusion, then for fMor(C)f \in Mor(C) a morphism, the following are equivalent

  1. ff is an S 0S_0-local equivalence;

  2. ff belongs to the strongly saturated class SS generated by S 0S_0;

  3. the morphism LfL f is an equivalence.

This is HTT, prop. 5.5.4.15.

The main ingredient in the proof of this assertion is the following lemma, whose proof we give below in Proof of the localization lemma.

Proposition

(localization lemma)

Let CC be a locally presentable (∞,1)-category, and let SMor(C)S \subset Mor(C) be a strongly saturated collection of morphisms, generated from a small set S 0S_0.

Then for every object cCc \in C there exists a reflector, i.e. a morphism f:cdf : c \to d such that dd is an SS-local object and fSf \in S.

With that in hand we look at the proof of the above proposition:

Proof

(localization proposition)

The localization lemma gives for each object cCc \in C a reflector f:cdf : c \to d with dd SS-local. By one of the above lemmas, this already gives the reflective embedding

DRLC D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} C

of the full subcateory of SS-local objects in CC.

It remains to prove the statements about the role of SS in the localization:

First, by one of the above lemmas, we have that the S 0S_0-local equivalences are a strongly saturated class of morphisms containong S 0S_0. Hence they in particular contain SS. So the second claim implies the first.

That the first and the third condition are equivalent follows from noticing that for any local object dDd \in D the morphism Hom C(f,Rd)Hom_C(f,R d) is an equivalence precisely if Hom D(Lf,d)Hom_D(L f, d) is and then applying the Yoneda lemma (for instance in the homotopy category), which implies that if a morphism produces an equivalence when hommed into all objects, then it is itself an equivalence.

It remains to show that the third item implies the second. Let f:cdf : c \to d be a morphism such that Lf:LcLdL f : L c \to Ld is an equivalence. Consider the commuting triangle

c f d Lc Lf Ld. \array{ c &\stackrel{f}{\to}& d \\ \downarrow &\searrow& \downarrow \\ L c &\stackrel{L f}{\to}& L d } \,.

Since every reflector is in SS and the reflectors are the units of the reflective adjunction constructed from them, we have that the vertical morphisms in this diagram are in SS, and the bottom morphism is, since it is an equivalence by assumption. By applying the 2-out-of-3 property of SS twice it follows that ff is in SS.

Proof of the localization lemma

We here spell out the proof of

Proposition

(localization lemma)

Let CC be a locally presentable (∞,1)-category, and let SMor(C)S \subset Mor(C) be a strongly saturated collection of morphisms, generated from a small set S 0S_0.

Then for every object cCc \in C there exists a reflector, i.e. a morphism f:cdf : c \to d such that dd is an SS-local object and fSf \in S.

This is HTT, prop. 5.5.5.14.

Proof

Regard all (,1)(\infty,1)-categories as quasi-categories for the purpose of this proof. Write DFunc(Δ[1],C)D \subset Func(\Delta[1], C) for the full sub-quasicategory on the elements of SS. Consider the pullback (in sSet)

D c D {c} Func({0},C). \array{ D_c &\to& D \\ \downarrow && \downarrow \\ \{c\} &\to& Func(\{0\}, C) } \,.

Since SS is by assumption closed under pushouts in CC, we have for each morphism xyx \to y in DFunc({0},C)D \simeq Func(\{0\}, C) and each lift

x y x \array{ x &\to& y \\ \downarrow \\ x' }

of its source to Func(Δ[1],C)Func(\Delta[1], C) a lift of this morphism with this source, given by the the pushout square

x y x x xy \array{ x &\to& y \\ \downarrow && \downarrow \\ x' &\to& x' \coprod_x y }

in CC, regarded as a morphism in Func(Δ[1],C)Func(\Delta[1], C). By the universality of the pushout, one finds that this is a coCartesian lift. Hence DFunc({0},C)CD \to Func(\{0\}, C) \simeq C is a coCartesian fibration. Moreover, by the behaviour under pullback of Cartesian fibrations it follows that the above diagram is a homotopy pullback diagram in the Joya model structure sSet JoyalsSet_{Joyal}.

Use now that accessible quasi-categories are stable under homotopy pullback to conclude that D cD_c is accessible. Moreover, one can check that D cD_c has all small colimits. Together this means that D cD_c is a locally presentable (∞,1)-category. This implies in particular that D cD_c also has all small (∞,1)-limits and hence contains a terminal object, f:cdf : c \to d.

We now complete the proof by showing that f:cdf : c \to d being terminal in D cD_c implies that dd is an SS-local object. This is equivalent to showing that for t:abt : a \to b any element in SS, composition with tt induces an equivalence

Hom C(t,d):Hom C(b,d)Hom C(a,d). Hom_C(t,d) : Hom_C(b,d) \to Hom_C(a,d) \,.

This in turn may be checked by checking that all its homotopy fibers are contractible. By general statements about the homotopy fiber of functor categories the homotopy fiber of Hom C(t,d)Hom_C(t,d) over a point g:adg : a \to d of Hom C(a,d)Hom_C(a,d) is equivalent to the hom-object Hom C a/(t,g)Hom_{C_{a/}}(t,g) in the under-quasi-category C a/C_{a/}.

This in turn can be checked to be equivalent to Hom C d/(g *t,Id d)Hom_{C_{d/}}(g_* t, Id_d), where g *tg_* t is the (,1)(\infty,1)-categorical pushout

a t b g d g *t d ab \array{ a &\stackrel{t}{\to}& b \\ \downarrow^{\mathrlap{g}} && \downarrow \\ d &\underset{g_* t}{\to}& d \coprod_a b }

in CC. Notice that g *tg_* t, being a pushout of tSt \in S, is itself in SS.

Now pick a composite

d f σ g *t c g *tf d ab \array{ && d \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^\sigma& \searrow^{\mathrlap{g_* t}} \\ c &&\underset{g_* t \circ f}{\to}&& d \coprod_a b }

and observe that we have an isomorphism of simplicial sets

Hom C d/(g *t,Id d)Hom C f/(σ,s 1(f)) Hom_{C_{d/}}(g_* t, Id_{d}) \simeq Hom_{C_{f/}}(\sigma, s_1(f))

(where s 1s_1 is the corresponding degeneracy map).

Applying the expression for homotopy fibers of functor categories once again, this is found to be the homotopy fiber of

Hom C c/(g *tf,f)Hom C c/(f,f), Hom_{C_{c/}}(g_* t \circ f, f) \to Hom_{C_{c/}}(f,f) \,,

because Hom (C c/) f/(σ,s 1(f))=Hom C f/(σ,s 1(f))Hom_{(C_{c/})_{f/}}(\sigma , s_1(f)) = Hom_{C_{f/}}(\sigma , s_1(f)).

Finally we can use that ff is terminal in the full subcategory D cD_c of C c/C_{c/} that contains g *tfg_* t \circ f. This implies that the above morphism goes between contractible \infty-groupoids and hence has contractible homotopy fibers.

Transport of reflective subcategories

Proposition

Let f:𝒞𝒟f \colon \mathcal{C} \to \mathcal{D} be an (∞,1)-functor between presentable (∞,1)-categories, and let 𝒞 0𝒞\mathcal{C}^0 \hookrightarrow \mathcal{C} be a reflective sub-(,1)(\infty,1)-category. If ff has a right adjoint (∞,1)-functor f *f^*, then

𝒟 0(f *) 1(𝒞 0)𝒟 \mathcal{D}^0 \coloneqq (f^*)^{-1}(\mathcal{C}^0) \hookrightarrow \mathcal{D}

is also a reflective sub-(,1)(\infty,1)-category.

𝒞 0 𝒞 f * f * f 𝒟 0 𝒟. \array{ \mathcal{C}^0 &\hookrightarrow& \mathcal{C} \\ {}^{\mathllap{f^*}}\uparrow && {}^{\mathllap{f^*}}\uparrow\downarrow^{\mathrlap{f}} \\ \mathcal{D}^0 &\hookrightarrow& \mathcal{D} } \,.

This is (Lurie, lemma 5.5.4.17).

Proof

By prop. 2, 𝒞 0𝒞\mathcal{C}^0 \hookrightarrow \mathcal{C} is the inclusion of the SS-local objects for some class SS of morphisms of 𝒞\mathcal{C}. By adjunction it follows that 𝒟 0\mathcal{D}^0 is precisely the class of f(S)f(S)-local objects, and hence is a reflective subcategory, again by prop. 2.

Model category presentation

Proposition

Let CC be a left proper combinatorial simplicial model category which presents an (∞,1)-category 𝒞C \mathcal{C} \simeq C^\circ.

Then if 𝒞 0𝒞\mathcal{C}^0 \hookrightarrow \mathcal{C} is an accessible reflective inclusion with reflector L:𝒞𝒞 0L \colon \mathcal{C} \to \mathcal{C}^0, then there exists a corresponding left Bousfield localization

CididC C' \stackrel{\overset{id}{\leftarrow}}{\underset{id}{\to}} C

of the model category CC which presents this inclusion in that

  1. an object in CC' is a fibrant object precisely if it is fibrant as an object of CC and in addition its image in the homotopy category Ho(C)Ho(𝒞)Ho(C) \simeq Ho(\mathcal{C}) is in the inclusion Ho(𝒞 0)Ho(𝒞)Ho(\mathcal{C}^0) \hookrightarrow Ho(\mathcal{C});

  2. a morphism in CC' is a weak equivalence precisely if under Ho(L):Ho(C)Ho(𝒞)Ho(𝒞 0)Ho(L) \colon Ho(C) \simeq Ho(\mathcal{C}) \to Ho(\mathcal{C}^0) is an isomorphism.

This is (Lurie, prop. A.3.7.8).

Proof

Use that by the above discussion 𝒞 0\mathcal{C}^0 is the full subcategory on SS-local objects for a small set of morphisms. By the discussion at Bousfield localization of model categories this presents precisely such localizations.

Extra conditions

Extra conditions on a reflective sub-(,1)(\infty,1)-category of relevance are

Accessible reflective subcategories

The following proposition characterizes when a reflective subcategory of an accessible (∞,1)-category CC is accessible

Proposition

Let CC be an accessible (∞,1)-category and

DRLC D \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\hookrightarrow}} C

a reflective subcategory. Then the following conditions are equivalent:

  1. DD is itself accessible;

  2. The localization Loc:RL:CCLoc : R\circ L : C \to C is an accessible (∞,1)-functor.

  3. There exists a small set S 0S:=L 1(equiv.)S_0 \subset S := L^{-1}(equiv.) such that every SS-local object is also S 0S_0-local.

This is (Lurie, prop. 5.5.1.2 and prop. 5.5.4.2, part 3).

Proof

This is work…

Exact localizations

Recall that the reflective subcategory DLCD \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} C is exact – or LL an exact localization – if LL is a left exact functor in that it preserves finite limits. Accordinly we say:

Definition

An exact reflective sub-(,1)(\infty,1)-category is a reflective sub-(,1)(\infty,1)-category whose reflector is a left exact (∞,1)-functor, hence preserves finite (∞,1)-limits.

Recall also that by the above results, a reflective subcategory is characterized by the collection S=L 1(equiv)Mor(C)S = L^{-1}(equiv) \subset Mor(C) of those morphisms, that LL sends to equivalences in DD.

The following propositions say how the property that LL preserves finite limits is characterized by pullback-stability properties of SS.

Proposition

(recognition of exact localization)

A reflective sub-(,1)(\infty,1)-category DLCD \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} C such that CC has all finite limits is exact precisely if the collection S:=L 1(equiv)Mor(C)S := L^{-1}(equiv) \subset Mor(C) of morphisms that LL sends to equivalences is stable under pullback.

So if for every pullback diagram

X X f f Y Y \array{ X' &\to& X \\ \downarrow^{\mathrlap{f'}} && \downarrow^{\mathrlap{f}} \\ Y' &\to& Y }

we have that if L(f)L(f) is an equivalence then also L(f)L(f') is an equivalence.

This is HTT, prop. 6.2.1.1.

Proof

If LL preserves finite limits, then it preserves pullbacks, so that L(f)L(f') is a pullback of the equivalence L(f)L(f), hence itself an equivalence.

So it remains to check that, conversely, stability of SS under pullback implies that LL preserves finite limits. By a general characterization of left exact functors (see there) it suffices to check that LL preserves the terminal object and all pullbacks.

Since the terminal object is evidently SS-local, we have L**L * \simeq *.

Next we check that LL preserves products, because we will need this to show that all binary pullbacks are preserved. For that it is sufficient to check that the morphism L(x×y)L(x)×L(y)L(x \times y) \to L(x) \times L(y) induced from the units i x:xLxi_x : x \to L x and i y:yLyi_y : y \to L y is in SS. From inspection of the diagram

x i x Lx x×y (i x,Id) Lx×y y Id y \array{ x &\stackrel{i_x}{\to}& L x \\ \uparrow && \uparrow \\ x \times y &\stackrel{(i_x,Id)}{\to}& L x \times y \\ \downarrow && \downarrow \\ y &\stackrel{Id}{\to}& y }

one finds that x×yLx×Lyx \times y \to L x \times L y is a pullback of i xi_x. Hence is in SS, by assumption. Similarly in

Lx Id Lx Lx×y Lx×Ly y i y Ly \array{ L x &\stackrel{Id}{\to}& L x \\ \uparrow && \uparrow \\ L x \times y &\to& L x \times L y \\ \downarrow && \downarrow \\ y &\stackrel{i_y}{\to}& L y }

one see that Lx×yLx×LyL x \times y \to L x \times L y is a pullback of i yi_y and hence in SS. The composite of these two morphisms is a morphism x×yLx×Lyx \times y \to L x \times L y, which is in SS since SS is closed under composition. Applying LL hence yields an equivalence L(x×y)Lx×LyL(x \times y) \stackrel{\simeq}{\to} L x \times L y.

We now apply the same kind of argument to show that LL respects more generally pullbacks.

For that, first notice that for xyzx \to y \leftarrow z a diagram in CC, the pullback Lx× L yL zL x \times_{L_y} L_z of the image exists in CC, by assumption, but is easily seen to be SS-local and hence lands in DD. Therefore to show that we have an equivalence L(x× yz)Lx× Ly×LzL(x \; \times_y z) \simeq L x \; \times_{L y} \times L z it is sufficient to show that the natural morphism, x× yzLx× LyLzx \times_y z \to L x \times_{L y} L z induced from the morphism of diagrams

x i x Lx y i y Ly z i z Lz \array{ x &\stackrel{i_x}{\to}& L x \\ \downarrow && \downarrow \\ y &\stackrel{i_y}{\to}& L y \\ \uparrow && \uparrow \\ z &\stackrel{i_z}{\to}& L z }

in CC with the adjunction unit morphism on the horizonatals, is in SS. By passing along these units one at a time

x Id x i x Lx Id Lx f i yf Lf Lf y i y Ly Id Ly Id Ly g i yg i yg Lg z Id z Id z i z Lz \array{ x &\stackrel{Id}{\to}& x &\stackrel{i_x}{\to}& L x &\stackrel{Id}{\to}& L x \\ \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{i_y f}} && \downarrow^{L f} && \downarrow^{L f} \\ y &\stackrel{i_y}{\to}& L y &\stackrel{Id}{\to}& L y &\stackrel{Id}{\to}& L y \\ \uparrow_{\mathllap{g}} && \uparrow_{\mathrlap{i_y g}} && \uparrow_{\mathrlap{i_y g}} && \uparrow_{\mathrlap{L g}} \\ z &\stackrel{Id}{\to}& z &\stackrel{Id}{\to}& z &\stackrel{i_z}{\to}& L z }

this may be decomposed as a composite of three morphisms

x× yzx× LyzLx× LyzLx× LyLz. x \times_y z \to x \times_{L y} z \to L x \times_{L y} z \to L x \times_{L y} L z \,.

If we equivalently reformulate these pullbacks as equalizers then this is

x× yz x×z fp 1gp 2 y Id i y x× Lyz x×z i ygi yf Ly (i x,Id) Id Lx× Lyz Lx×z i ygLf Ly (Id,i z) Id Lx× LyLz Lx×Lz LgLf Ly \array{ x \times_y z &\to& x \times z & \stackrel{\overset{g p_2}{\to}}{\underset{f p_1 }{\to}} & y \\ \downarrow && \downarrow^{\mathrlap{Id}} && \downarrow^{\mathrlap{i_y}} \\ x \times_{L y} z &\to& x \times z &\stackrel{\overset{i_y f}{\to}}{\underset{i_y g}{\to}}& L y \\ \downarrow && \downarrow^{\mathrlap{(i_x, Id)}} && \downarrow^{Id} \\ L x \times_{L y} z &\to& L x \times z &\stackrel{\overset{L f}{\to}}{\underset{i_y g}{\to}}& L y \\ \downarrow && \downarrow^{\mathrlap{(Id, i_z)}} && \downarrow^{\mathrlap{Id}} \\ L x \times_{L y} L z &\to& L x \times L z &\stackrel{\overset{L f}{\to}}{\underset{L g}{\to}}& L y }

It is immediate to check that the two bottom left squares are pullback squares. So the two left vertical morphisms are pullbacks of (Id,i z)(Id, i_z) and (i x,Id)(i_x, Id), respectively. Of morphisms of this form we had seen above that they are in SS. Hence by the assumed pullback-stability of SS also x× LyzLx× LyzLx× LyLzx \times_{L y} z \to L x \times_{L y} z \to L x \times_{L y} L z is in SS.

So it remains to show that x× yzx× Lyzx \times_y z \to x \times_{L y} z is in SS. We claim that this morphism in turn may be expressed as a pullback

x× yz y x× Lyz × Lyy \array{ x \times_y z &\to& y \\ \downarrow && \downarrow \\ x \times_{L y} z &\to& \times_{L y} y }

of the diagonal yy× Lyyy \to y \times_{L y} y. To see this notice that cones qq over the corresponding pullback diagram are equivalently diagrams

q x y z Ly. \array{ && q \\ & \swarrow &\downarrow& \searrow \\ x &\to& y &\leftarrow& z \\ & \searrow &\downarrow& \swarrow \\ && L y } \,.

So now we need to show that the diagonal yy× Lyyy \to y \times_{L y} y is in SS.

To see this, notice that it has a left inverse y× Lyyyy \times_{L y} y \to y, given by any one of the two projections. So if finally we show that this is in SS, we are done, since SS satisfies 2-out-of-3. But this follows now from pullback stability of SS, because this projection is the pullback of yLyy \to L y along itself.

Proposition

(accessibility of exact localizations)

Let CC be a locally presentable (∞,1)-category with universal colimits. Assume moreover that finite limits commute with filtered colimits in CC (this holds for example if CC is an (∞,1)-topos). Let S 0Mor(C)S_0 \subset Mor(C) be a small set of morphisms, and SS the smallest strongly saturated class containing S 0S_0 and stable under pullbacks. Then SS is strongly generated by a small set of morphisms.

This is HTT, Prop. 6.2.1.2. This proposition is used to construct pullbacks of (∞,1)-topoi, c.f. HTT Prop. 6.3.4.6.

Examples

Inclusion of the terminal object

If CC has a terminal object, then the full subcategory on terminal objects is a reflective subcategory of CC.

coCartesian fiber over a reflective subcategory

Proposition

let p:C 1C 0p : C_1 \to C_0 be a coCartesian fibration and D 0C 0D_0 \stackrel{\leftarrow}{\hookrightarrow} C_0 a reflective (,1)(\infty,1)-subcategory of the base.

The restriction D 1C 1D_1 \hookrightarrow C_1 of C 1C_1 over D 0D_0, i.e. the strict (say in sSet if everything is modeled by quasi-categories) pullback

D 1 C 1 p D 0 C 0 \array{ D_1 & \stackrel{}{\hookrightarrow} & C_1 \\ \downarrow && \downarrow^{\mathrlap{p}} \\ D_0 &\hookrightarrow& C_0 }

is itself a reflective (,1)(\infty,1)-subcategory of C 1C_1.

Proof

By the above proposition on reflectors, it is sufficient to produce for every cC 1c \in C_1 there is a reflection morphism f:cdf : c \to d with dD 1d \in D_1.

Such ff is obtained by choosing any coCartesian lift of a reflector p(f):p(c)d¯p(f) : p(c) \to \bar d.

To see this, consider for every object eD 1e \in D_1 the diagram

Hom C 1(d,e) Hom C 1(f,e) Hom C 1(c,e) Hom C 0(p(d),p(e)) Hom C 0(p(f),p(e)) Hom C 0(p(c),p(e)). \array{ Hom_{C_1}(d,e) &\stackrel{Hom_{C_1}(f,e)}{\to}& Hom_{C_1}(c,e) \\ \downarrow && \downarrow \\ Hom_{C_0}(p(d), p(e)) &\stackrel{Hom_{C_0}(p(f) ,p(e))}{\to}& Hom_{C_0}(p(c), p(e)) } \,.

By assumption p(f)p(f) is a reflector, hence the bottom morphism is an equivalence. By one of the characterizations of coCartesian morphisms, the fact that ff is a coCartesian lift means that this diagram is a (homotopy) pullback diagram. This means that also the top horizontal morphism is an equivalence.

Locally presentable (,1)(\infty,1)-categories

If C=PSh (,1)(K)C = PSh_{(\infty,1)}(K) is the (∞,1)-category of (∞,1)-presheaves on some small (,1)(\infty,1)-category KK, then accessibly embedded reflective subcategory

DPSh (,1)(K) D \stackrel{\leftarrow}{\hookrightarrow} PSh_{(\infty,1)}(K)

(i.e. one where the inclusion is an accessible (∞,1)-functor) is a locally presentable (∞,1)-category.

(,1)(\infty,1)-Toposes

If C=PSh (,1)(K)C = PSh_{(\infty,1)}(K) is the (∞,1)-category of (∞,1)-presheaves on some small (,1)(\infty,1)-category KK, then an accessibly embedded exact reflective subcategory

Sh (,1)(K)LPSh (,1)(K) Sh_{(\infty,1)}(K) \stackrel{\overset{L}{\leftarrow}}{\underset{}{\hookrightarrow}} PSh_{(\infty,1)}(K)

is an (∞,1)-category of (∞,1)-sheaves on KK – an (∞,1)-topos. We have:

  • the collection of morphism S=L 1(equiv.)S = L^{-1}(equiv.) that are sent to weak equivalences are the analog of local isomorphisms of ordinary sheaf theory;

  • the SS-local objects are the ∞-stacks ;

  • the localizaton functor Loc=RL:PSh (,1)(K)PSh (,1)(K)Loc = R \circ L : PSh_{(\infty,1)}(K) \to PSh_{(\infty,1)}(K) is ∞-stackification .

\infty-Lie algebroids inside all \infty-Lie groupoids

Let K=Alg k opK = Alg_k^{op} be the opposite of the category of kk-associative algebras, regarded as a site with the fpqc-topology. Then an object in Sh (,1)(Alg k op)Sh_{(\infty,1)}(Alg_k^{op}) may be regarded as an algebraic \infty-groupoid. The infinitesimal version is an Lie ∞-algebroid, which may be identified with an object in (Alg k Δ) op(dgAlg k) op(Alg_k^\Delta)^{op} \simeq (dgAlg_k)^{op} – the opposite of the category of cosimplicial algebras. The simplicial model structure on cosimplicial algebras, presents this as an (,1)(\infty,1)-category (Alg k Δ) (Alg_k^\Delta)^\circ

The Yoneda embedding induces an inclusion

((Alg k Δ) op) Sh (,1)(Alg k op). ((Alg_k^\Delta)^{op})^\circ \stackrel{}{\hookrightarrow} Sh_{(\infty,1)}(Alg_k^{op}) \,.

which is a reflective embedding. It exhibits localization at A 1A^1-cohomology, where A 1=Speck[x]A^1 = Spec k[x] is the algebraic line object.

This is discussed at rational homotopy theory in an (∞,1)-topos.

References

The general theory is discussed in section 5.2.7 of

A Coq-formalization of left-exact reflective sub-(,1)(\infty,1)-categories in homotopy type theory is in

Revised on May 26, 2014 00:39:11 by Urs Schreiber (89.204.130.238)