equivalences in/of $(\infty,1)$-categories
The notion of reflective $(\infty,1)$-subcategory is the generalization of the notion of reflective subcategory from category theory to (∞,1)-category theory.
(local objects, local equivalences)
A full and faithful (∞,1)-functor
exhibits $D$ as a reflective sub-(∞,1)-category (of $C$) if it has a left adjoint (∞,1)-functor $L : C \to D$.
If $L$ moreover is a left exact functor in that it preserves finite (∞,1)-limits, then the embedding is called exact.
The (∞,1)-functor $R$ or its composite
may be understood as exhibiting a localization of $C$ at those morphisms that $L$ sends to equivalences in $D$. If $L$ preserves finite limits (is a left exact functor), then this is an exact localization
One finds, as discussed below, that reflective subcategories may be entirely characterized by the class of morphisms that the localization functor $Loc : C \to C$ sends to weak equivalences.
(local objects, local equivalences)
Let $S \subset Mor(C)$ be a class of morphisms.
An object $c \in C$ is called an $S$-local object if for all morphisms $f : x \to y$ in $S$ the induced morphism
is an equivalence (of ∞-groupoids).
An morphism $f : x \to y$ in $C$ is called an $S$-local morphism or $S$-local equivalence if for all $S$-local objects $c \in C$ we have that
is an equivalence (of ∞-groupoids).
Notice that the class of $S$-equivalences always contains $S$ itself. Hence passing from a collection $S$ to its class $\bar S$ of $S$-equivalences is a kind of saturation procedure. This is formalized by the following definition, whose justification is given by the propositions below.
(strongly saturated class of morphisms)
For $C$ an (∞,1)-category with small (∞,1)-colimits, a class $S \subset C_1$ of morphisms in $C$ is said to be strongly saturated if its satisfies the following three conditions
It is stable under pushouts along arbitrary morphisms of $C$;
The full sub-(∞,1)-category of the (∞,1)-category of (∞,1)-functors $Func(\Delta[1], C)$ on $S$ has all (∞,1)-colimits;
it satisfies the 2-out-of-3 property.
Notice that this definition has some immediate consequences:
The identity $Id_\emptyset$ on the initial object of $C$, which is the initial object in $Func(\Delta[1],C)$ is in $S$, since it is the colimit of the empty diagram. Moreover, every equivalence is a pushout of $Id_{\emptyset}$ so
Given any collection $\{S_i\}_i$ of strongly saturated classes of morphisms in $C$, their intersection is clearly also strongly saturated. Therefore for every collection $S$ of morphisms, there is a smallest strongly saturated class $\bar S$ containing it. We say that $S$ generates the strongly saturated class $\bar S$. If $S$ is a small set, then $\bar S$ is said to be of small generation.
The smallest strongly saturated class of morphism in $C$ is that containing only the equivalences of $C$.
Of importance are the strongly saturated classes arising as follows.
For $C$ and $D$ two $(\infty,1)$-categories that have small (∞,1)-colimits, and for $F : C \to D$ an (∞,1)-functor that preserves small $(\infty,1)$-colimits, given a strongly saturated class of morphisms $S$ in $D$, its preimage $F^{-1}(S)$ is a strongly saturated class in $C$.
In particular the class of morphisms in $C$ sent to equivalences by $F$ is strongly saturated.
The class of $S_0$-local equivalences for $S_0$ any class of morphisms is strongly saturated.
For each object $c \in C$ let $j(c) : C \to \infty Grpd^{op}$ be the functor represented by $c$. Let $S_c$ be the class of morphisms sent by $j(c)$ to weak equivalences in ∞Grpd. Since $j(c)$ preserves small colimits, this is a strongly saturated class, by the above lemma. Now observe that $S$ is the intersection $S = \cap_c S_c$ where $c$ ranges over the $S_0$-local objects.
In the following this language of local morphisms is used to characterize reflective $(\infty,1)$-subcategories.
The following proposition characterizes the reflectors of a reflective $(\infty,1)$-subcategory. (You may read this proposition as the characterization of adjoint functors via universal arrows (via this prop.), 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 $(\infty,1)$-adjunction.)
(universality of reflection counit)
Let $C$ be an (∞,1)-category and $D \hookrightarrow C$ a full sub-(∞,1)-category. Then this inclusion has a left adjoint (∞,1)-functor precisely if
for every object $c \in C$ there is a localization or reflection: a morphism $f : c \to \bar c$ such that $\bar c \in D\hookrightarrow C$ and such that for all $e \in D \hookrightarrow C$ we have that
is an equivalence (of ∞-groupoids).
This appears as HTT, prop. 5.2.7.8.
We produce an evident cograph realization $K$ of the inclusion and check that it being also a coCartesian fibration, hence exhibiting $R$ as a right adjoint, is equivalent to the second statement.
Let $K \subset C \times \Delta[1]$ be the full subcategory on those objects $(c,i)$ for which $c \in D$ if $i = 1$. Let $p : K \to \Delta[1]$ be the induced projection. One checks that this is the correspondence which is associated to the inclusion functor $D \hookrightarrow C$.
Therefore by the properties of adjoint (∞,1)-functors, we have that the inclusion functor has a left adjoint precisely if $p$ is not only a Cartesian fibration but also a coCartesian fibration.
To see that this is the case precisely if every $c$ has a reflection $f : c \to d$, recall the characterization of coCartesian morphisms $\tilde f : (c,0) \to (d,1)$ as those making the squares
being homotopy pullback squares, for all $(e,i) \in K$. Now in $\Delta[1]$ all hom-objects are either empty or are points, so that the bottom morphism becomes the identity on the point if $i = 1$. Since for $i = 0$ everything becomes entirely trivial we consider the case that $i =1$ and hence $e \in D$.
In that case the homotopy-pullback property is equivalent to the top morphism being an equivalence, hence to
being an equivalence. This way the reflectors are identified precisely with the coCartesian morphisms in $K \to \Delta[1]$ 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.
Let
be a localization of the $(\infty,1)$-category $C$ and let
be the corresponding localization (∞,1)-monad. Write $S \subset Mor(C)$ for the collection of morphisms that $Loc$ sends to equivalences.
Then
an object $c \in C$ is an $S$-local object precisely if it is in the essential image of $Loc$ (equivalent to an object of the form $Loc x$);
every $S$-local morphism is already in $S$.
This is HTT, prop 5.5.4.2.
The reasoning is entirely analogous to the 1-categorical case (see for instance localization, reflective subcategory and geometric embedding).
First notice that because $D \hookrightarrow C$ is a full and faithful (∞,1)-functor we have that the counit $L R \stackrel{\simeq}{\to} Id_D$ is an equivalence. From this it follows that precomposition with the unit $i_z : z \to Loc z$ of morphisms in the image of $Loc$ is a weak equivalence: for all $z,x \in C$ we have
If $z$ is itself in the image of $Loc$, then this means that precomposition with the unit $z \to Loc z$ is an isomorphism on hom-sets in the homotopy category of $Loc C$, hence by the Yoneda lemma is itself an isomorphism in the homotopy category, hence $i_z : z \to Loc z$ is a weak equivalence if $z$ is itself in the image of $Loc$.
Applying this statement to the naturality square for the natural transformation $Id \to Loc$ on $i_s$
we find that $Loc i_s \simeq i_{Loc s}$, hence that $Loc i_s$ is a weak equivalence, and hence that $i_s$ is in $S$, for all $s \in C$.
Now to show that for all $x \in X$ the object $Loc x$ is $S$-local, let $f : y \to z$ be in $S \subset Mor(C)$ and consider the induced square
Here the vertical morphisms are equivalences by the above remark, and the top morphism is an equivalence by the assumption that $f$ in in $S$. It follows that the bottom morphism is an equivalence. This says that $Loc x$ is $S$-local, for all $x \in C$.
Conversely, to show that for $s \in C$ an $S$-local object, we have that $s$ is in the essential image of $Loc$ use that since $i_s : s \to Loc s$ is in $S$, we have an equivalence $Hom_C(i_s, s) : Hom_C(Loc s, s) \stackrel{\simeq}{\to} Hom_C(s,s)$. The pre-image of the identity under this equivalence is hence a left-inverse $Loc s \to s$ of $s \to Loc s$. But this means that $Loc s \to s$ is itself in $S$ (since the morphisms in $S$ evidently satisfy 2-out-of-three), hence by applying the same argument again, we find that the left inverse $Loc s \to s$ has itself a left inverse. That implies that it is actually an inverse of $s \to Loc s$, hence that this is an equivalence. So this shows that the $S$-local $s$ is indeed in the essential iamge of $Loc$.
Finally, to show that every $S$-local morphism is already in $S$, let $f : x \to y$ be such an $S$-local morphism and consider the square
By the above we know now that the vertical morphisms here are also $S$-local. It follows that the image of $Loc f : Loc x \to Loc y$ on the homotopy category of $Ho(Loc C)$ corepresents an isomorphism, hence by the Yoneda lemma that $Lof f$ is a weak equivalence. Hence $f$ is indeed in $S$.
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 $(\infty,1)$-subcategories by specifying collections of local morphisms.
(localization proposition)
Let $C$ be a presentable (∞,1)-category and $S_0$ be a small set of morphisms of $C$.
Then the full sub-(∞,1)-category
on $S_0$-local objects is a reflective $(\infty,1)$-subcategory.
If $L : C \to D$ denotes the left adjoint (∞,1)-functor of the inclusion, then for $f \in Mor(C)$ a morphism, the following are equivalent
$f$ is an $S_0$-local equivalence;
$f$ belongs to the strongly saturated class $S$ generated by $S_0$;
the morphism $L f$ 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.
(localization lemma)
Let $C$ be a locally presentable (∞,1)-category, and let $S \subset Mor(C)$ be a strongly saturated collection of morphisms, generated from a small set $S_0$.
Then for every object $c \in C$ there exists a reflector, i.e. a morphism $f : c \to d$ such that $d$ is an $S$-local object and $f \in S$.
With that in hand we look at the proof of the above proposition:
(localization proposition)
The localization lemma gives for each object $c \in C$ a reflector $f : c \to d$ with $d$ $S$-local. By lemma , this already gives the reflective embedding
of the full subcateory of $S$-local objects in $C$.
It remains to prove the statements about the role of $S$ in the localization:
First, by one of the above lemmas, we have that the $S_0$-local equivalences are a strongly saturated class of morphisms containong $S_0$. Hence they in particular contain $S$. So the second claim implies the first.
That the first and the third condition are equivalent follows from noticing that for any local object $d \in D$ the morphism $Hom_C(f,R d)$ is an equivalence precisely if $Hom_D(L f, d)$ 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 $f : c \to d$ be a morphism such that $L f : L c \to Ld$ is an equivalence. Consider the commuting triangle
Since every reflector is in $S$ and the reflectors are the units of the reflective adjunction constructed from them, we have that the vertical morphisms in this diagram are in $S$, and the bottom morphism is, since it is an equivalence by assumption. By applying the 2-out-of-3 property of $S$ twice it follows that $f$ is in $S$.
We here spell out the proof of
(localization lemma)
Let $C$ be a locally presentable (∞,1)-category, and let $S \subset Mor(C)$ be a strongly saturated collection of morphisms, generated from a small set $S_0$.
Then for every object $c \in C$ there exists a reflector, i.e. a morphism $f : c \to d$ such that $d$ is an $S$-local object and $f \in S$.
This is HTT, prop. 5.5.5.14.
Regard all $(\infty,1)$-categories as quasi-categories for the purpose of this proof. Write $D \subset Func(\Delta[1], C)$ for the full sub-quasicategory on the elements of $S$. Consider the pullback (in sSet)
Since $S$ is by assumption closed under pushouts in $C$, we have for each morphism $x \to y$ in $D \simeq Func(\{0\}, C)$ and each lift
of its source to $Func(\Delta[1], C)$ a lift of this morphism with this source, given by the the pushout square
in $C$, regarded as a morphism in $Func(\Delta[1], C)$. By the universality of the pushout, one finds that this is a coCartesian lift. Hence $D \to Func(\{0\}, C) \simeq C$ 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 Joyal model structure $sSet_{Joyal}$.
Use now that accessible quasi-categories are stable under homotopy pullback to conclude that $D_c$ is accessible. Moreover, one can check that $D_c$ has all small colimits. Together this means that $D_c$ is a locally presentable (∞,1)-category. This implies in particular that $D_c$ also has all small (∞,1)-limits and hence contains a terminal object, $f : c \to d$.
We now complete the proof by showing that $f : c \to d$ being terminal in $D_c$ implies that $d$ is an $S$-local object. This is equivalent to showing that for $t : a \to b$ any element in $S$, composition with $t$ induces an equivalence
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 $Hom_C(t,d)$ over a point $g : a \to d$ of $Hom_C(a,d)$ is equivalent to the hom-object $Hom_{C_{a/}}(t,g)$ in the under-quasi-category $C_{a/}$.
This in turn can be checked to be equivalent to $Hom_{C_{d/}}(g_* t, Id_d)$, where $g_* t$ is the $(\infty,1)$-categorical pushout
in $C$. Notice that $g_* t$, being a pushout of $t \in S$, is itself in $S$.
Now pick a composite
and observe that we have an isomorphism of simplicial sets
(where $s_1$ 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
because $Hom_{(C_{c/})_{f/}}(\sigma , s_1(f)) = Hom_{C_{f/}}(\sigma , s_1(f))$.
Finally we can use that $f$ is terminal in the full subcategory $D_c$ of $C_{c/}$ that contains $g_* t \circ f$. This implies that the above morphism goes between contractible $\infty$-groupoids and hence has contractible homotopy fibers.
Let $f \colon \mathcal{C} \to \mathcal{D}$ be an (∞,1)-functor between presentable (∞,1)-categories, and let $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ be a reflective sub-$(\infty,1)$-category. If $f$ has a right adjoint (∞,1)-functor $f^*$, then
is also a reflective sub-$(\infty,1)$-category.
This is (Lurie, lemma 5.5.4.17).
By prop. , $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ is the inclusion of the $S$-local objects for some class $S$ of morphisms of $\mathcal{C}$. By adjunction it follows that $\mathcal{D}^0$ is precisely the class of $f(S)$-local objects, and hence is a reflective subcategory, again by prop. .
Let $C$ be a left proper combinatorial simplicial model category which presents an (∞,1)-category $\mathcal{C} \simeq C^\circ$.
Then if $\mathcal{C}^0 \hookrightarrow \mathcal{C}$ is an accessible reflective inclusion with reflector $L \colon \mathcal{C} \to \mathcal{C}^0$, then there exists a corresponding left Bousfield localization
of the model category $C$ which presents this inclusion in that
an object in $C'$ is a fibrant object precisely if it is fibrant as an object of $C$ and in addition its image in the homotopy category $Ho(C) \simeq Ho(\mathcal{C})$ is in the inclusion $Ho(\mathcal{C}^0) \hookrightarrow Ho(\mathcal{C})$;
a morphism in $C'$ is a weak equivalence precisely if under $Ho(L) \colon Ho(C) \simeq Ho(\mathcal{C}) \to Ho(\mathcal{C}^0)$ is an isomorphism.
This is (Lurie, prop. A.3.7.8).
Use that by the above discussion $\mathcal{C}^0$ is the full subcategory on $S$-local objects for a small set of morphisms. By the discussion at Bousfield localization of model categories this presents precisely such localizations.
Extra conditions on a reflective sub-$(\infty,1)$-category of relevance are
The following proposition characterizes when a reflective subcategory of an accessible (∞,1)-category $C$ is accessible
Let $C$ be an accessible (∞,1)-category and
a reflective subcategory. Then the following conditions are equivalent:
$D$ is itself accessible;
The localization $Loc : R\circ L : C \to C$ is an accessible (∞,1)-functor.
There exists a small set $S_0 \subset S := L^{-1}(equiv.)$ such that every $S$-local object is also $S_0$-local.
This is (Lurie, prop. 5.5.1.2 and prop. 5.5.4.2, part 3).
This is work…
Recall that the reflective subcategory $D \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} C$ is exact – or $L$ an exact localization – if $L$ is a left exact functor in that it preserves finite limits. Accordinly we say:
An exact reflective sub-$(\infty,1)$-category is a reflective sub-$(\infty,1)$-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 $S = L^{-1}(equiv) \subset Mor(C)$ of those morphisms, that $L$ sends to equivalences in $D$.
The following propositions say how the property that $L$ preserves finite limits is characterized by pullback-stability properties of $S$.
(recognition of exact localization)
A reflective sub-$(\infty,1)$-category $D \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} C$ such that $C$ has all finite limits is exact precisely if the collection $S := L^{-1}(equiv) \subset Mor(C)$ of morphisms that $L$ sends to equivalences is stable under pullback.
So if for every pullback diagram
we have that if $L(f)$ is an equivalence then also $L(f')$ is an equivalence.
This is HTT, prop. 6.2.1.1.
If $L$ preserves finite limits, then it preserves pullbacks, so that $L(f')$ is a pullback of the equivalence $L(f)$, hence itself an equivalence.
So it remains to check that, conversely, stability of $S$ under pullback implies that $L$ preserves finite limits. By a general characterization of left exact functors (see there) it suffices to check that $L$ preserves the terminal object and all pullbacks.
Since the terminal object is evidently $S$-local, we have $L * \simeq *$.
Next we check that $L$ 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 $L(x \times y) \to L(x) \times L(y)$ induced from the units $i_x : x \to L x$ and $i_y : y \to L y$ is in $S$. From inspection of the diagram
one finds that $x \times y \to L x \times L y$ is a pullback of $i_x$. Hence is in $S$, by assumption. Similarly in
one see that $L x \times y \to L x \times L y$ is a pullback of $i_y$ and hence in $S$. The composite of these two morphisms is a morphism $x \times y \to L x \times L y$, which is in $S$ since $S$ is closed under composition. Applying $L$ hence yields an equivalence $L(x \times y) \stackrel{\simeq}{\to} L x \times L y$.
We now apply the same kind of argument to show that $L$ respects more generally pullbacks.
For that, first notice that for $x \to y \leftarrow z$ a diagram in $C$, the pullback $L x \times_{L_y} L_z$ of the image exists in $C$, by assumption, but is easily seen to be $S$-local and hence lands in $D$. Therefore to show that we have an equivalence $L(x \; \times_y z) \simeq L x \; \times_{L y} \times L z$ it is sufficient to show that the natural morphism, $x \times_y z \to L x \times_{L y} L z$ induced from the morphism of diagrams
in $C$ with the adjunction unit morphism on the horizonatals, is in $S$. By passing along these units one at a time
this may be decomposed as a composite of three morphisms
If we equivalently reformulate these pullbacks as equalizers then this is
It is immediate to check that the two bottom left squares are pullback squares. So the two left vertical morphisms are pullbacks of $(Id, i_z)$ and $(i_x, Id)$, respectively. Of morphisms of this form we had seen above that they are in $S$. Hence by the assumed pullback-stability of $S$ also $x \times_{L y} z \to L x \times_{L y} z \to L x \times_{L y} L z$ is in $S$.
So it remains to show that $x \times_y z \to x \times_{L y} z$ is in $S$. We claim that this morphism in turn may be expressed as a pullback
of the diagonal $y \to y \times_{L y} y$. To see this notice that cones $q$ over the corresponding pullback diagram are equivalently diagrams
So now we need to show that the diagonal $y \to y \times_{L y} y$ is in $S$.
To see this, notice that it has a left inverse $y \times_{L y} y \to y$, given by any one of the two projections. So if finally we show that this is in $S$, we are done, since $S$ satisfies 2-out-of-3. But this follows now from pullback stability of $S$, because this projection is the pullback of $y \to L y$ along itself.
(accessibility of exact localizations)
Let $C$ be a locally presentable (∞,1)-category with universal colimits. Assume moreover that finite limits commute with filtered colimits in $C$ (this holds for example if $C$ is an (∞,1)-topos). Let $S_0 \subset Mor(C)$ be a small set of morphisms, and $S$ the smallest strongly saturated class containing $S_0$ and stable under pullbacks. Then $S$ 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.
If $C$ has a terminal object, then the full subcategory on terminal objects is a reflective subcategory of $C$.
let $p : C_1 \to C_0$ be a coCartesian fibration and $D_0 \stackrel{\leftarrow}{\hookrightarrow} C_0$ a reflective $(\infty,1)$-subcategory of the base.
The restriction $D_1 \hookrightarrow C_1$ of $C_1$ over $D_0$, i.e. the strict (say in sSet if everything is modeled by quasi-categories) pullback
is itself a reflective $(\infty,1)$-subcategory of $C_1$.
By the above proposition on reflectors, it is sufficient to produce for every $c \in C_1$ there is a reflection morphism $f : c \to d$ with $d \in D_1$.
Such $f$ is obtained by choosing any coCartesian lift of a reflector $p(f) : p(c) \to \bar d$.
To see this, consider for every object $e \in D_1$ the diagram
By assumption $p(f)$ is a reflector, hence the bottom morphism is an equivalence. By one of the characterizations of coCartesian morphisms, the fact that $f$ is a coCartesian lift means that this diagram is a (homotopy) pullback diagram. This means that also the top horizontal morphism is an equivalence.
If $C = PSh_{(\infty,1)}(K)$ is the (∞,1)-category of (∞,1)-presheaves on some small $(\infty,1)$-category $K$, then accessibly embedded reflective subcategory
(i.e. one where the inclusion is an accessible (∞,1)-functor) is a locally presentable (∞,1)-category.
If $C = PSh_{(\infty,1)}(K)$ is the (∞,1)-category of (∞,1)-presheaves on some small $(\infty,1)$-category $K$, then an accessibly embedded exact reflective subcategory
is an (∞,1)-category of (∞,1)-sheaves on $K$ – an (∞,1)-topos. We have:
the collection of morphism $S = L^{-1}(equiv.)$ that are sent to weak equivalences are the analog of local isomorphisms of ordinary sheaf theory;
the $S$-local objects are the ∞-stacks ;
the localizaton functor $Loc = R \circ L : PSh_{(\infty,1)}(K) \to PSh_{(\infty,1)}(K)$ is ∞-stackification .
Let $K = Alg_k^{op}$ be the opposite of the category of $k$-associative algebras, regarded as a site with the fpqc-topology. Then an object in $Sh_{(\infty,1)}(Alg_k^{op})$ may be regarded as an algebraic $\infty$-groupoid. The infinitesimal version is an Lie ∞-algebroid, which may be identified with an object in $(Alg_k^\Delta)^{op} \simeq (dgAlg_k)^{op}$ – the opposite of the category of cosimplicial algebras. The simplicial model structure on cosimplicial algebras, presents this as an $(\infty,1)$-category $(Alg_k^\Delta)^\circ$
The Yoneda embedding induces an inclusion
which is a reflective embedding. It exhibits localization at $A^1$-cohomology, where $A^1 = Spec k[x]$ is the algebraic line object.
This is discussed at rational homotopy theory in an (∞,1)-topos.
The general theory is discussed in section 5.2.7 of
A Coq-formalization of left-exact reflective sub-$(\infty,1)$-categories in homotopy type theory is in