Higher Sheaves and Left-Exact Localizations of \infty-Topoi
aa r X i v : . [ m a t h . C T ] J a n Higher Sheaves and Left-Exact Localizations of ∞ -Topoi Mathieu Anel ∗ ,Georg Biedermann † ,Eric Finster ‡ ,and Andr´e Joyal § Abstract
We propose a definition of higher sheaf with respect to an arbitrary set of maps Σ in an ∞ -topos E .We then show that the associated reflection E → Sh ( E , Σ ) is left-exact so that the subcategory of sheaveswith respect to Σ is itself an ∞ -topos. Furthermore, we show that the reflection E → Sh ( E , Σ ) may becharacterized as the left-exact localization generated by Σ. In the course of the proof, we study theinteraction of various types of factorization systems, and make essential use of the notion of a modality ,that is, a factorization system whose left class is stable by base change. Contents R -equivalences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.6 Congruence Classes and Higher Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 The theory of ∞ -topoi ressembles ordinary topos theory in many aspects, but there are some importantdifferences. A striking example is the absence, in the case of ∞ -topoi, of any concrete notion of sheaf resem-bling the classical one. Recall that any Grothendieck 1-topos E arises as a reflective left-exact localization ofthe category of presheaves of sets P ( C ) ∶= [ C op , Set ] on a small category C , and that any such localizationuniquely corresponds to a Grothendieck topology on C . The objects in the reflective subcategory of P ( C ) are then the sheaves for the topology. Furthermore, the Grothendieck topology itself may be viewed as a ∗ Department of Philosophy, Carnegie Mellon University, [email protected] † Departamento de Matem´aticas y Estad´ıstica, Universidad del Norte, [email protected] ‡ Cambridge University, ericfi[email protected] § CIRGET, UQ `AM. [email protected] monomorphisms in P ( C ) , and an equivalent characterization of the sheaves is thatthey are exactly the objects which are local for this class. In other words, the left-exact localizations of P ( C ) can always be presented as the inversion of some monomorphisms.In the theory of ∞ -topoi, the category of sets is replaced by the ∞ -category of ∞ -groupoids S . It remainstrue that every ∞ -topos E is a left-exact localization of the category of presheaves P ( C ) ∶= [ C op , S ] on asmall ∞ -category C , but the collection of all such localizations can no longer be completely described usingGrothendieck topologies. More precisely, it is not true anymore that every left-exact localization of P ( C ) is generated by inverting monomorphisms. Recall from [HTT, Proposition 6.5.2.19.], that every left-exactlocalization P ( C ) → E is the composite of a topological localization P ( C ) → E ′ followed by a cotopological localization E ′ → E . While topological part can indeed still be described entirely as inverting a class ofmonomorphisms, the cotopological part is a localization which inverts no monomorphisms. The classicaltheory of sheaves and Grothendieck topologies can therefore be applied to the topological localizations, butnot to the cotopological ones. Our goal in the present paper is to introduce a general notion of sheaf withrespect to an arbitrary set of maps Σ ⊆ E (not necessarily monomorphisms) in an ∞ -topos E such that thecategory of sheaves is exactly the left-exact localization generated by Σ.As the rest of the introduction will be solely concerned with the higher categorical situation, we willfrom now on drop the prefix “ ∞ ” when refering to ∞ -topoi and ( ∞ , ) -categories, and speak explicityof 1-categories and 1-topoi if the occasion arises. See Section 2.1 for a more complete description of ourconventions with respect to higher category theory.We will need the following construction: for a map u ∶ A → B in E , we denote by ∆ ( u ) ∶ A → A × B A itsdiagonal and by ∆ n ( u ) its n -th iterated diagonal (with the convention that ∆ ( u ) ∶= u ). For a set of mapsΣ, we write ∆ ( Σ ) = { ∆ ( u ) ∣ u ∈ Σ } for the set of all diagonals of maps in Σ and we define the diagonalclosure of Σ by ∆ ∞ ( Σ ) = { ∆ n ( u ) ∣ u ∈ Σ , n ∈ N } . Definition 3.6.6.
Let Σ be a set of maps in a topos E . We say that an object X ∈ E is a Σ -sheaf if, forevery u ∶ A → B ∈ ∆ ∞ ( Σ ) and every base change u ′ ∶ A ′ → B ′ of u , the map Map ( B ′ , X ) → Map ( A ′ , X ) isinvertible. We write Sh ( E , Σ ) for the full subcategory of Σ-sheaves.Our main theorem is then the following: Theorem 3.6.8.
Let Σ be a set of maps in a topos E . Then the subcategory Sh ( E , Σ ) is reflective and thereflector ρ ∶ E → Sh ( E , Σ ) is left-exact. In particular, Sh ( E , Σ ) is a topos. Furthermore, the reflector ρ invertsthe maps in Σ universally among cocontinuous and left-exact functors.In practice, we will view Definition 3.6.6 as a composite of several, simpler definitions, and the proof ofTheorem 3.6.8 arises from a careful examination of the interactions of these more basic notions. For example,the reader will no doubt have recognized the condition on the equivalence induced on mapping spaces as aninstance of the notion of local object , well known in the literature on localizations. Recall that an object X of a category E is said to be local for a set of maps Σ ⊆ E if the map Map ( u, X ) ∶ Map ( B, X ) → Map ( A, X ) is invertible for every map u ∶ A → B in Σ. We write Loc ( E , Σ ) ⊆ E for the full subcategory of E spanned bythe Σ-local objects.Perhaps less well known is the notion of a modal object in an topos. We say that an object X ∈ E is modal with respect to a set of maps Σ ⊆ E if it is local with respect to every base change of every map inΣ. We write Mod ( Σ , E ) ⊆ E for the full subcategory spanned by the Σ-modal objects. It is worth notingthat if G ⊆ E is a set of generators of the presentable category E , then it suffices to consider base changesalong maps having their domain in G . It follows that the modal objects with respect to a set of maps Σ canthemselves be described as the local objects with respect to another set of maps, albeit a slightly larger one.In view of the preceding discussion, our definition of Σ-sheaf in a topos E may now be rephrased asfollows: an object X ∈ E is a Σ-sheaf if it is modal for ∆ ∞ ( Σ ) , the diagonal closure of Σ. Furthermore, wehave a nested sequence of subcategoriesSh ( E , Σ ) ⊆ Mod ( E , Σ ) ⊆ Loc ( E , Σ ) ⊆ E E . The inclusions are, in general, all proper, and one of the main objectives ofthe present article is to understand when they coincide.An important technical tool throughout this work is the connection between the three reflective sub-categories just described and the theory of factorization systems . Recall briefly that a factorization system S = ( L , R ) on a category E consists of two classes of maps L and R , which are orthogonal and for whichevery map f ∶ A → B ∈ E admits a factorization f = A u Ð→ S ( f ) v Ð→ B where v ∈ L and u ∈ R . (See Section 3.1for precise definitions.) Let us write R ( B ) ⊆ E / B for the full subcategory of the slice category of E at B consiting of those objects whose structure map lies in R . Then the association f ↦ v determines a reflection E / B → R ( B ) . In other words, the factorization system S provides us with a family of reflective subcate-gories, one for each slice of E . In particular, if E has a terminal object 1, we obtain a distinguished reflectivesubcategory R ( ) .A basic observation, then, is that when E is a topos and Σ is a set of maps of E , then each of thesubcategories Sh ( Σ , E ) , Mod ( Σ , E ) and Loc ( Σ , E ) introduced above can be realized as a subcategory of theform R ( ) for a particular type of factorization system generated by Σ (even if the factorisation system isnot necessarily unique, as we shall see). These various types of factorization systems are distinguished bythe closure properties of their left classes. As a first step, let us note that any factorization system hasthe property that its left class contains all the isomorphisms, is closed under composition and formation ofcolimits in the arrow category (Lemma 3.1.11). Following [HTT, Definition 5.5.5.1.], we refer to classes ofmorphisms satisfying these properties as saturated classes .Recall that if E is a presentable category, then every set of maps Σ generates a factorization system whoseleft class L can be identified as the saturation of Σ, that is, its closure under the properties listed above. Wedenote this class by Σ s . It is easily seen in this case that the subcategory R ( ) coincides with Loc ( E , Σ ) .A related, but distinct, construction arises from the fact that the subcategory Loc ( E , Σ ) is reflective. Thisleads to another factorization system associated to Σ: namely, the one whose left maps are those inverted bythe associated reflector E → Loc ( E , Σ ) . The left class of this factorization system has the additional propertythat, in addition to being saturated, it is closed under the familiar 3-for-2 axiom. Following [HTT, Definition5.5.4.5.], we say that it is strongly saturated and we call the factorization systems which arise in this manner reflective for obvious reasons. Furthermore, it may be shown that the left class of the reflective factorizationsystem generated by Σ is exactly the strongly saturated closure of Σ, which we denote Σ ss .We say that a factorization system S = ( L , R ) is a modality if the class L is stable by base change.The subcategory Mod ( E , Σ ) from above may then be recovered as the subcategory R ( ) associated to themodality generated by Σ. In addition to being saturated, the left class of a modality is stable by base changeby definition. We call a saturated class with this property acyclic and it can be shown that the left class ofthe modality generated by Σ is its acyclic closure, which we denote Σ a . Modalities play a key role in thepresent work, and we discuss some motivation for this approach below.Finally, we say that a factorization system S = ( L , R ) is a left-exact modality if L is stable under theformation of all finite limits in the arrow category. A saturated class of maps with this property will bereferred to as a congruence class . As we will see, if we know in advance that S is a modality, then it sufficesthat S additionally be reflective in order that it be left exact. Every set of maps Σ generates a smallestcongruence class denoted Σ c , and one central goal of the current work is to give a concrete description of Σ c .We summarize the previous discussion in Table 1.As it happens, Theorem 3.6.8 can be very simply rephrased in terms of modalities, since as we willshow left-exact modalities are in bijective correspondence with left-exact localizations (see Proposition 3.4.8).Hence we can attempt to understand these localizations by understanding how left-exact modalities differfrom ordinary modalities. Concretely, if we would like to understand what the left-exact modality generatedby a collection of maps is, we can first attempt to answer a related question: under what conditions is themodality generated by a collection of maps already left exact? This question turns out to have a relativelystraightforward answer: 3able 1: Types of Factorization Systems ( L , R ) unique factorizationsystem reflectivefactorization system modality left-exact modality L closureproperties saturated class strongly saturatedclass acyclic class congruence class(stable undercolimits) (stable undercolimits and 3-for-2) (stable undercolimits andbase change) (stable undercolimits andfinite limits) R ( ) Local objects Modal objects SheavesLoc ( Σ , E ) Mod ( Σ , E ) Sh ( Σ , E ) Left classgenerated byΣ Σ s Σ ss Σ a Σ c AssociatedLocalizations E → Loc ( E , Σ ) universal forcocontinuousfunctors E → Sh ( E , Σ ) universal for lexcocontinuousfunctors Theorem 3.5.7.
Let ( L , R ) be a modality generated by a set Σ of maps in a topos E . If ∆ ( Σ ) ⊆ L , thenthe modality ( L , R ) is left-exact.Furthermore, the theorem immediately provides us with a method of generating left-exact modalities:starting with some set of maps Σ, we may pass to its diagonal closure ∆ ∞ ( Σ ) , since then the condition ofthe theorem is automatic. And indeed, it is not hard to see that the modality generated by ∆ ∞ ( Σ ) is exactlythe left-exact modality generated by Σ (since any left-exact modality will necessarily contain the diagonalsof all its generating maps).The proof of Theorem 3.5.7 leads us to consider a fourth kind of factorization system, namely the cartesianfactorization systems described in Section 3.2. A cartesian factorization system is one which is reflective,and where the left class L is stable by base change along maps in R . These factorization systems play twoimportant but distinct roles in the proof. A first observation is that, if a factorization system is at the sametime a modality and cartesian, then it is necessarily a lex modality. In a certain sense, this observation issomewhat trivial since a cartesian factorization system is, by definition, reflective, and as we have pointedout above, a modality which is reflective is automatically left-exact. (See Theorem 3.4.7). But the forceof this observation comes from the fact that the left class of a cartesian factorization system admits thefollowing relatively simple characterization: Proposition 3.2.10.
A factorization system ( L , R ) in a category with finite limits E is cartesian if and onlyif the pullback functor u ∗ ∶ R ( B ) → R ( A ) is an equivalence of categories for every map u ∶ A → B in L .In view of the discussion above, this means that we can show that a modality is left-exact by showingthat for each map u ∈ L the associated base change functor is an equivalence of categories. In fact, it turnsout to be useful to stabilize this condition by base change, leading to the following definition: Definition 3.5.1.
Let ( L , R ) be a modality in a category with finite limits E . We shall say that a map u ∶ A → B in E is an R - equivalence if the functor u ⋆ ∶ R ( B ) → R ( A ) is an equivalence of categories. We shallsay that u is a fiberwise R -equivalence if every base change of u is an R -equivalence.4he usefulness of the notion of fiberwise R -equivalence comes from the following proposition, which showsthat the hypotheses of Theorem 3.5.7, namely that we have ∆ ( Σ ) ⊆ L for Σ a set of generators for a modality ( L , R ) , turn out to assert exactly that every generating map u ∈ Σ is a fiberwise R -equivalence. Proposition 3.5.2.
Let ( L , R ) be a modality in a category with finite limits E . Then a map u ∶ A → B in E is a fiberwise R -equivalence if and only if the maps u and ∆ ( u ) both belong to L .We would like to conclude from this that every map f ∈ L is a fiberwise R -equivalence. And since the leftclass of a modality generated by a set of maps Σ may be identified as Σ a , that is, the acyclic class generatedby Σ, this follows as soon as we have the following: Proposition 3.5.6.
Let ( L , R ) be a modality in a topos E . Then the class of fiberwise R -equivalences isacyclic.That the fiberwise R -equivalences contain identities and are closed under composition and base changeis immediate from the definition. Hence the only remaining task is to show that they are closed undercolimits. It is at this point that the second use of the theory of cartesian factorization systems makesits appearance. As it happens, a rich source of cartesian factorization systems comes from the theory ofGrothendieck fibrations (Section 2.4). A crucial observation, then, is that for a topos E and small indexingcategory K , the colimit functor colim ∶ [ K , E ] → E is a Grothendieck fibration, and hence gives rise to acartesian factorization system in which the right maps are exactly the cartesian natural transformations ofdiagrams. This can be seen as a reformulation of the property of descent in a topos. A careful calculationin this fibration of diagrams finally leads to the desired result.Finally, we would like to make some remarks of the use of the concept of modality in this work. Indeed,the appearance of the intermediate notion of modal object and modality may seem surprising given that ourstated goal is to understand left-exact localizations, and modalities do not have a simple analog in terms oflocalization theory. There are at least two reasons, however, to recommend the approach taken here. A firstis that, though modalities have not received a great deal of attention in the literature on category theoryand homotopy theory, they do arise quite naturally from the logical perspective on topos theory, a point ofview whose higher analog goes by the name Homotopy Type Theory . From the perspective of the internallanguage, modalities appear as operations on types similar to the operators of modal logic, which is wherethe concept gets its name. Indeed, many of the early properties of modalities were first worked out in inthis setting [HoTT, RSS]. Furthermore, our original approach to Theorem 3.5.7 was very much inspired byideas from type theory, and in fact, the theorem can be given a completely type-theoretic proof. Of course,other approaches are also possible: the language of the present paper is entirely categorical, as is that of[AS, Theorem 3.4.16], which presents an account based on the small object argument and sheafificationtechniques.A second motivation is that, owing to the ubiquity with which they appear in various parts of homotopytheory and higher category theory, we feel modalities deserve to be better known. For example, the factor-ization system given by factoring a map as an n -connected map followed by an n -truncated one, well knownin classical homotopy, is a modality. As is the factorization system obtained from Quillen’s + -construction,whose left class consists of the so-called acyclic maps [Raptis]. Even the A -local objects of motivic homotopytheory are part of a modality on the Nisnevich topos [MV]. Furthermore, arbitrary modalities in a topos maybe seen as generalizations of the notion of closed class introduced by Dror-Farjoun. (See Remark 3.3.10).In short, these objects appear frequently (if not always explicitly) in the literature, and we hope the presentpaper will contribute to their adoption and study. Acknowledgments:
The authors would like to thank Mike Shulman, Egbert Rijke and Simon Henry foruseful discussions on the material of this paper. The first author gratefully acknowledges the support of theAir Force Office of Scientific Research through grant FA9550-20-1-0305 and MURI grant FA9550-15-1-0053.The fourth author acknowledge the support of the Natural Sciences and Engineering Research Council ofCanada through grant 70733401. 5
Preliminaries
Throughout the paper, we use the language of higher category theory. The word category refers to ( ∞ , ) -cat-egory, and all constructions are assumed to be homotopy invariant. When necessary, we shall refer to anordinary category as a and to an ordinary Grothendieck topos as a . Furthermore, wework in a model independent style, which is to say, we do not choose an explicit combinatorial model for ( ∞ , ) -categories such as quasicategories, but rather give arguments which we feel are robust enough to holdin any model.We use the word space to refer generically to a homotopy type or ∞ -groupoid and write S for the categoryof spaces. We shall denote by C ( A, B ) or by Map C ( A, B ) the space of maps between two objects A and B of a category C and f ∶ A → B to indicate that f ∈ C ( A, B ) . We write A ∈ C to indicate that A is an objectof C . The opposite of a category C is denoted C op and defined by the fact that C op ( B, A ) ∶= C ( A, B ) with itscategory structure inherited from C . We write C / A the slice category of C with respect to an object A . If f ∶ X → A is a morphism of C , we often write ( X, f ) ∈ C / A , as it is frequently convenient to have both theobject and structure map visible when working in a slice category. If a category C has a terminal object, wedenote it by 1.Every category C has a homotopy category ho C which is a 1-category with the same objects as C , butwhere ho C ( A, B ) = π C ( A, B ) . We shall say that a morphism f ∶ A → B in C is invertible , or that it is an isomorphism if the morphism is invertible in the homotopy category ho C . We write A ≃ B to indicate thattwo objects are isomorphic. We make a small exception to this terminology with regard to categories andspaces: for these objects, we continue to employ the more traditional term equivalence .We shall say that a functor F ∶ C → D is essentially surjective if for every object X ∈ D there exists anobject A ∈ C together with an isomorphism X → F A . We shall say that a functor F ∶ C → D is fully faithful if the induced map C ( A, B ) → D ( F A, F B ) is an equivalence for every A, B ∈ C . A functor F ∶ C → D is an ifit is fully faithful and essentially surjective. We denote the category of functors from C to D alternatively by [ C , D ] or D C as seems appropriate from the context. For a small category C , we will write P ( C ) ∶= [ C op , S ] for the category for presheaves on C .We say an object is unique if the space it inhabits is contractible. For example, the inverse of anisomorphism is unique in this sense. We assume that all subcategories and classes of maps in a category aredefined by properties which are invariant under isomorphism, and consequently adopt the convention thatall subcategories are replete.We will denote by [ ] the poset { < } , regarded as a category. If C is a category, then the functorcategory C [ ] can be described as the category of arrows of C . By construction, an object of C [ ] is a map u ∶ A → B in C , and a morphism ( f, g ) ∶ u → u ′ in C [ ] is a commutative square A A ′ B B ′ u f u ′ g For any class of maps A in a category C , we shall write A for the full subcategory of C [ ] whose objects arethe maps in A . This construction permits us to succinctly describe closure properties of the class A in termsof the associated subcategory A . For example, we say that A is closed under colimits or limits if this is trueof A .Given a commutative square A BC D f g hk
6n a category C with finite limits we refer to the canonically induced map ( f, g ) ∶ A → C × D B as the cartesiangap map of the square.Finally, for an object A of a category C with finite limits, we will write ∆ ( A ) = ( A , A ) ∶ A → A × A forthe canonical map, which we refer to as the diagonal of A . More generally, the diagonal of a map u ∶ A → B is defined to be the canonical map ∆ ( u ) = ( A , A ) ∶ A → A × B AA A × B A AA B ∆ ( u ) A A f p p ⌜ uu induced by the universal property of the pullback. This construction can be iterated, and we use the notation∆ i ( u ) for the i -th iterated diagonal of a map or object. We shall say that a functor F ∶ E → F inverts a map f ∈ E if the map F ( f ) ∈ F is invertible. Similarly, F inverts a class of maps Σ ⊆ E if it inverts every map in Σ. A functor F ∶ E → F is said to be a Σ- localisation if it is initial in the category of functors which inverts Σ. More precisely, for any category G , let us denoteby [ E , G ] Σ the full subcategory of [ E , G ] spanned by the functors E → G inverting the class Σ. If a functor F ∶ E → F inverts a class Σ, then the composition functor ( − ) ○ F ∶ [ F , G ] → [ E , G ] induces a functor ( − ) ○ F ∶ [ F , G ] → [ E , G ] Σ and F is a Σ-localization if and only if this functor is an equivalence of categories for every category G . Weshall say that a functor F ∶ E → F is a localisation if it is a Σ-localisation with respect to some class of mapsΣ ⊆ E . If Σ is a class of maps in a category E , then the codomain of any Σ-localization is unique up toequivalence of categories, and we denote this category generically by E [ Σ − ] .The notion of localization introduced in the proceeding paragraph is the most general, but certain vari-ations also appear naturally. For example, if E and F are cocomplete categories and Σ ⊆ E is a class ofmaps in E , then we can ask that a cocontinuous functor F ∶ E → F invert the maps in Σ universally amongcocontinuous functors . If we write [ E , F ] cc for the category of colimit preserving functors and [ E , F ] Σcc forthe full-subcategory of those functors which invert the maps in Σ, the we will say that F is a cocontinuous Σ -localization if the induced functor ( − ) ○ F ∶ [ F , G ] cc → [ E , G ] Σcc is an equivalence of categories for every cocomplete category G . We will write E [ Σ − ] cc for the codomain ofa generic cocontinuous Σ-localization.Similarly, when E is a topos and Σ is a class of maps of E , then it is natural to consider the notion of a cocontinuous and left-exact Σ -localization which inverts the maps in Σ universally among colimit and finitelimit preserving functors F ∶ E → F , and is defined completely analogously. In this case, we denote the targetof such a localization by E [ Σ − ] lexcc . Remark 2.2.1.
The previous notions of localization need not coincide. For example, if Σ = { S n + → } isthe singleton set consisting of the terminal map from the ( n + ) -sphere to the point in S , then S [ Σ − ] cc isthe category of n -truncated spaces, while S [ Σ − ] lexcc is the terminal category.7hile arbitrary localizations can be difficult to describe in general, those which arise from reflectivesubcategories tend to be considerably more tractable. Recall that a full subcategory E ′ of a category E issaid to be reflective if the inclusion functor E ′ ↪ E has a left adjoint ρ ∶ E → E ′ called the reflector , or the reflection . More generally, we shall say that a functor ρ ∶ E → F is a reflector if it has a fully faithful rightadjoint ι ∶ F → E . In this case, the essentialy image of ι coincides with the subcategory spanned by thoseobjects X such that η ( X ) ∶ X → ιρ ( X ) is invertible, and this subcategory is equivalent to F itself. A reflector ρ is always a localization with respect to the class of morphisms that it inverts, or equivalently the class ofall unit maps η ( X ) ∶ X → ιρ ( X ) .If E ′ is a full subcategory of a category E , we shall say that a map r ∶ X → X ′ reflects the object X into E ′ or simply that r is a reflecting map if X ′ ∈ E ′ and the mapMap ( r, Z ) ∶ Map ( X ′ , Z ) → Map ( X, Z ) is invertible for every object Z ∈ E ′ .Note that a full subcategory E ′ ⊆ E is reflective if and only if, for every object X ∈ E , there exists an object ρ ( X ) ∈ E ′ and a reflecting map η ( X ) ∶ X → ρ ( X ) . Indeed, the choice of a reflecting map η ( X ) ∶ X → ρ ( X ) for each object X ∈ E determines a endofunctor ρ ∶ E → E together with a natural transformation η ∶ Id → ρ (the functoriality of this construction follows form the universal property of the reflecting map). Remark 2.2.2. [HTT, Definition 5.2.7.2] defines a localization to be what we have here called a reflection .In favorable cases, the two notions coincide as we will see below. We prefer, in any case, to distinguish them.See [HTT, Warning 5.2.7.3].
Definition 2.2.3.
An object X in a category E is said to be local with respect to a map u ∶ A → B in E ifthe map Map ( u, X ) ∶ Map ( B, X ) → Map ( A, X ) is invertible. The object X is said to be local with respect to a class of maps Σ ⊆ E if it is local with respectto every map in Σ. We shall denote by Loc ( E , Σ ) the full subcategory of E spanned by the Σ-local objects.This notion is particularly useful in presentable categories [HTT, 5.5]. Recall that a category E is saidto be presentable if it is equivalent to a category Loc ( P ( K ) , Σ ) for a small category K and a (small) set ofmaps Σ ⊆ P ( K ) . Proposition 2.2.4.
Let Σ be a set of maps in a presentable category E . Then the subcategory Loc ( E , Σ ) ⊆ E of Σ -local objects is presentable, reflective, and the reflection E → Loc ( E , Σ ) is equivalent to the localization E → E [ Σ − ] cc .Proof. This follows from [HTT, Propositions 5.5.4.15 and 5.5.4.20].If E is a cocomplete category, we shall say that a class of objects G ⊆ E generates strongly the category E , or that G is a class of generators , if every objects in E is the colimit of a diagram of objects in G .As an example, let P ( K ) = [ K op , S ] be the category of presheaves on a small category K . Then the setof representable presheaves R ( K ) generates strongly P ( K ) . Every presentable category E admits a set ofgenerators.If E is a cocomplete category, then so is the category E / B for any object B ∈ E . If G ⊆ E is a class ofobjects, let us denote by G / B the class of objects ( G, g ) ∈ E / B with G ∈ G . Lemma 2.2.5.
If a cocomplete category E is strongly generated by a class of objects G ⊆ E , then the cocompletecategory E / B is strongly generated by the class G / B ⊆ E / B for every object B ∈ E .Proof. If ( X, f ) is an object of E / B , then the object X is the colimit of a diagram F ∶ K → E of objectsof G , since G strongly generates E . If ι k ∶ F ( k ) → X denotes the inclusion for each object k ∈ K , considerthe functor F ′ ∶ K → E / B defined by putting F ′ ( k ) = ( F ( k ) , f ι k ) for every object k ∈ K . The family ofmorphisms ι k ∶ ( F ( k ) , f ι k ) → ( X, f ) is defining a colimit cone ι ′ ∶ F ′ → ( X, f ) since the family of morphisms8 ∶ F ( k ) → X is defining a colimit cone ι ∶ F → X . Thus, ( X, f ) is the colimit of the diagram F ′ ∶ K → E / B .We have ( F ( k ) , f ι k ) ∈ G / B for every object k ∈ K , since F ( k ) ∈ G for every object k ∈ K . This shows that ( X, f ) is the colimit of a diagram of objects in E / B . We have proved that E / B is strongly generated by theclass G / B . If E is a category and K is a small category, then the diagonal functor δ ∶ E → [ K , E ] takes an object A ∈ E tothe the constant functor δA ∶ K → E with value A . If F is a diagram K → E , then a natural transformation α ∶ F → δA , is a cone with base F and apex A . A cone α ∶ F → δA with apex A ∈ E is the same thing asa diagram K → E / A that we shall denote α / A . Let us denote by K ⋆ K . Let i K ∶ K → K ⋆ C ∶ K ⋆ → E is the same thing as a cone α ∶ F → δA , where F = C ○ i K and A = F ( ) . If the category E iscocomplete, then the restriction functor i ⋆ ∶ [ K ⋆ , E ] → [ K , E ] has a left adjoint γ = i ! ∶ [ K , E ] → [ K ⋆ , E ] which takes a diagram F ∶ K → E to its colimit cone γ ( F ) ∶ F → δ colim F .If the category E has pullbacks, then we define the base change of a cone α ∶ F → δA along a map g ∶ B → A in E to be the base change of α along the natural transformation δ ( g ) ∶ δ ( B ) → δ ( A ) : g ⋆ ( F ) FδB δA g ⋆ ( α ) p αδ ( g ) computed in the functor category. If E is moreover cocomplete, colimits are said to be universal if for anysmall category K , the base change g ⋆ ( α ) ∶ g ⋆ ( F ) → δB of a colimit cone α ∶ F → δA ∶ K ⋆ → E along anymap g ∶ B → A in E is again a colimit cone.A natural transformation u ∶ F → G between two functors F, G ∶ K → E is cartesian if the naturalitysquare F ( i ) F ( j ) G ( i ) G ( j ) u ( i ) F ( r ) u ( j ) G ( r ) is cartesian for every morphism r ∶ i → j in K . Colimits in E are said to be effective (or van Kampen ) if, forany cartesian transformation f ∶ F → G ∶ K → E of K -diagrams, the following square is cartesian for everyobject k ∈ K , F ( k ) colim ( F ) G ( k ) colim ( G ) γ ( F )( k ) f ( k ) colim ( f ) γ ( G )( k ) Definition 2.3.1 (Rezk) . A cocomplete and finitely complete category is said to have descent if its colimitsare universal and effective. A topos is a presentable category with descent.Examples of topoi are the category S of spaces, the category of diagrams [ C , E ] where C is a small categoryand E a topos, as well as the slice category E / A for any object A of a topos E .As a topos E is both complete and cocomplete, given a class of maps Σ in E , we may consider eitherthe cocomplete E → E [ Σ − ] cc or cocomplete and left-exact E → E [ Σ − ] lexcc localizations of E with respect toΣ. Recall from Remark 2.2.1 that the two need not coincide. But it is the latter plays a crucial role in thetheory of topoi because owing to the following: 9 heorem 2.3.2 (Rezk, [HTT, Section 6.1.3]) .
1. A presentable category E is a topos if and only if E is a left-exact localization of a presheaf category2. Any left-exact localization of a topos is a topos In light of Theorem 2.3.2, it is clearly a central problem of higher topos theory to understand the left-exact localization generated by a set of maps Σ in a topos E . On the other hand, as we have seen inProposition 2.2.4, the cocotinuous localization generated by Σ has a straigtforward description in terms oflocal objects. Our strategy, therefore, will be to understand the left-exact localization generated by a classΣ in terms of the cocontinuous localization generated by a class Σ ′ related to Σ. We shall say that a map f ∶ X → Y in a topos E is surjective , or that f is a surjection , ifthe base change functor f ⋆ ∶ E / Y → E / X is conservative. We shall say that a family of maps { g i ∶ Y i → Y } i ∈ I is surjective if the induced map ⟨ g i ∶ i ∈ I ⟩ ∶ ⊔ i ∈ I Y i → Y is surjective. Equivalently, a family of maps { g i ∶ Y i → Y } i ∈ I is surjective if and only if the induced functor E / Y → ∏ i ∈ I E / Y i is conservative. Remark 2.3.4.
Surjective maps as defined above are called as effective epimorphism in [HTT][Section 6.2.3]and covers in [ABFJ2].
Example 2.3.5.
Some useful examples of the previous definition are the following:a) A map between two spaces f ∶ X → Y is surjective if and only if the map π ( f ) ∶ π ( X ) → π ( Y ) is surjective. A pointed object ( X, x ) in a topos E is connected if and only if the map x ∶ → X issurjective.b) If X ∈ S is a space, then the family of maps { x ∶ x Ð→ X } x ∈ X is surjective.c) If E is a topos, K a small category, and F ∶ K → E a K -diagram in E , then the collection of inclusionmaps { ι k ∶ F ( k ) → colim ( F )} k ∈ Ob ( K ) is surjective. Definition 2.3.6 ([HTT, Proposition 6.2.3.14]) . We shall say that a class of maps A in a topos E is local ifthe following two conditions hold:1. A is closed under base change2. Given a map f ∶ X → Y ∈ E and a surjective family { g i ∶ Y i → Y } i ∈ I of Y , if the base change of f along g i belongs to A for every i ∈ I , then f itself belongs to A We shall refer to condition 2 of the definition by saying that a local class descends along surjective families . Example 2.3.7. a) The class of isomorphisms in a topos E is local: if { g i ∶ Y i → Y } i ∈ I is surjective and the base change g ∗ i ( f ) ∶ Y i × Y X → Y i of a map f ∶ X → Y is invertible for every i ∈ I , then f is invertible.b) In view of Example 2.3.5 (b), if A is a local class in the category of spaces S , then a map f ∶ X → Y belongs to A if and only if the map f − ( y ) → A for every y ∶ → Y . In particular, a map f ∶ X → Y in S is invertible if and only the map f − ( y ) → y ∈ Y .10 .4 Grothendieck Fibrations Recall that a functor F ∶ E → B induces a functor F / A ∶ E / A → B / F A for every object A ∈ E . By definition, F / A ( X, f ) = ( F ( X ) , F ( f )) for every object ( X, f ) of E / A . Definition 2.4.1.
We shall say that a functor F ∶ E → B is a (generalized) Grothendieck fibration if thefunctor F / A ∶ E / A → B / F A induced by F is a reflector for every object A in E . Remark 2.4.2.
The notion of Grothendieck fibration introduced above is sligthly more general than thenotion of cartesian fibration in the sense of [HTT, Section 2.4]. It is closely related to Street’s notion of fibra-tion betweem 1-categories [Street] [Gray]. A functor between quasi-categories F ∶ E → B is a Grothendieckfibration if and only if it admits a factorization F = ˜ F ○ W ∶ E → E ′ → B , where W is a categorical equivalenceand F ′ is a cartesian fibration. Example 2.4.3.
Some basic examples of Grothendieck fibrations:a) The unique functor E → E b) Any functor E → B , if B is a groupoidc) The forgetful functor E / B → E for any object B in a category E d) The codomain functor E [ ] → E for any category with finite limits E e) The colimit functor [ K , E ] → E where E is a topos and K is a small category Proposition 2.4.12f) A left-exact reflector φ ∶ E → E ′ , where E and E ′ are categories with finite limits Proposition 2.4.5 Lemma 2.4.4.
Let E be a category with finite limits and F ∶ E → F a functor with a right adjoint G ∶ F → E .Then for every B ∈ E , the induced functor F / B ∶ E / B → F / F B also has a right adjoint.Proof.
Let us write η ∶ Id → GF for the unit of the adjunction. Then we may define a functor G / B ∶ F / F B → E / B as follows: given some X = ( X, f ) in F / F B , we define G / B ( X, f ) to be the base change ( B × GF B
GX, p ) of the map G ( f ) ∶ GX → GF B along the map η ( B ) ∶ B → GF B as in the following diagram: B × GF B
GX GXB GF B p p G ( f ) η ( B ) That this construction is indeed right adjoint to the functor F B follows easily from the adjunction F ⊣ G together with the universal property of the pullback.Furthermore, the counit of the adjunction F / B ⊣ G / B is the map F / B G / B ( X, f ) → ( X, f ) of F / F B whichcorresponds to the map p ∶ B × GF X GX → GX under the adjunction F ⊣ G . And if u ∶ A → B is a map in E ,then the unit ( A, u ) → G / B F / B ( A, u ) of the adjunction F / B ⊣ G / B is the cartesian gap map A → B × GF B
GF A in E / B of the square: A GF AB GF B u η ( A ) GF ( u ) η ( B ) Proposition 2.4.5.
Let E be a category with finite limits and let φ ∶ E → E ′ ⊆ E be a left-exact reflector.Then the functor φ ∶ E → E ′ is a Grothendieck fibration. In particular, every left-exact localization ρ ∶ E → E ′ of a topos E is a Grothendieck fibration. roof. Since E has finite limits and φ is a reflector, we are in the situation of Lemma 2.4.4. We thus obtain,for every B ∈ E , a right adjoint ψ / B ∶ E ′/ φB → E / B to the induced functor φ / B ∶ E / B → E ′/ φB . By thedefinition of Grothendieck fibration, it therefore suffices to show that each of the ψ / B is fully faithful. For f ∶ X → φB ∈ E ′/ φB the description of the counit given in the proof above corresponds, in the case at hand,to the upper horizontal map in the following commutative diagram (where we have slightly abused notationby omitting mention of the inclusion ψ ∶ E ′ ↪ E ) φ ( B × φB X ) XφB φB f But this map is an isomorphism since φ is a reflection which preserves finite limits. Let F ∶ E → B be a Grothendieck fibration. We shall say that a map f ∶ X → Y in E is F -vertical if the map F ( f ) is invertible. We shall say that a map f ∶ X → Y is F -cartesian , or F -horizontal ,the following square is cartesian for every object K ∈ E :Map ( K, X ) Map ( K, Y ) Map ( F K, F X ) Map ( F K, F Y ) F f ○− FF ( f )○− Lemma 2.4.7.
Let F ∶ E → B be a Grothendieck fibration.1. The composite of two F -cartesian morphisms f ∶ X → Y and g ∶ Y → Z is F -cartesian.2. For every object Y ∈ E and every map u ∶ B → F Y there exists a F -cartesian morphism p ∶ E → Y together with an isomorphism i ∶ F E → B such that ui = F ( p ) .3. Every morphism f ∶ X → Y in E is the composite of a F -vertical morphism v ∶ X → E followed by a F -cartesian morphism p ∶ E → Y . This decomposition is unique.Proof. To prove (1), simply note that if f ∶ X → Y and g ∶ Y → Z are F -cartesian, then by definition, thetwo squares of the following diagram are cartesian for every object K ∈ E Map ( K, X ) Map ( K, Y ) Map ( K, Z ) Map ( F K, F X ) Map ( F K, F Y ) Map ( F K, F Z ) F f ○− F g ○− FF ( f )○− F ( g )○− Hence the composite square is cartesian, which shows that the map gf ∶ X → Z is F -cartesian as well.For (2), note that by definition, the functor F / Y ∶ E / Y → B / F Y induced by F is a reflector. Let us write G / Y for its fully faithful right adjoint. Define ( E, p ) ∶= G / Y ( B, u ) . Now, since F / Y is a reflection, we havean isomorhism i ∶ F / Y ( E, p ) = ( F E, F ( p )) → ( B, u ) as claimed. That p ∶ E → Y is cartesian is easily deducedfrom the fact that, for any m ∶ K → Y , one has an equivalenceMap E / Y (( K, m ) , ( E, p )) ≃ Map B / FY (( F K, F ( m )) , ( B, u )) obtained from the adjuction and postcomposition with the isomorphism i .Finally, to prove (3) observe that by property (2) applied to the map F ( f ) ∶ F X → F Y , there existsa F -cartesian morphism p ∶ E → Y together with an isomorphism i ∶ F E → F X such that F ( f ) i = F ( p ) .12y the definition of cartesian morphism, there therefore exists a unique map v ∶ X → E such that pv = f and F ( v ) = i − . The map v is F -vertical, since F ( v ) is invertible. The proof of the uniqueness of thedecomposition f = pv is left to the reader.If F ∶ E → B is a functor, then for every object A ∈ E , the following square of functors commutes E / A EB / F A B A ! F / A F ( F A ) ! where the horizontal functors are forgetful functors. Lemma 2.4.8. If F ∶ E → B is a Grothendieck fibration, then so is the functor F / A ∶ E / A → B / F A for everyobject A in E . Moreover, a map h in E / A is F / A -cartesian (resp. F / A -vertical) if and only if the map A ! ( h ) in E is F -cartesian (resp. F -vertical).Proof. If ( B, f ) ∈ E / A , then ( E / A ) /( B,f ) = E / B and the functor ( F / A ) /( B,f ) ∶ ( E / A ) /( B,f ) → ( B / F A ) / F A ( B,f ) isisomorphic to the functor F / B ∶ E / B → B / F B . Hence the former is a reflector, since the latter is a reflector.This shows that F / A is a Grothendieck fibration. Given a map h ∶ X → B , a straightforward if tediouscalculuation shows that the map h ∶ ( X, f h ) → ( B, f ) in the category E / A is F / A -cartesian if and only ifthe map h ∶ X → B in E is F -cartesian. Finally, by definition, a map h ∶ ( X, f h ) → ( B, f ) in the category E / A is F / A -vertical if and only if the map F / A ( h ) = F ( h ) ∶ ( F X, F ( f h )) → ( F B, F ( f )) in the category B / F A is invertible. But the map F ( h ) ∶ ( F X, F ( f h )) → ( F B, F ( f )) is invertible if and only if the map F ( h ) ∶ F X → F B in the category B is invertible, since the forgetful functor B / F A → B is conservative.Hence a map h ∶ ( X, f h ) → ( B, f ) in the category E / A is F / A -vertical, if and only if the map h ∶ X → B inthe category E is F -vertical. Lemma 2.4.9.
Suppose F ∶ E → B is a Grothendieck fibration which admits a right adjoint G ∶ B → E andlet η ∶ Id → GF be the unit of the adjunction F ⊣ G . Then a morphism f ∶ X → Y in E is F -cartesian if andonly if the following square is cartesian: X GF XY GF Y η ( X ) f GF ( f ) η ( Y ) (1) Proof.
The square (1) is cartesian if and only if the following square is cartesian for every object K in E :Map ( K, X ) Map ( K, GF X ) Map ( K, Y ) Map ( K, GF Y ) f ○− η ( X )○− GF ( f )○− η ( Y )○− (2)On the other hand, we have a natural isomorphism θ X ∶ Map ( F K, F X ) ≃ Ð→ Map ( K, GF X ) by adjunction, from which we deduce that the right hand square in the following diagram is (trivially)cartesian. Map ( K, X ) Map ( F K, F X ) Map ( K, GF X ) Map ( K, Y ) Map ( F K, F Y ) Map ( K, GF Y ) Ff ○− F ( f )○− θ X GF ( f )○− F θ Y (3)13t follows that the square (2) is cartesian if and only if the left hand square of diagram (3) is cartesian, whichproves the lemma. Lemma 2.4.10.
Let F ∶ E → B be a Grothendieck fibration. Then the functor F takes a cartesian square of F -cartesian maps X UY V uf gv (4) to a cartesian square in B . Remark 2.4.11.
In fact, the same conclusion holds if we only suppose that the vertical maps of the cartesiansquare (4) are F -cartesian. This stronger result will be proved below as Lemma 3.2.5 after the introductionof the notion of a cartesian factorization system. Proof.
The functor F / V ∶ E / V → B / F V is a Grothendieck fibration by Lemma 2.4.8 and the following squareof functors commutes E / V EB / F V B V ! F / V F ( F V ) ! where the horizontal arrows are the forgetful functors. Note that the square (4) which we would like to provecartesian is the image by the forgetful functor V ! ∶ E / V → E of the following square ( X, vf ) (
U, g )( Y, v ) ( V, V ) uf gv By Lemma 2.4.8, a map h in E / V is F / V -cartesian if and only the map V ! ( h ) in E is F -cartesian. Moreover, asquare in the category E / V (resp. B / F V ) is cartesian if and only if its image by the functor V ! (resp. ( F V ) ! )is cartesian. But the object ( V, V ) is terminal in the category E / V and the functor F / V ∶ E / V → B / F V is areflector since F is a Grothendieck fibration. It follows from these observations that we may suppose that F ∶ E → B is a reflector and that the object V is terminal in the category E . In this case, the functor F has afully faithful right adjoint G ∶ B → E , since it is a reflector, and the essential image E ′ ⊆ E of the functor G isa reflective subcategory E ′ ⊆ E with reflector R ∶ E → E ′ equivalent to the functor F ∶ E → B . Hence we maysuppose that B = E ′ and that F = R ∶ E → E ′ , so that we have reduced to the case of a reflective subcategory E ′ ⊆ E whose reflector is a Grothendieck fibration.Now, let η ∶ Id → R be the unit of the reflector. First, we observe that 1 ∈ E ′ since the terminal objectis local with respect to every map in E and we may identify E ′ = Loc ( E , Σ ) , where Σ is the class of mapsinverted by the reflector R ∶ E → E ′ . Next, let us show that if a map p Z ∶ Z → E is R -cartesian, then Z ∈ E ′ . Note that the following square is cartesian by Lemma 2.4.9, since p Z is R -cartesian: Z RZ R η ( Z ) p Z R ( p Z ) η ( ) (5)But the map η ( ) is invertible, since 1 ∈ E . Hence the map η ( Z ) is invertible, since the square (5) is cartesianso that we have Z ∈ E ′ . It follows that the square X UY uf (6)14elongs to the subcategory E ′ since every map in the square is R -cartesian by hypothesis, and R -cartesianmaps compose by Lemma 2.4.7. Hence the square (6) is isomorphic to its image by the functor R ∶ E → E ′ ,since the endofunctor R ∣ E ′ ∶ E ′ → E ′ induced by R is isomorphic to the identity functor E ′ → E ′ . But thesquare (6) is cartesian in the subcategory E ′ , since it is cartesian in E and the inclusion E ′ ↪ E is fullyfaithful. Thus the image of the square (6) by the functor R ∶ E → E ′ is cartesian and completes the proof. A fundamental example of Grothendieck fibration arises from the category of diagrams in a topos, as wenow describe.
Proposition 2.4.12. If E is a topos and K is a small category, then the colimit functor colim ∶ [ K , E ] → E is a Grothendick fibration. A natural transformation u ∶ A → B in [ K , E ] is horizontal (that is, cartesianwith respect to the functor colim ) if and only it is cartesian as a natural transformation.Proof. The functor colim ∶ [ K , E ] → E has a right adjoint, namely the diagonal functor δ ∶ E → [ K , E ] sendingand object X ∈ E to the constant K -diagram at X . Since the category [ K , E ] has finite limits, we are inthe situation of Lemma 2.4.4. We have, therefore, a right adjoint δ / B ∶ E / colim ( B ) → [ K , E ] / B to the inducedfunctor colim / B ∶ [ K , E ] / B → E / colim ( B ) for every diagram B ∶ K → E in [ K , E ] .Unfolding the definition, the functor δ / B takes an object X = ( X, f ) of E / colim ( B ) to the object δ / B ( X ) of [ K , E ] / B defined by the pullback square δ / B ( X ) δ ( X ) B δ colim ( B ) p p δ ( f ) η ( B ) The counit of the adjunction colim / B ⊣ δ / B is the map colim ( δ / B ( X )) → X that corresponds to the map p ∶ δ / B ( X ) → δ ( X ) via the adjunction colim ⊣ δ . The cone p ∶ δ / B ( X ) → δ ( X ) is a colimit cone since colimitsare universal in E and the cone η ( B ) ∶ B → δ colim ( B ) is a colimit cone. Hence the map colim ( δ / B ( X )) → X is invertible. This shows that the counit of the adjunction colim / B ⊣ δ / B is invertible and hence that thefunctor colim / B is a reflector. It follows by definition that the functor colim ∶ [ K , E ] → E is a Grothendieckfibration.Now, a map u ∶ A → B in [ K , E ] is horizontal if and only if the unit η ( A, u ) ∶ A → δ / B colim / B ( A, u ) of theadjunction colim / B ⊣ δ / B is invertible. By construction, η ( A, u ) is the cartesian gap map of the followingsquare: A δ colim ( A ) B δ colim ( B ) u γ ( A ) δ colim ( u ) γ ( B ) (7)Hence the map η ( A, u ) is invertible if and only if the square (7) is cartesian. But it is easily seen by descentin the topos E that the square (7) is cartesian if and only if the natural transformation u ∶ A → B is cartesian.This shows that the map u ∶ A → B is horizontal if and only if it is a cartesian natural transformation.15 Higher Sheaves and Left-Exact Localizations
Definition 3.1.1.
Let u ∶ A → B and f ∶ X → Y be two maps in a category E . We say that u is leftorthogonal to f , or that f is right orthogonal to u , if every commutative square A XB Y xu fy has a unique diagonal filler B → X . We shall denote this relation by u ⊥ f . Equivalently, the condition u ⊥ f means that the following square is cartesianMap ( B, X ) Map ( A, X ) Map ( B, ) Map ( A, Y ) Map ( u,X ) Map ( B,f ) Map ( A,f ) Map ( u,Y ) If E has a terminal object 1, we shall say that an object A ∈ E is left orthogonal to a map f ∶ X → Y , andwrite A ⊥ f , if the map A → f . We shall say that a map u ∶ A → B is left orthogonal to an object X ∈ E , and write u ⊥ X , if u left orthogonal to the map X →
1. Finally, we shall say that anobject A ∈ E is left orthogonal to an object X ∈ E , and write A ⊥ X , if the map A → X → Lemma 3.1.2.
Let E be a category with a terminal object . Then an object X ∈ E is right orthogonal to amap u ∶ A → B if and only if X is local with respect to the map u .Proof. A map u ∶ A → B is left orthogonal to the map p ∶ X → ( B, X ) Map ( A, X ) Map ( B, ) Map ( A, ) Map ( u,X ) Map ( B,p ) Map ( A,p ) Map ( u, ) But the map Map ( u, ) is invertible, since Map ( B, ) = = Map ( A, ) . Hence the square is cartesian if andonly if the map Map ( u, X ) is invertible. This shows that u ⊥ p if and only if the object X is local withrespect to the map u .If A and B are two classes of maps in a category E , we shall write A ⊥ B to mean that we have u ⊥ v forevery u ∈ A and v ∈ B . We shall denote by A ⊥ (resp. ⊥ A ) the class of maps in E that are right orthogonal(resp. left orthogonal) to every map in A . We have A ⊆ ⊥ B ⇔ A ⊥ B ⇔ A ⊥ ⊇ B Recall that if A is a class of maps in a category E , then A denotes the full subcategory of E [ ] whoseobjects are the maps in A . Definition 3.1.3.
We say that a pair S = ( L , R ) of classes of maps in a category E is a factorization system if the following three conditions hold:1. L ⊥ R
2. every map f ∶ X → Y in E admits a factorization f = pu ∶ X → E → Y with u ∈ L and p ∈ R L of a factorization system S = ( L , R ) is said to be the left class of the factorization systemand the class R to be the right class . A factorization f = pu ∶ X → E → Y with u ∈ L and p ∈ R is calledan S - factorization of the map f ∶ X → Y ; the data of such a factorization is unique, and we therefore write L ( f ) ∶= u , R ( f ) ∶= p and S ( f ) ∶= E for a specific choice depending on f . X YS ( f ) L ( f ) f R ( f ) Moreover, the operations f ↦ L ( f ) , f ↦ R ( f ) and f ↦ S ( f ) are functorial . When the category E hasa terminal object, the we shall write ∥ X ∥ S for S ( X → ) . We shall see in Proposition 3.1.16 that theendo-functor ∥ − ∥ S ∶ E → E is reflecting the category E into a full subcategory. Example 3.1.4. a) If
Map ( E ) denotes the class of all maps in a category E and Iso ( E ) denotes the class of isomorphisms,then the pair ( Map ( E ) , Iso ( E )) and the pair ( Iso ( E ) , Map ( E )) are (trivial) examples of factorizationsystems.b) A map between two spaces f ∶ X → Y is said to be injective if the square X XX Y ff (8)is a pullback. Equivalently, f ∶ X → Y is injective if it is the composite of a homotopy equivalence f ′ ∶ X → Y ′ with the inclusion of union of connected components Y ′ ⊆ Y . If Surj is the class of(homotopy) surjective maps in S (see Example 2.3.5 (a)) and Inj is the class injective maps, then thepair ( Surj , Inj ) is a factorization system in S .c) More generally, we say that a map f ∶ X → Y in a topos E is injective , or that it is a monomorphism ,if the square (8) is a pullback. If Inj ( E ) is the class injective maps in E and Surj ( E ) is the class ofsurjective maps (Definition 2.3.3), then the pair ( Surj ( E ) , Inj ( E )) is a factorization system in E . Thus,every map f ∶ X → Y in E admits a factorization f = up ∶ X → J → Y with p ∶ X → J a surjective mapand u ∶ J → Y an injective map, X YJ p f u
The subobject ( J, u ) of Y is the image Im ( f ) of the map f . Lemma 3.1.5.
A factorization system S = ( L , R ) in a category E induces a factorization system S / B = ( L / B , R / B ) in the category E / B for every object B ∈ E . By definition, a morphism f ∶ ( X, p ) → ( Y, q ) in E / B belongs to L / B (resp. R / B ) if and only if the map f ∶ X → Y belongs to L (resp. R ). Let A be a class of maps in a category E . If K is a small category, let us denote by A K the class of naturaltransformations α ∶ f → g in [ K , E ] that are objectwise in A , that is, for which the map α ( k ) ∶ f ( k ) → g ( k ) is in A for every object k ∈ K . Lemma 3.1.6.
A factorization system ( L , R ) in a category E induces a factorization system ( L K , R K ) inthe category [ K , E ] for every small category K .Proof. [HTT, Corollary 5.2.8.18] 17e record here for future reference a number of standard facts about the closure properties of the rightand left classes of a factorization system. Recall that if A is a class of maps in a category E , then A denotesthe full subcategory of E [ ] whose objects are the maps in A . We shall say that the class A is closed undercolimits is the full subcategory A ⊆ E [ ] is closed under colimits. The closure of A under finite colimits, orfinite finite limits, etc, is defined similarly. Proposition 3.1.7.
Let ( L , R ) be a factorization system in a category E .1. The classes L and R contain the isomorphisms and are closed under composition2. The class L is closed under cobase change3. The class R is closed under base change4. The class L is right cancellable: vu ∈ L & u ∈ L ⇒ v ∈ L
5. The class R is left cancellable: vu ∈ R & v ∈ R ⇒ u ∈ R
6. The class L is closed under colimits7. The class R is closed under limits8. The intersection L ∩ R is the class of isomorphisms9. L ⊥ = R and L = ⊥ R Proof.
See [HTT, Propositions 5.2.8.6 and 5.2.8.11]
Definition 3.1.8.
Let E be a cocomplete category. We shall say that a class of maps L ⊆ E is saturated ifthe following conditions hold:1. The class L contains the isomorphisms and is closed under composition2. The class L is closed under colimits Remark 3.1.9.
The definition of saturated class given as [HTT, Definition 5.5.5.1] includes the conditionthat a saturated class be stable by cobase change. The following lemma, however, shows that this additionalcondition is automatic.
Lemma 3.1.10. [AS]
Let E be a category with finite limits and let Q ⊆ E be a class of maps which containsthe isomorphisms and is closed under pushouts in E [ ] . Then the class Q is closed under cobase change.Proof. If the following square is a pushout in the category E A A ′ B B ′ u f u ′ g then the following square is also a pushout in the category E [ ] ,1 A A ′ u u ′( A ,u ) ( f,f ) ( A ′ ,u ′ )( f,g ) We have 1 A , A ′ ∈ Q , since the class Q contains the isomorphisms. Thus, u ∈ Q ⇒ u ′ ∈ Q , since the subcategory Q is closed under pushouts. 18 emma 3.1.11. If A is a class of maps in a cocomplete category E , then the class ⊥ A is saturated. The leftclass L of any factorization system ( L , R ) in E is saturated.Proof. The first statement follows from [HTT, Proposition 5.2.8.6.]. The second statement follows from thefirst, since L = ⊥ R by Proposition 3.1.7 Definition 3.1.12.
Let E be a cocomplete category and Σ ⊆ E a class of maps. Then Σ is contained ina smallest saturated class Σ s . We shall say that Σ s is the saturated class generated by Σ. If L ⊆ E is asaturated class in E then we say L is of small generation if L = Σ s for a set of maps Σ ⊆ E . Lemma 3.1.13. If Σ is a class of maps in a cocomplete category E , then ( Σ s ) ⊥ = Σ ⊥ .Proof. We have ( Σ s ) ⊥ ⊆ Σ ⊥ , since Σ ⊆ Σ s . Conversely, the class ⊥ ( Σ ⊥ ) is saturated by Lemma 3.1.11. ThusΣ s ⊆ ⊥ ( Σ ⊥ ) , since Σ ⊆ ⊥ ( Σ ⊥ ) . It follows that Σ s ⊥ Σ ⊥ and hence that Σ ⊥ ⊆ ( Σ s ) ⊥ . Proposition 3.1.14.
Let Σ be a set of maps in a presentable category E . If Σ s ⊆ E is the saturated classgenerated by Σ , then the pair ( Σ s , Σ ⊥ ) is a factorization system.Proof. The saturated class Σ s is of small generation, since Σ is a set of maps. Hence the pair ( Σ s , ( Σ s ) ⊥ ) is a factorization system by [HTT, Proposition 5.5.5.7.]. We have ( Σ s ) ⊥ = Σ ⊥ by Lemma 3.1.13. Hence thepair ( Σ s , Σ ⊥ ) = ( Σ s , ( Σ s ) ⊥ ) is a factorization system.We shall say that the factorization system ( Σ s , Σ ⊥ ) of Proposition 3.1.14 is generated by the set of mapsΣ ⊆ E . If S = ( L , R ) is a factorization system in a category E , then for every object B ∈ E , let us denote by R ( B ) thefull subcategory of E / B whose objects are the maps p ∶ E → B in R . For every X = ( X, f ) ∈ E / B applicationof the functor S produces an object ( S ( f ) , R ( f )) ∈ R ( B ) . X S ( f ) B f L ( f ) R ( f ) Moreover, in view of the commutativity of the above triangle, the map L ( f ) ∶ X → S ( f ) determines amorphism L ( f ) ∶ ( X, f ) → ( S ( f ) , R ( f )) in E / B . Proposition 3.1.15.
Let ( L , R ) be a factorization system in a category E and let B ∈ E . Then for everyobject ( X, f ) ∈ E / B , the morphism L ( f ) ∶ ( X, f ) → ( S ( f ) , R ( f )) reflects the object ( X, f ) into the subcategory R ( B ) . Hence the subcategory R ( B ) ⊆ E / B is reflective. A map r ∶ ( X, f ) → ( X ′ , f ′ ) in E / B exhibits thereflection of ( X, f ) into the subcategory R ( B ) if and only if r ∈ L and f ′ ∈ R .Proof. We have to show that if g ∶ Y → B is a map in R , then for every commutative triangle X YB f h g there exists a unique map h ′ ∶ S ( f ) → Y such that gh ′ = R ( f ) and h ′ L ( f ) = h . X S ( f ) YB f L ( f ) R ( f ) h ′ g h ′ may be obtained as the unique diagonal filler of the following square: X YS ( f ) B L ( f ) h g R ( f ) It remains to prove the last statement of the proposition. If a map h ∶ ( X, f ) → ( Y, g ) in E / B exhibits thereflection of ( X, f ) into the subcategory R ( B ) , let us show that h ∈ L and g ∈ R . We saw above that themap L ( f ) ∶ ( X, f ) → (∥ X ∥ S , R ( f )) in E / B exhibits the reflection of ( X, f ) into the subcategory R ( B ) . Itfollows that the map h ∶ ( X, f ) → ( Y, g ) of E / B is isomorphic to the map L ( f ) ∶ ( X, f ) → (∥ X ∥ S , R ( f )) andhence that h ∈ L . Similarly, g ∈ R .The special case where B = ∈ E is the terminal object will occur frequently. For every object X ∈ E , weshall denote the unique map X → p X ∶ X →
1. We shall write ∥ X ∥ S for the object S ( p X ) and write L ( X ) for the map L ( p X ) ∶ X → ∥ X ∥ S . X ∥ X ∥ S p X L ( X ) Proposition 3.1.16.
Let S = ( L , R ) be a factorization system in a category E with a terminal object .Then the subcategory R ( ) is reflective. Moreover,1. For every object X ∈ E the map L ( X ) ∶ X → ∥ X ∥ S exhibits the reflection of the object X into thesubcategory R ( )
2. A map r ∶ X → X ′ in E exhibits the reflection of X into the subcategory R ( ) if and only if X ′ ∈ R ( ) and r ∈ L
3. An object X ∈ E belongs to R ( ) if and only it is L -local4. Every map in R ( ) belongs to R
5. The functor ∥ − ∥ S ∶ E → R ( ) inverts every map in L Proof.
1. Follows from Proposition 3.1.15 since L ( X ) ∶= L ( p X ) ∶ X → ∥ X ∥ S .2. Follows from Proposition 3.1.15.3. By definition, an object X ∈ E belongs to R ( ) if and only if the map p X ∶ X → R = L ⊥ .But the map p X ∶ X → L ⊥ if and only if the object X is L -local by Lemma 3.1.2.4. If f ∶ X → Y is a map in R ( ) then the maps p X ∶ X → p Y ∶ Y → R . It follows that f ∈ R , since p Y f = p X and the class R is left cancellable by Proposition 3.1.7.5. If f ∶ X → Y belongs to L , let us show that the map ∥ f ∥ S ∶ ∥ X ∥ S → ∥ Y ∥ S is invertible. The followingsquare commutes by naturality X ∥ X ∥ S Y ∥ Y ∥ Sf L ( X ) ∥ f ∥ S L ( Y ) Moreover, the maps L ( X ) and L ( Y ) belong to L by construction. Since L is closed under composition,we have L ( Y ) f ∈ L from which we deduce ∥ f ∥ S L ( X ) = L ( Y ) f ∈ L as well. It follows that the map20 f ∥ S belongs to L , since the class L is right cancellable by Proposition 3.1.7. But we have ∥ f ∥ S ∈ R by (4). Thus, ∥ f ∥ S ∈ L ∩ R and this shows that ∥ f ∥ S is invertible by Proposition 3.1.7. Lemma 3.1.17.
Let ( L , R ) be a factorization system generated by a set of maps Σ in a presentable category E . Then R ( ) = Loc ( E , Σ ) and the category R ( ) is presentable.Proof. An object X ∈ E belongs to R ( ) if and only if the map X → R . But R = Σ ⊥ byProposition 3.1.14. Hence X → R if and only the object X is right orthogonal to every map u ∈ Σ. By Lemma 3.1.2, the object X is right orthogonal to u if and only X is local with respect to u .Thus, R ( ) = Loc ( E , Σ ) . The category Loc ( E , Σ ) is presentable, since the category E is presentable and Σ isa set.If u ∶ A → B is a map in a category E , then the pushforward functor u ! ∶ E / A → E / B is defined byputting u ! ( X, f ) = ( X, uf ) for every map f ∶ X → A . If S = ( L , R ) is a factorization system in E , then thepushforward functor descends to a functor u ♯ ∶ R ( A ) → R ( B ) defined by putting u ♯ ( X, f ) ∶= ( S ( uf ) , R ( uf )) for every map f ∶ X → A in R . X S ( uf ) A B L ( uf ) f R ( uf ) u It follows from the definition that the following square commutes E / A E / B R ( A ) R ( B ) u ! u ♯ where the vertical functors are reflectors.If the category E has finite limits, then the base change functor u ⋆ ∶ E / B → E / A (defined by putting u ⋆ ( Y, g ) = ( A × B Y, p ) for every map g ∶ Y → B ) is right adjoint to the functor u ! ∶ E / A → E / B . If S = ( L , R ) is a factorization system in E , then the functor u ⋆ ∶ E / B → E / A induces a base change functor u ⋆ ∶ R ( B ) → R ( A ) , since the class R is closed under base change by Proposition 3.1.7. Proposition 3.1.18.
Let ( L , R ) be factorization system in a category with finite limits E . For any map u ∶ A → B in E , the functor u ⋆ ∶ R ( B ) → R ( A ) is right adjoint to the functor u ♯ ∶ R ( A ) → R ( B ) .Proof. The adjunction u ! ⊣ u ⋆ is a natural equivalenceMap B ( u ! ( X ) , Y ) ≃ Ð→ Map A ( X, u ⋆ ( Y )) for every X = ( X, f ) ∈ E / A and Y = ( Y, g ) ∈ E / B . If ( Y, g ) ∈ R ( B ) , then u ⋆ ( Y, g ) ∈ R ( A ) . By Proposition 3.1.15,the map L ( uf ) ∶ u ! ( X ) → u ♯ ( X ) reflects the object u ! ( X ) = ( X, uf ) of E / B into R ( B ) . Hence the map − ○ L ( uf ) ∶ Map B ( u ♯ ( X ) , Y ) ≃ Ð→ Map B ( u ! ( X ) , Y ) is invertible. The adjunction u ♯ ⊣ u ⋆ follows. 21 .2 Cartesian Factorization Systems The theory of cartesian factorization systems in 1-categories is developped in [RT], [My]. For cartesianfactorization systems in ∞ -categories, see [Lan].Recall that a class of maps A in a category E is said to have the if every commutativetriangle YX Z gf h having two sides in A in fact has all three sides in A . Equivalently, A is closed under composition and bothleft and right cancellable. Definition 3.2.1.
We shall say that a factorization system ( L , R ) in a category E is cartesian if the class L satisfies the 3-for-2 property and the base change of a map in L along any map in R exists and belongs to L . Lemma 3.2.2.
If a factorization system ( L , R ) in a category E is cartesian, then so is the factorizationsystem ( L / B , R / B ) in the category E / B for every object B ∈ E . Let F ∶ E → B be a Grothendieck fibration. If L ⊆ E is the class of F -vertical maps and R ⊆ E is the class of F -horizontal maps, then the pair ( L , R ) is a cartesian factorization system.Proof. We first show that L ⊥ R . To see this, let u ∶ A → B ∈ L and f ∶ X → Y ∈ R and consider the followingdiagram Map ( B, X ) Map ( A, X ) Map ( B, Y ) Map ( A, Y ) Map ( F B, F X ) Map ( F A, F X ) Map ( F B, F Y ) Map ( F A, F Y ) −○ uf ○− f ○−−○ u −○ F ( u ) F ( f )○− F ( f )○−−○ F ( u ) where the vertical arrows of the cube are the maps induced by the functor F . The left hand face and righthand faces are cartesian since the map f is F -cartesian by hypothesis. On the other hand, the bottom faceof the cube is trivially cartesian, since the map F ( u ) is invertible. It follows that the top face of the cubeis cartesian, which shows that u ⊥ f . Since u and f were aribitrary, we have L ⊥ R . Finally, every map f ∶ X → Y in E admits a factorization f = pv ∶ X → E → Y with v ∈ L and p ∈ R by Lemma 2.4.7. This showsthat the pair ( L , R ) is a factorization system.To see that this factorization system is cartesian, note first that the class L has the 3-for-2 property, sincea map u ∈ E belongs to L if and only if the map F ( u ) is invertible. Hence it only remains then to check thatthe base change of any map u ∶ A → B in L along any map q ∶ B ′ → B in R exists and is in L . Given such a u and q , first note that the morphism F ( u ) ∶ F A → F B is invertible, since u belongs to L . We now considerthe composite F ( u ) − ○ F ( q ) ∶ F B ′ → F A . By Lemma 2.4.7, there exists a F -cartesian morphism p ∶ A ′ → A together with an isomorphism i ∶ F ( A ′ ) → F ( B ′ ) such that F ( u ) − ○ F ( q ) ○ i = F ( p ) . But now, since q is F -cartesian and F ( q ) ○ i = F ( u ○ p ) , we deduce the existence of a unique morphism u ′ ∶ A ′ → B ′ such that22 ○ u ′ = u ○ p and F ( u ′ ) = i . Moreover, clearly u ′ ∈ L since F ( u ) = i and i is invertible by assumption. Weclaim that the square A ′ AB ′ B pu ′ uq is cartesian, which will complete the proof. For this, it suffices to show that the top face of the followingcube is cartesian for every object K ∈ E .Map ( K, A ′ ) Map ( K, A ) Map ( K, B ′ ) Map ( K, B ) Map ( F K, F A ′ ) Map ( F K, F A ) Map ( F K, F B ′ ) Map ( F K, F B ) p ○− u ′ ○− u ○− q ○− F ( u ′ )○− F ( u )○− F ( q )○− But the front face and back faces of the cube are cartesian since both q and p are F -cartesian. Furthermore,the the bottom face of the cube is trivially cartesian, since the maps F ( u ′ ) and F ( u ) are invertible. It followsthat the top face is also cartesian as required. Proposition 3.2.4.
Let E be a category with finite limits and let φ ∶ E → E ′ ⊆ E be a left-exact reflector.Then the class L φ of maps inverted by φ is the left class of a cartesian factorization system ( L φ , R φ ) . If η ∶ Id → φ is the unit of the reflector, then a map f ∶ X → Y in E belongs to R φ if and only if the followingsquare is cartesian X φ ( X ) Y φ ( Y ) . η ( X ) f φ ( f ) η ( Y ) Moreover, R φ ( ) = E ′ .Proof. The functor φ ∶ E → E ′ ⊆ E is a Grothendieck fibration by Proposition 2.4.5 and hence we obtain acartesian factorization system from Proposition 3.2.3. It remains, therefore, only to justify the descriptionof the φ -horizontal maps (the identification of R φ ( ) we leave to the reader).By definition, a map f ∶ X → Y is φ -horizontal if and only if the squareMap ( K, X ) Map ( K, Y ) Map ( φK, φX ) Map ( φK, φY ) φ f ○− φφ ( f )○−
23s cartesian for every K ∈ E . Since φ is a reflector, on the other hand, the above square factors asMap ( K, X ) Map ( K, Y ) Map ( K, φX ) Map ( K, φY ) Map ( φK, φX ) Map ( φK, φY ) −○ η X f ○− −○ η Y φ ( f )○−≃ ≃ φ ( f )○− with the lower vertical maps equivalences as indictated. It follows that the outer square is cartesian if andonly if the upper square is cartesian, and this latter condition is equivalent to the cartesianess of the square(19) by Yoneda.The fact that the horizontal and vertical maps associated to a Grothendieck fibration determine a carte-sian factorization system permits us to strengthen the result of Lemma 2.4.10 as follows: Lemma 3.2.5.
Let E be a category with finite limits and F ∶ E → B be a Grothendieck fibration. Then thefunctor F respects the base change of an F -cartesian map u ∶ A → B along any map g ∶ Y → B .Proof. We have to show that if a map u ∶ A → B in E is F -cartesian, then the image by the functor F of acartesian square X AY B fv ug (9)is a cartesian square. If L ⊆ E is the class of F -vertical maps and R ⊆ E is the class of F -horizontal maps, thenthe pair S = ( L , R ) is a cartesian factorization system by Proposition 3.2.3. By pulling back the factorization g = R ( g ) L ( g ) ∶ Y → S ( g ) → B along the map u ∶ A → B we obtain a factorization of the cartesian square (9)as the composite of two cartesian squares X E AY S ( g ) B pv qw u L ( g ) R ( g ) (10)The map w ∶ E → S ( g ) belongs to R , since the map u ∶ A → B belongs to R and the class R is closedunder base change by Proposition 3.1.7. Similarly, the map q ∶ E → A belongs to R , since the map R ( q ) belongs to R . It then follows by Lemma 2.4.10 that the functor F takes the right hand square of diagram(10) to a cartesian square. On the other hand, the left hand square in (10) is cartesian since the outer andright squares are. From this it follows that p ∈ L since we have seen that the factorization system ( L , R ) iscartesian and we have both L ( g ) ∈ L and w ∈ R . But then both F ( p ) and F ( L ( g )) are invertible so that theimage of the left square by F is trivially cartesian. We have seen above that factorization systems give rise to reflective subcategories. When the factorizationsystem is cartesian, the associated subcategories have an additional property, which we now describe.
Definition 3.2.6.
We shall say that a reflective subcategory E ′ ⊆ E is fibering if the reflector R ∶ E → E ′ is aGrothendieck fibration. Example 3.2.7.
Let E be a category with finite limits and let φ ∶ E → E ′ ⊆ E be a left-exact reflector. Thenthe subcategory E ′ ⊆ E is fibering. 24 emma 3.2.8. A reflective subcategory E ′ ↪ E is fibering if and only if the base change of a reflecting map Y → Y ′ along any map X ′ → Y ′ in E ′ exists and is a reflecting map X → X ′ .Proof. Let us write η ∶ Id → R for the unit of the reflector R ∶ E → E ′ .( ⇒ ) Suppose that the reflector R ∶ E → E ′ is a Grothendieck fibration. We must show that the base changeof a reflecting map r ∶ Y → Y ′ along any map u ∶ X ′ → Y ′ in E ′ exists and is itself a reflecting map. Up toisomorphism, is suffices to suppose that Y ′ = RY and r = η ( Y ) . Now, since R is a Grothendieck fibration,Lemma 2.4.7 asserts the existence of an R -cartesian morphism p ∶ X → Y together with an isomorphism i ∶ RX → X ′ such that ui = R ( p ) . Since p is R -cartesian, and since, as a reflector, R admits a right adjoint,it follows from Lemma 2.4.9 that the following square is cartesian X YRX RY η ( X ) p η ( Y ) R ( p ) Hence the map η ( X ) ∶ X → RX is a base change of the map η ( Y ) along the map R ( p ) . It follows that themap i ○ η ( X ) ∶ X → X ′ is a base change of the map η ( Y ) along the map u = R ( p ) ○ i − ∶ X ′ → Y ′ . Moreover,the map i ○ η ( X ) ∶ X → X ′ reflects the object X into E ′ , since the map η ( X ) ∶ X → RX reflects the object X into E ′ .( ⇐ ) Conversely, by Definition 2.4.1, we are to show that the functor R / A ∶ E / A → E ′/ RA induced by thefunctor R is a reflector for every object A ∈ E . First of all, we observe that this functor admits a right adjoint R A ∶ E ′/ RA → E / A . Indeed, given some ( Y, g ) ∈ E ′/ RA , the map η ( A ) admits a base change along g ∶ Y → RA since the subcategory E ′ is fibering, and we may therefore define R A ( Y, g ) = ( A × RA Y, p ) . A × RA Y YA RA p p gη ( A ) But since p is the base change of a reflecting map by construction, it is also a reflecting map and hence wehave R ( A × RA Y ) = Y . But then R / A R A ( Y, g ) = ( Y, g ) which shows that the counit of the adjunction is anisomorphism. Consequently, R / A is a reflector as required. Proposition 3.2.9.
Let E be a category with a terminal object . If ( L , R ) is a cartesian factorizationsystem in E , then the subcategory R ( ) is fibering.Proof. Recall from Proposition 3.1.16 that a map y ∶ Y → Y ′ in E reflects the object Y into R ( ) if and only Y ′ ∈ R ( ) and y ∈ L . Every map v ∶ X ′ → Y ′ in R ( ) belongs to R by Proposition 3.1.16. Hence the basechange of a reflecting map y ∶ Y → Y ′ along any map v ∶ X ′ → Y ′ in R ( ) exists and belongs to L , since thefactorization system ( L , R ) is cartesian. X YX ′ Y ′ ux yv (11)The map x ∶ X → X ′ in the cartesian square (11) reflects the object X into R ( ) , since X ′ ∈ R ( ) and x ∈ L .This shows that the subcategory R ( ) is fibering. As described in the introduction, the following characterization of cartesian factorization systems in termsof their associated base change functors will be a key tool in the proof of Theorem 3.6.8 below.25 roposition 3.2.10.
A factorization system ( L , R ) in a category with finite limits E is cartesian if and onlyif the pullback functor u ⋆ ∶ R ( B ) → R ( A ) is an equivalence of categories for every map u ∶ A → B in L .Proof. Recall from Proposition 3.1.18 that the functor u ⋆ ∶ R ( B ) → R ( A ) has a left adjoint u ♯ ∶ R ( A ) → R ( B ) defined by putting u ♯ ( X, f ) = ( S ( uf ) , R ( uf )) X S ( uf ) A B f L ( uf ) R ( uf ) u (12)for a map f ∶ X → A in R .( ⇒ ) We will show that both the unit and counit of the adjunction u ♯ ⊣ u ⋆ are invertible. The unit η ∶ Id → u ⋆ u ♯ at an object ( X, f ) ∈ R ( A ) may be identified with the cartesian gap map η ( X, f ) ∶ X → A × B S ( uf ) ofthe following square X S ( uf ) A B f L ( uf ) R ( uf ) u and hence fits into the following diagram where we put η ( X, f ) ∶= ( f, L ( uf )) . X A × B S ( uf ) S ( uf ) A B fη ( X,f ) p p R ( uf ) u Now, as the map p is a base change of u along R ( uf ) by construction, and the factorization system ( L , R ) is cartesian, we have p ∈ L . It follows then that η ( X, f ) ∈ L , since the class L has the 3-for-2 property and p η ( X, f ) = L ( uf ) belongs to L . The rest of the argument follows from Proposition 3.1.7. Indeed, we have p ∈ R since it is a base change of a map in R . But then η ( X, f ) ∈ R , since p η ( X, f ) = f ∈ R and the class R has the left cancellation property. Thus, η ( X, f ) ∈ L ∩ R and is therefore invertible.To see that the counit ǫ ∶ u ♯ u ⋆ → Id is invertible, suppose that we are given g ∶ Y → B ∈ R . Then u ⋆ ( Y, g ) = ( A × B Y, p ) is defined by the following cartesian square A × B Y YA B p p gu Hence we have p ∈ L , since u ∈ L and g ∈ R and ( L , R ) is cartesian. It follows that L ( up ) = p and R ( up ) = g . Thus, u ♯ ( A × B Y, p ) = ( Y, g ) which shows that the counit of the adjunction u ♯ ⊣ u ⋆ is invertibleas claimed.( ⇐ ) We first show that the base change of any map u ∶ A → B in L along a map g ∶ Y → B in R exists,and that the resulting map is again in L . Note that since the functor u ⋆ ∶ R ( B ) → R ( A ) is an equivalenceof categories, so is its left adjoint u ♯ ∶ R ( A ) → R ( B ) . Now given some u and g as above, it follows from theessential surjectivity of u ♯ that there exists ( X, f ) ∈ R ( A ) such that u ♯ ( X, f ) = ( Y, g ) . That is, we have acommutative square X YA B f h gu h ∶= L ( uf ) and g = R ( uf ) . Futhermore, as pointed out in the first part of the proof, the unit of theadjunction u ♯ ⊣ u ⋆ is exactly the cartesian gap map of this square. It follows then that since this adjunctionis an equivalence, the unit is invertible, which shows that the above square is cartesian.It remains to show that the class L has the 3-for-2 property. Since L is already the left class of a factor-ization system, it suffices to show that the class L has the left cancellation property by Proposition 3.1.7.Let u ∶ A → B be a map in E and v ∶ B → C in L and suppose that we have vu ∶ A → B ∈ L . We will showthat u ∈ L by showing that R ( u ) is invertible. By composing the factorization u = R ( u ) L ( u ) ∶ A → S ( u ) → B with the map v ∶ B → C , we obtain a commutative triangle A S ( u ) C L ( u ) vu v R ( u ) from which we see that the composite v R ( u ) ∶ S ( u ) → C belongs to L , since the class L has the rightcancellation property by Proposition 3.1.7 and we have vu ∈ L and L ( u ) ∈ L . It follows that v ♯ ( S ( u ) , R ( u )) = ( C, C ) . But the functor v ♯ ∶ R ( B ) ↔ R ( C ) is an equivalence of categories, since v ∈ L . Hence the object ( S ( u ) , R ( u )) is terminal in the category R ( B ) since the object ( C, C ) is terminal in the category R ( C ) .This shows that R ( u ) must be invertible and completes the proof. Remark 3.2.11.
The hypothesis that the category E has finite limits can in fact be removed from Proposition 3.2.10.Indeed, the proposition may be equivalently formulated by asserting that the pushforward functor u ♯ is anequivalence, a condition which makes no mention of finite limits. As we will not have occasion to use theresult at this level of generality, and as the proof is considerably longer, we omit it here. Definition 3.3.1.
Let E be a category with finite limits. We shall say that a factorization system ( L , R ) in E is a modality if its left class L is closed under base change.The right class of a factorization system is always closed under base change by Proposition 3.1.7. Hence both classes L and R of a modality are stable under base change and it follows that the factorization of amap is itself stable by base change. Example 3.3.2. a) If E is a category with finite limits, then the factorization systems ( Map ( E ) , Iso ( E )) and ( Iso ( E ) , Map ( E )) of Example 3.1.4 (a) are modalities.b) If E is a topos, then the factorization system ( Surj ( E ) , Inj ( E )) of Example 3.1.4 (c) is a modality.c) More generally, if E is a topos, then the factorization system ( Conn n ( E ) , Trun n ( E )) consisting of the n -connected and n -truncated maps is a modality. [ABFJ2][Example 3.4.2]d) Let E be a category with finite limits and let φ ∶ E → E ′ ⊆ E be a left exact reflector. Then the cartesianfactorization system ( L φ , R φ ) of Proposition 3.2.4 is a modality. Definition 3.3.3.
Let E be a category with finite limits. We shall say that a map u ∶ A → B in E is fiberwiseleft orthogonal to another map f ∶ X → Y , or that f is fiberwise right orthogonal to u , if every base change u ′ of u is left orthogonal to f . We shall denote this relation by u ñ f .If A and B are two classes of maps in a category with finite limits E , we shall write A ñ B to mean thatwe have u ñ f for every u ∈ A and f ∈ B . We shall denote by A ñ (resp. ñ A ) the class of maps in E that arefiberwise right orthogonal (resp. fiberwise left orthogonal) to every map in A . We have A ⊆ ñ B ⇔ A ñ B ⇔ A ñ ⊇ B emma 3.3.4. Let E be a category with finite limits. Then a factorization system ( L , R ) in E is a modalityif and only if L ñ R , in which case R = L ñ and L = ñ R .Proof. [ABFJ1, Proposition 2.6.2]Recall from Lemma 3.1.5 that a factorization system ( L , R ) in a category E induces a factorization system ( L / B , R / B ) in the category E / B for every object B ∈ E ; recall additionally from Lemma 3.1.6 that ( L , R ) induces a factorization system ( L K , R K ) on the category of diagrams [ K , E ] for any small category K . Ineach case, the latter is a modality as soon as the original factorization system ( L , R ) is, a fact which is easilychecked. We record these statements for future reference in the following two lemmas. Lemma 3.3.5.
Let E be a category with finite limits. If a factorization system ( L , R ) in E is a modality,then so is the factorization system ( L / B , R / B ) in the slice category E / B for every object B ∈ E . Lemma 3.3.6. If ( L , R ) is a modality in a category with finite limits E , then the factorization system ( L K , R K ) is a modality in the category [ K , E ] for any small category K . Recall from Definition 2.3.6 that a class of maps is said to be local if it is stable by base change anddescends along any surjective family of maps. The following proposition says that modalities provide us witha rich source of local classes:
Proposition 3.3.7.
In a topos E , the left and the right classes of a modality ( L , R ) are local.Proof. [ABFJ2, Proposition 3.6.5] Lemma 3.3.8.
Let ( L , R ) be a modality in a topos E . If a map u ∶ A → B in E is surjective and ∆ ( u ) ∈ L ,then u ∈ L .Proof. If ∆ ( u ) ∈ L , then the projection p in the following pullback square belongs to L , A × B A AA B p p uu since p ∆ ( u ) = A and the class L has the right cancellation property by Proposition 3.1.7. Thus, u ∈ L bydescent (Proposition 3.3.7), since u ∶ A → B is surjective. Let E be a topos. We shall say that a class of maps L ⊆ E is acyclic if the followingconditions hold:1. The class L contains the isomorphisms and is closed under composition2. The class L is closed under colimits3. The class L is closed under base changesBy definition, a class L is acyclic if and only if it is saturated and closed under base changes. An acyclicclass L is closed under cobase changes, since a saturated class is closed under base changes by Lemma 3.1.10. Remark 3.3.10.
Every acyclic class is local. In the category of spaces S , the concept of an acyclic class ofmaps L ⊆ S is equivalent to that of a closed class of spaces introduced in [DF95]. In view of Example 2.3.5 (b),a map f ∶ X → Y belongs to L ⊆ S if and only if the map f − ( y ) → L for every y ∶ → Y . Thatis, the class of maps L is determined entirely by a class of spaces. Proposition 3.3.11.
In a topos, the left class of a modality ( L , R ) is acyclic. roof. The class L is saturated by Lemma 3.1.11 and it is closed under base change by Definition 3.3.1. Lemma 3.3.12.
In a topos E , every saturated class L ⊆ E contains a largest acyclic class L ′ ⊆ L . A map u ∶ A → B belongs to L ′ if and only if every base change u ′ ∶ A ′ → B ′ of u belongs to L .Proof. Let L ′ be the class of maps u ∶ A → B in E having all their base changes in L . Obviously, L ′ ⊆ L .Moreover, the class L ′ is closed under base change by construction. It remains to show that the class L ′ is saturated. The class L ′ contains the isomorphisms and is closed under composition, since this is true ofthe class L . It remains to show that the subcategory L ′ ⊆ E [ ] is closed under colimits. If K is a smallcategory, then a diagram K → E [ ] is the same thing as a natural transformation u ∶ A → B between twodiagrams A, B ∶ K → E . If the map u ( k ) ∶ A ( k ) → B ( k ) belongs to L ′ for every object k ∈ K , let us show thatthe map colim ( u ) ∶ colim A → colim B belongs to L ′ . We have the following commutative square of naturaltransformations A δ colim ( A ) B δ colim ( B ) γ ( A ) u δ colim ( u ) γ ( B ) (13)where δ colim ( A ) represents the constant diagram K → E with value colim ( A ) ∈ E , and similarly for δ colim ( B ) . The natural transformations γ ( A ) and γ ( B ) are colimit cones. The map u ( k ) ∶ A ( k ) → B ( k ) belongs to L for every object k ∈ K , since L ′ ⊆ L . Thus, colim ( u ) ∈ L , since the subcategory L is closedunder colimits. It remains to show that every base change colim ( u ) is in L . For this, we have to show thatin every pullback square D C colim ( A ) colim ( B ) w g colim ( u ) the map w belongs to L . By pulling back the square (13) along the map δ ( g ) ∶ δC → δ colim ( B ) we obtainthe following cube of natural transformations, where every vertical face is a cartesian square. A ′ B ′ δD δCA Bδ colim ( A ) δ colim ( B ) u ′ γ ( A ′ ) γ ( B ′ ) δ ( w ) δ ( g ) uγ ( A ) γ ( B ) δ colim ( u ) The natural transformations γ ( A ′ ) ∶ A ′ → δD and γ ( B ′ ) ∶ B ′ → δC are colimit cones, since colimit areuniversal in a topos. It follows that w = colim ( u ′ ) . The map u ′ ( k ) ∶ A ′ ( k ) → B ′ ( k ) is a base change of themap u ( k ) ∶ A ( k ) → B ( k ) for every object k ∈ K . Hence the map u ′ ( k ) belongs to L , since u ( k ) belongs to L ′ . Thus, w = colim ( u ′ ) belongs to L , since the subcategory L ⊆ E [ ] is closed under colimits. This showsthat colim ( u ) ∈ L ′ . Hence the subcategory L ′ ⊆ E [ ] is closed under colimits. We have proved that the class L ′ is saturated. This shows that the class L ′ is acyclic, since it is closed under base change. If L ′′ ⊆ L is anacylic class, let us show that L ′′ ⊆ L ′ . If u ∶ A → B is a map in L ′′ , then L ′′ contains every base change u ′ ofthe map u , since L ′′ is closed under base change. Thus, L contains every base change u ′ of u , since L ′′ ⊆ L .Hence we have u ∈ L ′ and this shows that L ′′ ⊆ L ′ . 29 roposition 3.3.13. If A is a class of maps in a topos E , then the class ñ A is acyclic.Proof. The class ⊥ A is saturated by Lemma 3.1.11. By Lemma 3.3.12 the saturated class ⊥ A contains alargest acyclic class ( ⊥ A ) ′ . Moreover, a map u ∶ A → B belongs to ( ⊥ A ) ′ if and only if every base change u ′ ∶ A ′ → B ′ of u belongs to ⊥ A . In other words, ( ⊥ A ) ′ = ñ A . Hence the class ñ A is acyclic. Definition 3.3.14.
Let E be a topos and Σ a class of maps of E . Then Σ is contained in a smallest acyclicclass Σ a ⊆ E . We shall say that Σ a is the acyclic class generated by Σ. We shall say that an acyclic class L ⊆ E is of small generation if L = Σ a for a set of maps Σ ⊆ E . Lemma 3.3.15. If Σ is a class of maps in a topos E , then ( Σ a ) ñ = Σ ñ .Proof. We have ( Σ a ) ñ ⊆ Σ ñ , since Σ ⊆ Σ a . Conversely, the class ñ ( Σ ñ ) is acyclic by Proposition 3.3.13.Thus Σ a ⊆ ñ ( Σ ñ ) , since Σ ⊆ ñ ( Σ ñ ) . It follows that Σ a ñ Σ ñ and hence that Σ ñ ⊆ ( Σ a ) ñ .Let u ∶ A → B be map in a topos E . Consider the functor u ♭ ∶ E / B → E [ ] which takes an object ( X, f ) of E / B to the base change u ♭ ( f ) of the map u along the map f , X × B A AX B p u ♭ ( f ) uf Lemma 3.3.16.
The functor u ♭ ∶ E / B → E [ ] is cocontinuous.Proof. It suffices to show that the functor u ♭ has a right adjoint. Let us first consider the case where B = δ ∶ E → E [ ] (which takes an object X ∈ E to the map 1 X ∶ X → X ). If p A ∶ A →
1, then for every X ∈ E , we have ( p A ) ♭ ( X ) = δ ( X ) × p A . Hence the functor ( p A ) ♭ ∶ E → E [ ] is thecomposite of the functor δ ∶ E → E [ ] followed by the functor ( − ) × p A ∶ E [ ] → E [ ] . The functor δ is leftadjoint to the domain functor dom ∶ E [ ] → E . Moreover, the functor ( − ) × p A ∶ E [ ] → E [ ] has a right adjoint [ p A , − ] ∶ E [ ] → E [ ] since the category E [ ] is cartesian closed. It follows that the functor g ↦ dom ([ p A , g ]) is right adjoint to the functor p ♭ A . This proves the lemma in the case B =
1. In the general case, observe firstthat the functor ( A, u ) ♭ ∶ E / B → ( E / B ) [ ] defined by the object ( A, u ) of E / B has a right adjoint by the firstpart of the proof. Moreover, if F B ∶ ( E / B ) [ ] → E [ ] is the forgetful functor, then we have u ♭ = F B ○ ( A, u ) ♭ .The forgetful functor F B is left adjoint to the base change functor B ⋆ ∶ E [ ] → ( E / B ) [ ] which takes a map g ∶ U → V in E to the map B × g ∶ B × U → B × V of E / B . This shows that the functor u ♭ = F B ○ ( A, u ) ♭ has aright adjoint. Definition 3.3.17.
Let G be a class of objects in a topos E . We shall say that the base change u ′ ∶ A ′ → B ′ of a map u ∶ A → B in E is a G - base change if B ′ ∈ G . Lemma 3.3.18.
Let G be a class of generators in a topos E . If a class of maps Σ ⊆ E is closed under G -basechange, then Σ a = Σ s and Σ ñ = Σ ⊥ .Proof. Obviously, Σ s ⊆ Σ a . Let us show that Σ a ⊆ Σ s . The saturated class L ∶= Σ s contains a largest acyclicclass L ′ ⊆ L by Lemma 3.3.12. Let us show that Σ ⊆ L ′ . If a map u ∶ A → B belongs to Σ then every G -base change of u belongs to L , since Σ ⊆ L . Let G / B ⊆ E / B be the class of objects ( G, g ) of E / B with G ∈ G . By Lemma 2.2.5, the category E / B is strongly generated by the class G / B since the category E isstrongly generated by the class G . Consider the functor u ♭ ∶ E / B → E [ ] which takes an object X = ( X, f ) of E / B to the map u ♭ ( X ) = f ⋆ ( u ) ∶ X × B A → X . The functor u ♭ is cocontinuous by Lemma 3.3.16. Let L bethe full subcategory of E [ ] spanned by the maps in L . The subcategory L ⊆ E [ ] is closed under colimits,since the class L is saturated and a saturated class is closed under colimits. Hence the full subcategory ( u ♭ ) − ( L ) ⊆ E / B is closed under colimits, since the functor u ♭ is cocontinuous. We have u ♭ ( G, g ) ∈ L forevery object ( G, g ) ∈ E / B , since every G -base change of the map u ∶ A → B belongs to L . In other words,30 / B ⊆ ( u ♭ ) − ( L ) . Thus, E / B ⊆ ( u ♭ ) − ( L ) , since the class G / B strongly generates E / B by Lemma 2.2.5. Thisshows that we have u ♭ ( X, f ) ∈ L for every map f ∶ X → B . In other words, the base change of the map u ∶ A → B along any map f ∶ X → B belongs to L . Thus, u ∈ L ′ by Lemma 3.3.12. We have proved thatΣ ⊆ L ′ . It follows that Σ a ⊆ L ′ , since the class L ′ is acyclic. Thus, Σ a ⊆ Σ s , since Σ a ⊆ L ′ ⊆ L = Σ s . Theequality Σ a = Σ s is proved. By Lemma 3.1.13 we have ( Σ s ) ⊥ = Σ ⊥ and by Lemma 3.3.15 we have ( Σ a ) ñ = Σ ñ .Thus, Σ ñ = Σ ⊥ . Theorem 3.3.19.
Let Σ be a set of maps in a topos E . If Σ a ⊆ E is the acyclic class generated by Σ , thenthe pair ( Σ a , Σ ñ ) is a modality.Proof. The topos E admits a set of generators G since it is presentable. Let us denote by Λ the closure ofΣ over G -base changes (Definition 3.3.17). The class Λ is a set, since the classes G and Σ are sets. Hencethe pair ( Λ s , Λ ⊥ ) is a factorization system by Proposition 3.1.14, since Λ is a set. But Λ a = Λ s and Λ ñ = Λ ⊥ by Lemma 3.3.18, since Λ is closed under G -base change. Hence the pair ( Λ a , Λ ñ ) is a factorization system,since the pair ( Λ s , Λ ⊥ ) is a factorization system. We have Λ a ñ ( Λ a ) ñ ⊆ Λ ñ , since Λ ⊆ Λ a . Hence thefactorization system ( Λ a , Λ ñ ) is a modality by Lemma 3.3.4. Let us show that ( Σ a , Σ ñ ) = ( Λ a , Λ ñ ) . Wehave Σ a ⊆ Λ a , since Σ ⊆ Λ. Moreover, we have Λ ⊆ Σ a , since Σ a is closed under base change; thus, Λ a ⊆ Σ a .This shows that Σ a = Λ a . Let us show that Σ ñ = Λ ñ . We have Σ ñ = ( Σ a ) ñ by Lemma 3.3.15. But we have ( Σ a ) ñ = ( Λ a ) ñ , since Σ a = Λ a . Thus, Σ ñ = ( Λ a ) ñ = Λ ñ , since the pair ( Λ a , Λ ñ ) is a modality. This showsthat ( Σ a , Σ ñ ) = ( Λ a , Λ ñ ) . Hence the pair ( Σ a , Σ ñ ) is a modality. Definition 3.3.20.
We shall say that the modality ( Σ a , Σ ñ ) of Theorem 3.3.19 is generated by the set ofmaps Σ ⊆ E . Definition 3.3.21. If u ∶ A → B is a map in a topos E , we shall say that an object X ∈ E is u - modal if itis local with respect to every base change u ′ of u . If Σ is a class of maps in E , we shall say that an object X ∈ E is Σ- modal if it is u -modal for every map u ∈ Σ. We shall denote by Mod ( E , Σ ) the full subcategoryof E spanned by Σ-modal objects. It follows from Lemma 3.3.15 that we have Mod ( E , Σ ) = Mod ( E , Σ a ) . Lemma 3.3.22.
Let ( L , R ) be a modality generated by a set of maps Σ in a topos E . Then R ( ) = Mod ( E , Σ ) and the category R ( ) is presentable.Proof. By Definition 3.3.20, we have ( L , R ) = ( Σ a , Σ ñ ) . The topos E admits a set of generators G , since atopos is presentable. Let us denote by Λ the closure of Σ over G -base changes (Definition 3.3.17). The class Λis a set, since the classes G and Σ are sets. We saw in the proof of Theorem 3.3.19 that ( Σ a , Σ ñ ) = ( Λ s , Λ ⊥ ) .Hence the factorization system ( L , R ) is generated by the set Λ ⊆ E . It follows from Lemma 3.1.17 that R ( ) = Loc ( E , Λ ) and that the category R ( ) is presentable. By Lemma 3.1.2, an object X belongs to R ( ) = Loc ( E , Λ ) if and only if the map p X ∶ X → ⊥ . But we have Λ ⊥ = Σ ñ . Thus, an object X belong the R ( ) if and only if the map p X ∶ X → ñ . Thus, an object X belongs to R ( ) ifand only if it is Σ-local. Among the modalities ( L , R ) in a topos E , we may distinguish those for which the subcategory R ( ) is againa topos. As we have seen above, this will be the case as soon as the associated reflector preserves finitelimits. This motivates the following definition: Definition 3.4.1.
Let E be a category with finite limits. We shall say that a modality ( L , R ) in E is left-exact or simply lex if the class L is closed under finite limits as a subcategory of E [ ] . Remark 3.4.2.
It can be shown that a factorisation system ( L , R ) in a category with finite limits is aleft-exact modality if and only if the functor S ∶ E [ ] → E preserves finite limits.31 xample 3.4.3. Let E be a category with finite limits and let φ ∶ E → E ′ ⊆ E be a left-exact reflector. Wesaw in Proposition 3.2.4 that the class of maps L φ ⊆ E inverted by the functor φ is the left class of a cartesianfactorization system S φ ∶= ( L φ , R φ ) . We shall see in Proposition 3.4.8 that the factorization system S φ is infact a left-exact modality. Lemma 3.4.4.
If a modality ( L , R ) in a category with finite limits E is left-exact, then so is the modality ( L / B , R / B ) in the category E / B for every object B ∈ E . The notion of left-exact modality admits many equivalent characterizations (for example, at least 13 aregiven in [RSS]). We will content ourselves with those given by Theorem 3.4.7, which are most relevant tothe current setup. The proof of the theorem will require a couple of preparatory lemmas concerning classesof arrows and their closure properties for finite limits, which we turn to next.
Lemma 3.4.5 ([AS]) . Let E be a category with finite limits and let Q ⊆ E be a class such that the fullsubcategory Q ⊆ E [ ] whose objects are the maps in Q contains the isomorphisms of E (as objects) and isclosed under fiber products. Then the class Q has the left cancellation property: if v and vu belongs to Q ,then u ∈ Q .Proof. Observe that if u ∶ A → B and and v ∶ B → C are two maps in E , then the following square in thecategory E [ ] is cartesian. u vu B v ( u, B ) ( A ,v ) ( u, C )( B ,v ) We have 1 B ∈ Q , since Q contains the isomorphisms. Thus, if v and vu belongs to Q , then u ∈ Q since thesubcategory Q ⊆ E [ ] is closed under fiber products by the hypothesis. Lemma 3.4.6.
Let E be a category with finite limits and let L ⊆ E be a class of maps which contains theisomorphisms and is closed under base changes. Write L ( B ) for the full subcategory of E / B whose objectsare the maps A → B in L . If the sub-category L ( B ) ⊆ E / B is closed under fiber products for every object B ∈ E , then class L is closed under finite limits.Proof. Since the class L contains the isomorphisms, it contains the terminal object of the category E [ ] .It suffices therefore to show that the subcategory L is closed under fiber products. Consider, therefore, acartesian square in the category E [ ] z vu w gf qp (14)If u, v, w ∈ L , let us show that w ∈ L . The codomain functor cod ∶ E [ ] → E is a Grothendieck fibration, sincethe category E has finite limits. Hence the category E [ ] admits a factorisation system T = ( V , H ) , where V the class of cod -vertical morphisms and H the class of cod -horizontal morphisms (= cartesian squares in E ).The factorisation system T = ( V , H ) is a modality, since the class V is closed under base change. By pullingback the factorisation p = H ( p ) V ( p ) ∶ u → T ( p ) → v along the map q ∶ v → w , the cartesian square (14) isdecomposed into two cartesian squares: z v ′ vu T ( p ) w q V ( p ) H ( p ) (15)The morphism of H ( p ) ∶ T ( p ) → p of E [ ] is a cartesian square in E . Hence the map T ( p ) of E is a basechange of the map w . Thus, T ( p ) ∈ L , since w ∈ L and the class L is closed under base changes. The32orphism v ′ → v in the diagram (15) belongs to the class H , since the right class of a factorisation systemis closed under base change by Proposition 3.1.7. Hence the map v ′ ∈ E is a base change of the map v ∈ E .Thus, v ′ ∈ L , since v ∈ L and the class L is closed under base change. Hence it suffices to prove the lemmain the case of the left hand square of the diagram (15), since u, T ( p ) , v ′ ∈ L . Equivalently, we may supposethat the morphism p ∶ u → w in the original square (14) belongs to V . Let us then pullback the factorisation q = H ( q ) V ( q ) ∶ v → T ( q ) → w along the map p ∶ u → w . The cartesian square (14) is then decomposed intotwo cartesian squares: z vu ′ T ( q ) u w V ( q ) H ( q ) p (16)The argument above can be applied to show that u ′ , T ( p ) , v ∈ L . Hence it suffices to prove the lemma in thecase of the top square of the diagram (16), But the morphism u ′ → T ( q ) in the diagram (16) is a base changeof the morphism p ∶ u → w . Hence the morphism u ′ → T ( q ) belongs to V , since the morphism p ∶ u → w belongs to V by hypothesis and the class V is closed under base changes. Hence it suffices to prove the lemmafor the original square (14) with the extra hypothesis that the morphisms u → w and v → w belongs to V . Inwhich case the four maps cod ( p ) , cod ( q ) , cod ( f ) and cod ( g ) of the category E are invertible. Hence we maysuppose that cod ( z ) = cod ( u ) = cod ( v ) = cod ( w ) = B for some object B ∈ E and that the square (14) belongsto E / B instead of E [ ] . In which case we have u, v, w ∈ L ( B ) , since u, v, w ∈ L by hypothesis. Hence we have z ∶= u × w v ∈ L ( B ) , since L ( B ) is closed under finite limits by hypothesis. This proves that z ∈ L and hencethat L is closed under fiber products. The class L contains the terminal object of E [ ] , since this terminalobject is an isomrphism in E and L contains the isomorphisms. This completes the proof that L is closedunder finite limits. Theorem 3.4.7.
Let E be a category with finite limits. Then the following conditions on a modality S = ( L , R ) in E are equivalent:1. The modality ( L , R ) is left-exact2. The class L has the 3-for-2 property3. If a map u ∶ A → B belongs to L , then its diagonal ∆ ( u ) ∶ A → A × B A belongs to L
4. The factorization system ( L , R ) is cartesian5. The reflector ∥ − ∥ B ∶ E / B → R ( B ) preserves finite limits for every object B ∈ E Proof. (1 ⇒
2) The class L is closed under composition and right cancellable by Proposition 3.1.7. Henceit suffices to show that the class L is left cancellable. Since the modality ( L , R ) is left-exact, the class L isclosed under pullbacks in E [ ] so that Lemma 3.4.5 applies, proving the claim.(2 ⇒
3) If a map u ∶ A → B belongs to L , then so does the projection p ∶ A × B A → A , since the latter isa base change of the former. Thus, ∆ ( u ) ∶ A → A × B A belongs to L by 3-for-2, since p ∆ ( u ) = A is in L .(3 ⇒
2) Let u ∶ A → B and v ∶ B → C be two maps in E ; if vu ∈ L and v ∈ L , let us show that u ∈ L . Weshall use the following diagram of cartesian squares: A B × C A AB B × C B BB C u ( u, A )( a ) B × C u p ( b ) u ∆ ( v ) p p ( c ) vv ( b ) is cartesian, since the squares ( c ) and ( b + c ) are cartesian by construction. Hence the square(a) is cartesian, since the square ( a + b ) is (trivially) cartesian. The map ∆ ( v ) ∶ B → B × C B belongs to L by condition (4), since v ∈ L . Hence the map ( u, A ) ∶ A → B × C B belongs to L , since the square ( a ) iscartesian and the class L is closed under base change. Similarly, the projection p ∶ B × C A → B belongs to L , since the map vu ∶ A → C belongs to L and the square ( b + c ) of the diagram is cartesian. It follows that u belongs to L , since u = p ( u, A ) and L is closed under composition.(2 ⇒
4) The class L has the 3-for-2 property by assumption. It remains to show that the class L is stableby base change along maps in R . But since ( L , R ) is a modality, L is in fact stable by base change along any map. This shows that the factorization system ( L , R ) is cartesian.(4 ⇒
5) Let us show that the reflector ∥ − ∥ ∶ E → R ( ) preserves finite limits. We shall prove that thefunctor ∥ − ∥ takes a cartesian square in the category E X AY B fv ug (17)to a cartesian square. By pulling back the factorization g = R ( g ) L ( g ) ∶ Y → S ( g ) → B along the map u ∶ A → B we obtain a factorization of the cartesian square (17) as the composite of two cartesian squares X E AY S ( g ) B pv qw u L ( g ) R ( g ) (18)The functor ∥ − ∥ ∶ E → R ( ) is a Grothendieck fibration by Proposition 3.2.9, since the factorization system ( L , R ) is cartesian. It then follows by Lemma 3.2.5 that the functor ∥ − ∥ takes the right hand square ofdiagram (18) to a cartesian square. Let us show that the functor ∥ − ∥ takes the left hand square to a cartesiansquare. The map p ∶ X → E belongs to L , since the map L ( g ) belongs to L and the class L is closed underbase change. Hence the maps ∥ p ∥ and ∥ L ( g )∥ are invertible by Proposition 3.1.16. It follows that the functor ∥ − ∥ ∶ E → R ( ) takes the left hand square of diagram (18) to a cartesian square. This shows that the functor ∥ − ∥ ∶ E → R ( ) takes the square (17) to a cartesian square. Hence the functor ∥ − ∥ ∶ E → R ( ) preservesfinite limits, since it preserves terminal objects.More generally, let us show that the reflector ∥ − ∥ B ∶ E / B → R ( B ) preserves finite limits for everyobject B ∈ E . The factorization system S / B ∶= ( L / B , R / B ) is a modality by Lemma 3.3.5, since ( L , R ) is amodality by hypothesis. Moreover, the factorization system ( L / B , R / B ) is cartesian by Lemma 3.2.2, sincethe factorization system ( L , R ) is cartesian by hypothesis. It then follows by the first part of the argumentthat the reflector ∥ − ∥ B ∶ E / B → R ( B ) preserves finite limits.(5 ⇒
1) Let us show that the class L is closed under finite limits. By Lemma 3.4.6, it suffices to showthat the subcategory L ( B ) ⊆ E / B is closed under pullbacks for every object B ∈ E . A map u ∶ A → B in E belongs to L if and only the map R ( u ) ∶ S ( u ) → B is invertible. Hence the object ( A, u ) belongs to L ( B ) ifand only if the object ∥( A, u )∥ B = ( S ( u ) , R ( u )) of the category R ( B ) is terminal. But the full subcategory ofterminal objects of R ( B ) is obviously closed under pullbacks. It follows that the full subcategory L ( B ) ⊆ E / B is closed under pullbacks, since the functor ∥ − ∥ B ∶ E / B → R ( B ) is left-exact by hypothesis. Proposition 3.4.8.
Let E be a category with finite limits. If φ ∶ E → E ′ ⊆ E is a left exact reflector, then theclass L φ of maps inverted by φ is the left class of a left-exact modality S φ ∶= ( L φ , R φ ) and R φ ( ) = E ′ . If η ∶ Id → φ is the unit of the reflector, then a map f ∶ X → Y in E belongs to R φ if and only if the following quare is cartesian X φ ( X ) Y φ ( Y ) . η ( X ) f φ ( f ) η ( Y ) (19) Conversely, if S = ( L , R ) is a left-exact modality in E and φ is the reflector ∥ − ∥ S ∶ E → R ( ) , then S = S φ .The construction φ ↦ S φ is setting a one to one correpondance between the left-exact reflectors φ ∶ E → E ′ ⊆ E and the left-exact modalities S = ( L , R ) in E .Proof. The functor φ ∶ E → E ′ ⊆ E is a Grothendieck fibration by Proposition 2.4.5. Thus, if L ⊆ E is the classof φ -vertical maps and R ⊆ E is the class of φ -cartesian maps, then the pair ( L , R ) is a cartesian factorizationsystem by Proposition 3.2.3. By definition, we have L = L φ . The functor φ ∶ E → E ′ is left adjoint to theinclusion functor E ′ ⊆ E . It then follows by Lemma 2.4.9 that a map f ∶ X → Y in E is φ -cartesian if and onlyif the square (19) is cartesian. Let us show that R φ ( ) = E ′ . For every object X ∈ E , the map p X ∶ X → R φ if and only if the following square is cartesian X φ ( X ) η ( X ) p X φ ( p X ) if and only if the map η ( X ) is invertible, if and only if X ∈ E ′ . Thus, R φ ( ) = E ′ .Conversely, if S = ( L , R ) is a left-exact modality in E and φ is the reflector ∥ − ∥ S ∶ E → R ( ) , let us showthat S = S φ . The modality ( L , R ) is a cartesian factorization system by Theorem 3.4.7. It then follows fromProposition 3.2.9 that L = L φ . Thus, ( L , R ) = ( L φ , R φ ) since R = L ⊥ and R φ = L ⊥ φ by Proposition 3.1.7.Hence construction φ ↦ S φ is a one to one correpondance between the left-exact reflectors φ ∶ E → E ′ ⊆ E andthe left-exact modalities S = ( L , R ) in E . R -equivalences In this section, we give a criterion under which the modality generated by a set of maps Σ is left-exact. Thetheorem will follow from the careful analysis of the following class of maps:
Definition 3.5.1.
Let ( L , R ) be a modality in a category with finite limits E . We shall say that a map u ∶ A → B in E is a R - equivalence if the functor u ⋆ ∶ R ( B ) → R ( A ) is an equivalence of categories. We shallsay that u is a fiberwise R -equivalence if every base change of u is an R -equivalence.In fact, the class of fiberwise R -equivalences admits a very simple characterization, as shown by thefollowing proposition. Proposition 3.5.2.
Let ( L , R ) be a modality in a category with finite limits E . Then a map u ∶ A → B in E is a fiberwise R -equivalence if and only if the maps u and ∆ ( u ) both belong to L . We first prepare the proof with a couple of lemmas.
Lemma 3.5.3.
Let ( L , R ) be a modality in a category with finite limits E . Then a map u ∶ A → B in E belongs to L if and only if the functor u ⋆ ∶ R ( B ) → R ( A ) is fully faithful.Proof. The functor u ⋆ ∶ R ( B ) → R ( A ) has a left adjoint u ♯ ∶ R ( A ) → R ( B ) by Proposition 3.1.18. Hence thefunctor u ⋆ ∶ R ( B ) → R ( A ) if fully faithful if and only if the counit of the adjunction u ♯ ⊣ u ⋆ is invertible. Bydefinition, for every map g ∶ Y → B ∈ R we have u ⋆ ( Y, g ) = ( A × B Y, p ) . A × B Y YA B p p gu u belongs to L , let us show that the canonical map u ♯ u ⋆ ( Y, g ) → ( Y, g ) is invertible. The map p belongs to L , since u belongs to L and L is closed under base change. But g belongs to R by hypothesis.Thus, L ( up ) = p and R ( up ) = g . This shows that u ♯ ( A × B Y, p ) = ( Y, g ) and hence that the canonicalmap u ♯ u ⋆ ( Y, g ) → ( Y, g ) is the identity. We have proved that the counit of the adjunction u ♯ ⊣ u ⋆ isinvertible. Conversely, if the the counit u ♯ u ⋆ ( Y, g ) → ( Y, g ) is invertible for every ( Y, g ) ∈ R ( B ) , then it isin particular in the special case where ( Y, g ) = ( B, B ) . In this case, we have u ⋆ ( Y, g ) = ( A, A ) and hence u ♯ ( A, A ) = ( S ( u ) , R ( u )) shows that R ( u ) is invertible. It follows that u = R ( u ) L ( u ) belongs to L . Lemma 3.5.4.
Let ( L , R ) be a modality in a category with finite limits E and let u ∶ A → B be a map in E .If ∆ ( u ) ∈ L , then the functor u ♯ ∶ R ( A ) → R ( B ) is fully faithful.Proof. It suffices to show that for any f ∶ X → A , the canonical map X → u ∗ ( u ♯ ( X )) is an isomorphism. Tothis end, consider the following cubical diagram: u ∗ ( X ) u ∗ ( u ♯ ( X )) X u ♯ ( X ) A × B A AA B u ∗ L ( uf ) u ∗ ( f ) u ∗ R ( uf ) L ( uf ) R ( uf ) p p uuf Viewing the maps A f ←Ð X L ( uf ) ÐÐÐ→ u ♯ ( X ) as a span in the category E / B , we obtain the back face of the cubeby applying the functor u ∗ ∶ E / B → E / A . It follows immediately that the left, right, top and bottom facesof the cube are cartesian. Consequently, we have u ∗ L ( uf ) ∈ L and u ∗ R ( uf ) ∈ R since both of the classes L and R are stable by base change. We deduce that ( p ) ♯ ( u ∗ ( X )) = u ∗ ( u ♯ ( X )) by the definition of u ♯ . Finally, since ∆ ( u ) ∈ L we have that u ∗ is fully faithful by Lemma 3.5.3 and thus wehave a canonical isomorphism ∆ ( u ) ♯ ( ∆ ( u ) ∗ ( Y )) = Y for any Y → A × B A . We now simply calculate: X = ( p ) ♯ ( ∆ ( u ) ♯ ( X )) since p ○ ∆ ( u ) = A = ( p ) ♯ ( ∆ ( u ) ♯ (( ∆ ( u )) ∗ (( p ) ∗ ( X )))) since p ○ ∆ ( u ) = A = ( p ) ♯ (( p ) ∗ ( X )) since ∆ ( u ) ∗ is fully faithful = ( p ) ♯ ( u ∗ ( X )) since the left face is cartesian = u ∗ ( u ♯ ( X )) from aboveThis completes the proof. Proof of Proposition 3.5.2.
If the map u ∶ A → B is an R -equivalence, then u ∈ L by Lemma 3.5.3, since thefunctor u ⋆ ∶ R ( B ) → R ( A ) is fully faithful. Let us show that ∆ ( u ) ∈ L . The projection p ∶ A × B A → B is abase change of the map u ∶ A → B , since the following square is cartesian. A × B A AA B p p uu p ∶ A × B A → B is an R -equivalence, since u is a fiberwise R -equivalence. Hence the functor p ⋆ ∶ R ( A ) → R ( A × B A ) is an equivalence of categories. But we have ∆ ( u ) ⋆ p ⋆ = id , since p ∆ ( u ) = A . Hencethe functor ∆ ( u ) ⋆ ∶ R ( A × B A ) → R ( A ) is an equivalence of categories, since the functor p ⋆ is an equivalenceof categories. It then follows by Lemma 3.5.3 that ∆ ( u ) ∈ L .Conversely, if the maps u and ∆ ( u ) belong to L , let us show that u is a fiberwise R -equivalence. Weshall first prove that u is an R -equivalence. The functor u ⋆ is fully faithful by Lemma 3.5.3, since u ∈ L .Hence the counit of the adjunction u ♯ ⊣ u ⋆ is invertible. But the functor u ♯ is fully faithful by Lemma 3.5.4,since ∆ ( u ) ∈ L . Hence the unit of the adjunction u ♯ ⊣ u ⋆ is invertible. This shows that the functor u ⋆ ∶ R ( B ) → R ( A ) is is an equivalence of categories. We have proved that u is an R -equivalence. It remainsto show that every base change u ′ of u is an R -equivalence. If u ′ is a base change of u along a map p ∶ B ′ → B ,then the map ∆ ( u ′ ) is a base change of ∆ ( u ) , since the base change functor p ⋆ ∶ E / B → E / B ′ preserves limits.Hence the maps u ′ and ∆ ( u ′ ) belong to L , since L is closed under base change. Hence the map u ′ is an R -equivalence by the argument above. This shows that u is a fiberwise R -equivalence.Next, we aim to show that the class of fiberwise R -equivalences is acyclic in the sense of Section 3.3. Thefact that it is closed under isomorphisms, composition and base change is immediate from the definition, sothat it remains only to show that it is closed under colimits. For this, we will need the following: Lemma 3.5.5.
Let ( L , R ) be a modality in a topos E , let K be a small category and let X YA B vp qu (20) be a cartesian square in the category [ K , E ] . Suppose that the horizontal maps of the square belong to theclass L K and that the vertical maps belong to R K . If the natural transformation p ∶ X → A is cartesian, thenso is the natural transformation q ∶ Y → B .Proof. The following cube commutes for every morphism r ∶ i → j in K , since the square (20) commutes. X ( i ) X ( j ) Y ( i ) Y ( j ) A ( i ) A ( j ) B ( i ) B ( j ) X ( r ) p ( i ) v ( i ) p ( j ) v ( j ) Y ( r ) q ( j ) A ( r ) u ( i ) u ( j ) B ( r ) q ( i ) (21)The left and the right faces of the cube are cartesian, since the square (20) is cartesian. Let us show thatthe front face of the cube is cartesian. The back face is cartesian, since the natural transformation p ∶ X → A is cartesian. Hence the composite of the back face with the right hand face is cartesian. It follows thatthe composite of the left hand face with the front face is cartesian, since the cube commutes. Hence theboundary square of the following diagram is cartesian. X ( i ) Y ( i ) Y ( j ) A ( i ) B ( i ) B ( j ) v ( i ) p ( i ) Y ( r ) q ( i ) q ( j ) u ( i ) B ( r ) (22)37et us show that right hand face of the diagram is cartesian. For this, it suffices to show that its cartesiangap map g ∶ Y ( i ) → B ( i ) × B ( j ) Y ( j ) is invertible. The vertical maps of the diagram (22) belong to R ,since the vertical maps of the square (20) belong to R K . Hence the projection p ∶ B ( i ) × B ( j ) Y ( j ) → B ( i ) belongs to R , since the map q ( j ) ∶ Y ( j ) → B ( j ) belongs to R , and R is closed under base change. Hencethe gap map g is a morphism in the category in R ( B ( i )) . The maps u ( i ) and v ( i ) in the diagram (22)belong to L , since the horizontal maps of the square (20) belong to L K . Hence the base change functor u ( i ) ⋆ ∶ R ( B ( i )) → R ( A ( i )) is fully faithful by Lemma 3.5.3, since u ( i ) ∈ L by hypothesis. We will provethat g is invertible by showing that the map u ( i ) ⋆ ( g ) is invertible. But u ( i ) ⋆ ( g ) is the cartesian gap mapof the boundary square of the diagram (22), since the left hand square of this diagram is cartesian. But thecartesian gap map of the boundary square is invertible, since the boundary square is cartesian. It followsthat u ( i ) ⋆ ( g ) is invertible and hence g is invertible. We have proved that the front face of the cube (21) iscartesian. Thus shows that the natural transformation p ∶ X → A is cartesian. Proposition 3.5.6.
Let ( L , R ) be a modality in a topos E . Then the class of fiberwise R -equivalences isacyclic.Proof. Let us denote the class of fiberwise R -equivalences in E by W . As we have already remarked, W contains every isomorphism and is closed under composition and base change. Let us show that the thesubcategory W ⊆ E [ ] is closed under colimits. If K is a small category, then a diagram K → E [ ] is thesame thing as a natural transformation u ∶ A → B between two diagrams A, B ∶ K → E . If the map u ( k ) ∶ A ( k ) → B ( k ) is in W for every k ∶ K , let us show that the map colim ( u ) ∶ colim A → colim B is in W .We have the following commutative square of natural transformations A δ colim ( A ) B δ colim ( B ) γ ( A ) u δ colim ( u ) γ ( B ) (23)where δ colim ( A ) represents the constant diagram K → E with value colim ( A ) ∈ E and similarly for δ colim ( B ) . The natural transformations γ ( A ) and γ ( B ) are colimit cones. The map u ( k ) ∶ A ( k ) → B ( k ) belongs to L for every k ∶ K , since W ⊆ L by Proposition 3.5.2. Thus, colim ( u ) ∈ L , since the the subcat-egory L ⊆ E [ ] is closed under colimits by Proposition 3.1.7. Hence the functor colim ( u ) ⋆ ∶ R ( colim ( B )) → R ( colim ( A )) is fully faithful by Lemma 3.5.3. Let us show that the functor colim ( u ) ⋆ is essentially surjec-tive. For this, we have to show that every object X = ( X, p ) ∈ R ( colim ( A )) is the pullback of an object Y = ( Y, q ) ∈ R ( colim ( B )) along the map colim ( u ) ∶ colim ( A ) → colim ( B ) . We shall construct a map q ∶ Y → colim ( B ) in R together with a pullback square X Y colim ( A ) colim ( B ) p q colim ( u ) (24)For this we shall show that the square (24) is the colimit over k ∈ K of a square of natural transformationsin the category [ K , E ] , X ′ Y ′ A B vp ′ q ′ u The whole procedure depends on a step by step construction of the following cube, where k is a variable38bject of K : X ′ ( k ) Y ′ ( k ) X YA ( k ) B ( k ) colim ( A ) colim ( B ) v ( k ) p ′ ( k ) γ ( X ′ )( k ) q ′ ( k ) γ ( Y ′ )( k ) colim ( v ) qu ( k ) γ ( A )( k ) γ ( B )( k ) colim ( u ) p The bottom face of the cube is obtained from the square (23). The left hand face of the cube is obtained bypulling back the map p ∶ X → colim ( A ) along the canonical map γ ( A )( k ) ∶ A ( k ) → colim ( A ) , X ′ ( k ) XA ( k ) colim ( A ) γ ( X ′ )( k ) p ′ ( k ) pγ ( A )( k ) (25)This defines a diagram X ′ ∶ K → E together with a natural transformation p ′ ∶ X ′ → A . Let δX ∶ K → E bethe constant diagram with value X . The cone γ ( X ′ ) ∶ X ′ → δX defined by the maps γ ( X ′ )( k ) ∶ X ′ ( k ) → X is a colimit cone since colimits are universal in E and γ ( A ) ∶ A → δ colim ( A ) is a colimit cone. Thus,colim ( p ′ ) = p . The map p ′ ( k ) ∶ X ′ ( k ) → A ( k ) belongs to R for every k ∈ K , since the map p ∶ X → colim ( A ) belongs to R and the square (25) is cartesian. The modality ( L , R ) induces a modality ( L K , R K ) in thecategory of K -diagrams [ K , E ] by Lemma 3.3.6. The natural transformation p ′ ∶ X ′ → A belongs to R K ,since the map p ′ ( k ) ∶ X ′ ( k ) → A ( k ) belongs to R for every k ∈ K . The maps u ( k ) and ∆ ( u ( k )) belongs to L for every k ∈ K by Proposition 3.5.2, since u ( k ) ∈ W by the hypothesis. Hence the natural transformations u and ∆ ( u ) belongs to L K . It follows by Proposition 3.5.2 that the natural transformation u ∶ A → B is a R K -equivalence. Let us put ( Y ′ , q ′ ) = u ♯ ( X ′ , p ′ ) , v = L K ( up ′ ) and q ′ = R K ( up ′ ) . The resulting square X ′ Y ′ A B vp ′ q ′ u is the square (24). The square is cartesian, since u ∶ A → B is a R K -equivalence. The horizontal arrows ofsquare belong to the class L K and the vertical arrows belong to R K . Moreover, the natural transformation q ′ ∶ Y ′ → B is cartesian by Lemma 3.5.5, since the natural transformation p ′ ∶ X ′ → A is cartesian. Hence thefollowing square is cartesian by Lemma 3.2.5, since the colimit functor colim ∶ [ K , E ] → E is a Grothendieckfibration by Proposition 2.4.12. colim ( X ′ ) colim ( Y ′ ) colim ( A ) colim ( B ) colim ( v ) colim ( p ′ ) colim ( q ′ ) colim ( u ) (26)We saw above that X = colim ( X ′ ) and that p = colim ( p ′ ) . Let us put Y ∶= colim ( Y ′ ) and q ∶= colim ( q ′ ) .We then have colim ( u ) ⋆ ( Y, q ) = ( X, p ) , since the square (26) is cartesian. Let us show that the map q ∶ Y → colim ( B ) belongs to R . The family of maps γ ( B )( k ) → colim ( B ) for object k ∈ K is surjective, since39he natural transformation γ ( B ) → δ colim ( B ) is a colimit cone. Moreover, since the natural transformation q ′ ∶ Y ′ → B is cartesian, the following square is cartesian for every object k ∈ K by descent: Y ′ ( k ) colim ( Y ′ ) B ( k ) colim ( B ) γ ( Y ′ )( k ) q ′ ( k ) colim ( q ′ ) γ ( B )( k ) But we have q ′ ( k ) ∈ R for every object k ∈ K , since q ′ ∈ R K . Thus, q = colim ( q ′ ) ∈ R , since the class R islocal by Proposition 3.3.7. Hence the object ( Y, q ) belongs to R ( colim ( B )) . This shows that the functorcolim ( u ) ⋆ ∶ R ( colim ( B )) → R ( colim ( A )) is essentially surjective. We have proved that the functor colim ( u ) ⋆ is an equivalence of categories. Hence the map colim ( u ) ∶ colim ( A ) → colim ( B ) is an R -equivalence. Itremains to show that the map colim ( u ) is a fiberwise R -equivalence. For this, we have to show that in everypullback square D C colim ( A ) colim ( B ) w g colim ( u ) the map w is an R -equivalence. By pulling back the square (23) along the map δ ( g ) ∶ δC → δ colim ( B ) weobtain the following cube of natural transformations, where every vertical face is a cartesian square. A ′ B ′ δD δCA Bδ colim ( A ) δ colim ( B ) u ′ γ ( A ′ ) γ ( B ′ ) δ ( w ) δ ( g ) uγ ( A ) γ ( B ) δ colim ( u ) The natural transformations γ ( A ′ ) ∶ A ′ → δD and γ ( B ′ ) ∶ B ′ → δC are colimit cones, since a base changefunctor in a topos preserves colimits. Thus, w = colim ( u ′ ) . The map u ′ ( k ) ∶ A ′ ( k ) → B ′ ( k ) is a base changeof the map u ( k ) ∶ A ( k ) → B ( k ) for every object k ∈ K . Hence the map u ′ ( k ) is a fiberwise R -equivalence,since u ( k ) is a fiberwise R -equivalence. Thus, w = colim ( u ′ ) is an R -equivalence by the first par of the proof.This shows that every base change of the map colim ( u ) is an R -equivalence. Hence the map colim ( u ) is afiberwise R -equivalence. We have proved that the class of fiberwise R -equivalences is closed under colimits.This completes the proof that the class of fiberwise R -equivalences is acyclic. Theorem 3.5.7.
Let ( L , R ) be a modality generated by a set Σ of maps in a topos E . If ∆ ( Σ ) ⊆ L , then themodality ( L , R ) is left-exact.Proof. We again denote by W ⊆ E the class of fiberwise R -equivalences. In view of the fact that wehave ∆ ( Σ ) ⊆ L , it follows from Proposition 3.5.2 that we have Σ ⊆ W . As the class W is acyclic byProposition 3.5.6, we therefore have L ⊆ W . It follows that the functor u ⋆ ∶ R ( B ) → R ( A ) is an equivalenceof categories for every map u ∶ A → B in L and therefore that the factorization system ( L , R ) is cartesian byProposition 3.2.10. Hence the modality ( L , R ) is left-exact by Theorem 3.4.7. Corollary 3.5.8.
If a modality ( L , R ) in a topos generated by a set of monomorphisms, then it is left-exact.Proof. The diagonal ∆ ( u ) of a monomorphism u ∶ A → B is invertible. Thus, ∆ ( Σ ) ⊆ L , since everyisomorphism is in L by Proposition 3.1.7. 40 .6 Congruence Classes and Higher Sheaves Definition 3.6.1.
Let E be a topos. We shall say that a class of maps L ⊆ E is a congruence if the followingconditions hold:1. The class L contains the isomorphisms and is closed under composition2. The class L is closed under colimits3. The class L is closed under finite limits Example 3.6.2. a) The class of isomorphisms
Iso ( E ) in a topos E is a congruence class. It is the smallest congruence in E .b) If φ ∶ E → E ′ is a left exact cocontinuous functor between topoi, then the class of maps L ⊆ E invertedby φ is a congruence.c) A modality ( L , R ) in a topos E is left exact if and only if its left class L is a congruence. Lemma 3.6.3.
Every congruence class L is acyclic and has the 3-for-2 property.Proof. The class L is closed under base changes by the dual of Lemma 3.1.10. Moreover, L is right cancellableby Proposition 3.1.7 and it is left cancellable by Lemma 3.4.5.Every class of maps Σ ⊆ E in a topos E is contained in a smallest congruence Σ c ⊆ E the congruence generated by Σ. We shall say that a congruence L ⊆ E is of small generation if L = Σ c for a set of mapsΣ ⊆ L . Lemma 3.6.4.
Let φ ∶ E → E ′ be a left exact cocontinuous functor between topoi. Then the class of maps L ⊆ E inverted by φ is a congruence of small generation.Proof. The class L contains the isomorphisms, it is closed under composition and under colimits. It is alsoclosed under cobase changes by Lemma 3.1.10. Hence the class L is strongly saturated in the sense of [HTT,Definition 5.5.4.5.]. That is, it is saturated and it has the 3-for-2 property. Moreover, the strongly saturatedclass L is of small generation by [HTT, Proposition 5.5.4.16.]. Thus, L = Σ ss for a set of maps Σ ⊆ E .Let us show that L = Σ c . The class Σ c is strongly saturated, since a congruence is strongly saturated byLemma 3.6.3. Thus, L = Σ ss ⊆ Σ c , since Σ ⊆ Σ c . But the class L is a congruence, since the functor φ ∶ E → E ′ is left exact and cocontinuous. Thus Σ c ⊆ L , since Σ ⊆ L . This shows that L = Σ c .For Σ a set of maps in a topos E , we write∆ ∞ ( Σ ) = { ∆ i u ∣ u ∈ Σ , i ≥ } for the diagonal closure of Σ. That is, the collection of all diagonals of all maps in Σ. Proposition 3.6.5. If Σ is a set of maps in a topos E , then Σ c = ∆ ∞ ( Σ ) a .Proof. Let us put Λ ∶= ∆ ∞ ( Σ ) . The pair ( Λ a , Λ ñ ) is a modality by Theorem 3.3.19. Furthermore, themodality ( Λ a , Λ ñ ) is left-exact by Theorem 3.5.7 since we have ∆ ( Λ ) ⊆ Λ ⊆ Λ a . Hence the class Λ a is acongruence. Thus, Σ c ⊆ Λ a , since Σ ⊆ Λ a . Conversely, we have Λ ⊆ Σ c , since the congruence Σ c is closedunder finite limits. Hence we have Λ a ⊆ Σ c , since Σ c is acyclic by Lemma 3.6.3. Definition 3.6.6.
Let Σ be a set of maps in a topos E . We say that an object X ∈ E is a Σ -sheaf if the mapMap ( u ′ , X ) ∶ Map ( B ′ , X ) → Map ( A ′ , X ) is invertible for every base change u ′ ∶ A ′ → B ′ of a map u ∈ ∆ ∞ ( Σ ) .We write Sh ( E , Σ ) for the full subcategory of Σ-sheaves.The notion of Σ-sheaf above uses a condition that depends on the class of all base changes of the mapsin Σ. The following lemma shows that it suffices verify this condition for a set of maps.41 emma 3.6.7. Let Σ be a set of maps in a topos E . If G is a set of generators, then an object X ∈ E is a Σ -sheaf if and only if X is local with respect to the G -base changes of the maps in ∆ ∞ ( Σ ) .Proof. Every object in E is the colimit of a diagram of objects in G , since G is a set of generators. Let Λ bethe set of G -base changes of the maps in ∆ ∞ ( Σ ) . Let us show that Λ a = ∆ ∞ ( Σ ) a . We have Λ ⊆ ∆ ∞ ( Σ ) a ,since every map in Λ is a base change of a map in ∆ ∞ ( Σ ) . Thus, Λ a ⊆ ∆ ∞ ( Σ ) a . Conversely, let us showthat ∆ ∞ ( Σ ) ⊆ Λ a . The set Λ is closed under G -base changes by transitivity of base changes. Thus, Λ ⊥ = Λ ñ by Lemma 3.3.18. Moreover, the pair ( Λ a , Λ ñ ) is a modality by Theorem 3.3.19. Hence the class Λ a is localby 3.3.7. Let us show that ∆ ∞ ( Σ ) ⊆ Λ a . If u ∶ A → B is a map in ∆ ∞ ( Σ ) , then its codomain B is thecolimit of a diagram F ∶ K → E of objects F ( k ) ∈ G , since G is a set of generators. The collection of inclusions ι k ∶ F ( k ) → B is surjective by Example 2.3.5. The map ι ⋆ k ( u ) is the G -base change of the map u ∶ A → B along the map ι k ∶ F ( k ) → B . Hence we have ι ⋆ k ( u ) ∈ Λ for every object k ∈ K . It follows that we have u ∈ Λ a , since Λ ⊆ Λ a and the class Λ a is local. Thus, ∆ ∞ ( Σ ) ⊆ Λ a and hence ∆ ∞ ( Σ ) a ⊆ Λ a . The equalityΛ a = ∆ ∞ ( Σ ) a is proved. An object X ∈ E is a Σ-sheaf if and only if it is modal with respect to the set ofmaps ∆ ∞ ( Σ ) . Thus, an object X is a Σ-sheaf if and only if it is modal with respect to the set of maps Λ.But X is modal with respect to Λ if and only if it is local with respect to Λ, since Λ ñ = Λ ⊥ .As an example, let P ( K ) = [ K op , S ] be the category of presheaves on a small category K . The category P ( K ) is presentable and generated by the set of representable presheaves R ( K ) ⊂ P ( K ) . Hence given a setof maps Σ ⊆ P ( K ) , in order to check that a presheaf X on K is a Σ-sheaf, it suffices to check that is it localwith respect to the R ( K ) -base changes of the maps in ∆ ∞ ( Σ ) . Theorem 3.6.8.
Let Σ be a set of maps in a topos E . Then the subcategory Sh ( E , Σ ) is reflective and thereflector ρ ∶ E → Sh ( E , Σ ) is left-exact. In particular, Sh ( E , Σ ) is a topos. Furthermore, the reflector ρ invertsthe maps in Σ universally among cocontinuous and left-exact functors. In other words, we may identify thereflection ρ ∶ E → Sh ( E , Σ ) with the cocontinuous and left-exact Σ -localization E → E [ Σ − ] lexcc .Proof. Let us put Λ ∶= ∆ ∞ ( Σ ) . By definition, an object X is a Σ-sheaf if and only if it is modal withrespect to Λ. In other words, we have Sh ( E , Σ ) = Mod ( E , Λ ) . Now, the pair ( Λ a , Λ ñ ) is a modality byTheorem 3.3.19 and therefore the subcategroy Sh ( E , Σ ) = Mod ( E , Λ ) = R ( ) is reflective by Lemma 3.3.22.Furthermore, the modality ( Λ a , Λ ñ ) is left-exact by Theorem 3.5.7 since we have ∆ ( Λ ) ⊆ Λ ⊆ Λ a . It followsthat the reflector ρ = ∥ − ∥ S ∶ E → R ( ) = Sh ( E , Σ ) is left-exact by Theorem 3.4.7 and hence that the categorySh ( E , Σ ) is a topos.Finally, suppose we are given a cocontinuous and left-exact functor φ ∶ E → E ′ from E to some topos E ′ which inverts every map in Σ. We claim that there then exists a unique left-exact cocontinuous functor ψ ∶ Sh ( E , Σ ) → E ′ such that ψ ○ ρ = φ . E Sh ( E , Σ ) E ′ φρ ψ To see this, note that the class of isomorphisms J ⊆ E ′ is acyclic and hence also the class A ∶= φ − J sincethe functor φ preserves colimits and finite limits. We have Σ ⊆ A since the functor φ inverts the mapsin Σ by assumption. Observe that we have φ ( ∆ ( u )) = ∆ ( φ ( u )) for every map u ∈ E since the functor φ preserves finite limits and consequently that ∆ ( A ) ⊆ A since ∆ ( J ) ⊆ J . It follows that, Λ = ∆ ∞ ( Σ ) ⊆ A since Σ ⊆ A and therefore that Λ a ⊆ A , since the class A is acyclic. This shows that the functor φ ∶ E → E ′ inverts every map in Λ a . Hence there exists a unique cocontinuous functor ψ ∶ Loc ( E , Λ a ) → E ′ such that ψ ○ ρ = φ by Proposition 2.2.4. But now note that the functor ψ is left-exact since the functor ψ ○ ρ is leftexact and ρ is a left-exact reflector. Moreover, the uniqueness ψ is clear since the reflector ρ ∶ E → Sh ( E , Σ ) is a localization.In [Rezk], a model site is defined to be a pair ( K , Σ ) where K is a small category, Σ is a class of maps in P ( K ) satisfying the condition that the localization generated by Σ is left-exact. The results of this article,42owever, provide us with concrete methods of presenting topoi by means of generators and relations, andthus lead us to propose a modification of the notion of higher site. Specifically, we lift the requirement thatthe localization generated by Σ be left exact, since the definition of Σ-sheaf provides us with a means ofrecognizing the local objects for the left-exact localization which Σ generates. Our notion of site also extendsthat of [TV05] which is only suited for topological left-exact localizations. Definition 3.6.9. A site is defined to be a pair ( K , Σ ) , where K is a small category and Σ is a set of maps in P ( K ) . We shall say that a presheaf X on K is a sheaf on ( K , Σ ) if X is local with respect to the R ( K ) -basechanges of the maps in ∆ ∞ ( Σ ) . We shall denote the category of sheaves on ( K , Σ ) by Sh [ K , Σ ] .Every topos E is a left exact localization of a presheaf category P ( K ) [HTT, Definition 6.1.0.4.]. If L is the class of maps inverted by the localisation functor ρ ∶ P ( K ) → E , then we have E = Loc ( P ( K ) , L ) .The class L is a congruence of small generation by Lemma 3.6.4. Hence we have L = Σ c for a set of mapsΣ ⊆ P ( K ) . But we have Σ c = ∆ ∞ ( Σ ) a by Proposition 3.6.5. Thus, a presheaf X ∈ P ( K ) is local with respectto L if and only if it is a Σ-sheaf. Thus, Loc ( P ( K ) , L ) = Sh ( P ( K ) , Σ ) = Sh [ K , Σ ] . This shows that the topos E is equivalent to the category of sheaves on the site ( K , Σ ) . References [AS] M. Anel and C. L. Subramaniam,
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