arXiv:2101.02791v1 [math.CT] 7 Jan 2021 r hntesevsfrtetplg.Frhroe h Groth the Furthermore, topology. the for sheaves the then are Introduction 1 Contents nqeycrepnst a to corresponds uniquely h aeoyo rsevso sets of 1-topos presheaves Grothendieck any of that category Recall the one. classical the bling h hoyof theory The Introduction 1 Localizations Left-Exact and Sheaves Higher 3 Preliminaries 2 ieecs tiigeapei h bec,i h aeof case the in absence, the is example striking A differences. ∗ § ‡ † . etEatMdlte ...... Sheaves ...... Higher . . . . and . . . . . Classes . . . . Congruence ...... 3.6 . . . . Fiberwise ...... Modalities . 3.5 . . . Left-Exact . . . Classes . . . Acyclic . . . and . . . 3.4 . . Modalities ...... Systems . . . Factorization . . 3.3 . . Cartesian ...... Systems . 3.2 . . . Factorization ...... 3.1 ...... Fibrations . . . Grothendieck . Topoi . and . . Descent, 2.4 . Colimits, Presentability . and . Reflections, 2.3 . Localizations, notations and 2.2 Conventions 2.1 IGT UQ CIRGET, ericfi[email protected] University, Cambridge eatmnod ae´tcsyEstad´ıstica, Matem´aticas Universidad y de Departamento eateto hlspy angeMlo nvriy mat University, Mellon Carnegie Philosophy, of Department hti,afcoiainsse hs etcasi tbeby stable is class left whose system factorization a is, that ete hwta h soitdreflection associated the that show then We ihrsett sisl an itself is Σ to respect with neato fvrostpso atrzto ytm,and systems, factorization by of generated types various localization of left-exact interaction the as characterized ihrSevsadLf-xc oaiain of Localizations Left-Exact and Sheaves Higher epooeadfiiino ihrsefwt epc oa arbi an to respect with sheaf higher of definition a propose We ∞ M [email protected] AM. ` tpirsebe riaytpster nmn set,btth but aspects, many in theory topos ordinary ressembles -topoi R euvlne ...... -equivalences rtedektopology Grothendieck ∞ P tps utemr,w hwta h reflection the that show we Furthermore, -topos. 0 ( C ∶= ) er Biedermann Georg [ n Andr´e Joyal and C ahe Anel Mathieu op rcFinster Eric E Set , → Abstract on Sh ] ( C 1 e ot,[email protected] Norte, del nasalcategory small a on E h bet nterflciesbaeoyof subcategory reflective the in objects The . [email protected] , Σ ) aeesnilueo h oino a of notion the of use essential make slf-xc ota h uctgr fsheaves of subcategory the that so left-exact is .I h oreo h ro,w td the study we proof, the of course the In Σ. ‡ aechange. base ∞ , ∗ E , tpi faycnrt oinof notion concrete any of -topoi, rssa eetv eteatlclzto of localization left-exact reflective a as arises § † nic oooyisl a evee sa as viewed be may itself topology endieck , rr e fmp nan in Σ maps of set trary C n htaysc localization such any that and , ...... E ...... r r oeimportant some are ere → ...... ∞ Sh ...... -Topoi ( E , ∞ Σ ) modality -topos sheaf a be may E resem- , . . P 0 ( 16 C 41 35 31 27 22 16 11 6 1 9 7 6 ) distinguished class of monomorphisms in P0(C), and an equivalent characterization of the sheaves is that they are exactly the objects which are local for this class. In other words, the left-exact localizations of P0(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 remains true that every ∞-topos E is a left-exact localization of the category of presheaves P(C) ∶= [Cop, S] on a small ∞-category C, but the collection of all such localizations can no longer be completely described using Grothendieck 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-exact localization 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 of monomorphisms, the cotopological part is a localization which inverts no monomorphisms. The classical theory of sheaves and Grothendieck topologies can therefore be applied to the topological localizations, but not to the cotopological ones. Our goal in the present paper is to introduce a general notion of sheaf with respect to an arbitrary set of maps Σ ⊆ E (not necessarily monomorphisms) in an ∞-topos E such that the category 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 will from now on drop the prefix “∞” when refering to ∞-topoi and (∞, 1)-categories, and speak explicity of 1-categories and 1-topoi if the occasion arises. See Section 2.1 for a more complete description of our conventions 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 its diagonal and by ∆n(u) its n-th iterated diagonal (with the convention that ∆0(u) ∶= u). For a set of maps Σ, we write ∆(Σ) = {∆(u) S u ∈ Σ} for the set of all diagonals of maps in Σ and we define the diagonal closure of Σ by ∆∞(Σ) = {∆n(u) S 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, for every u ∶ A → B ∈ ∆∞(Σ) and every base change u′ ∶ A′ → B′ of u, the map Map(B′,X) → Map(A′,X) is invertible. 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 the reflector ρ ∶ E → Sh(E, Σ) is left-exact. In particular, Sh(E, Σ) is a topos. Furthermore, the reflector ρ inverts the 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 of Theorem 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 an instance 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 by the 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 in Σ. We write Mod(Σ, E) ⊆ E for the full subcategory spanned by the Σ-modal objects. It is worth noting that if G ⊆ E is a set of generators of the presentable category E, then it suffices to consider base changes along maps having their domain in G. It follows that the modal objects with respect to a set of maps Σ can themselves 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 as follows: an object X ∈ E is a Σ-sheaf if it is modal for ∆∞(Σ), the diagonal closure of Σ. Furthermore, we have a nested sequence of subcategories Sh(E, Σ) ⊆ Mod(E, Σ) ⊆ Loc(E, Σ) ⊆ E

2 each of which is reflective in E. The inclusions are, in general, all proper, and one of the main objectives of the 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 which u v every map f ∶ A → B ∈ E admits a factorization f = A Ð→ S(f) Ð→ B where v ∈ L and u ∈ R. (See Section 3.1 for 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 reflective subcategory R(1). A basic observation, then, is that when E is a topos and Σ is a set of maps of E, then each of the subcategories Sh(Σ, E), Mod(Σ, E) and Loc(Σ, E) introduced above can be realized as a subcategory of the form R(1) for a particular type of factorization system generated by Σ (even if the factorisation system is not necessarily unique, as we shall see). These various types of factorization systems are distinguished by the closure properties of their left classes. As a first step, let us note that any factorization system has the property that its left class contains all the isomorphisms, is closed under composition and formation of colimits in the arrow category (Lemma 3.1.11). Following [HTT, Definition 5.5.5.1.], we refer to classes of morphisms satisfying these properties as saturated classes. Recall that if E is a presentable category, then every set of maps Σ generates a factorization system whose left class L can be identified as the saturation of Σ, that is, its closure under the properties listed above. We denote this class by Σs. It is easily seen in this case that the subcategory R(1) coincides with Loc(E, Σ). A related, but distinct, construction arises from the fact that the subcategory Loc(E, Σ) is reflective. This leads to another factorization system associated to Σ: namely, the one whose left maps are those inverted by the associated reflector E → Loc(E, Σ). The left class of this factorization system has the additional property that, in addition to being saturated, it is closed under the familiar 3-for-2 axiom. Following [HTT, Definition 5.5.4.5.], 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 factorization system generated by Σ is exactly the strongly saturated closure of Σ, which we denote Σss. 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(1) associated to the modality generated by Σ. In addition to being saturated, the left class of a modality is stable by base change by definition. We call a saturated class with this property acyclic and it can be shown that the left class of the modality generated by Σ is its acyclic closure, which we denote Σa. Modalities play a key role in the present 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 the formation of all finite limits in the arrow category. A saturated class of maps with this property will be referred to as a congruence class. As we will see, if we know in advance that S is a modality, then it suffices that S additionally be reflective in order that it be left exact. Every set of maps Σ generates a smallest congruence class denoted Σc, and one central goal of the current work is to give a concrete description of Σc. 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 will show 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 differ from ordinary modalities. Concretely, if we would like to understand what the left-exact modality generated by a collection of maps is, we can first attempt to answer a related question: under what conditions is the modality generated by a collection of maps already left exact? This question turns out to have a relatively straightforward answer:

3 Table 1: Types of Factorization Systems

unique factorization reflective (L, R) modality left-exact modality system factorization system L closure strongly saturated saturated class acyclic class congruence class properties class (stable under (stable under (stable under (stable under colimits and colimits and colimits) colimits and 3-for-2) base change) finite limits) R(1) Local objects Modal objects Sheaves Loc(Σ, E) Mod(Σ, E) Sh(Σ, E) Left class generated by Σs Σss Σa Σc Σ E → Loc(E, Σ) E → Sh(E, Σ) Associated universal for universal for lex Localizations cocontinuous cocontinuous functors functors

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 the 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 of the theorem is automatic. And indeed, it is not hard to see that the modality generated by ∆∞(Σ) is exactly the left-exact modality generated by Σ (since any left-exact modality will necessarily contain the diagonals of all its generating maps). The proof of Theorem 3.5.7 leads us to consider a fourth kind of factorization system, namely the cartesian factorization 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 two important but distinct roles in the proof. A first observation is that, if a factorization system is at the same time a modality and cartesian, then it is necessarily a lex modality. In a certain sense, this observation is somewhat trivial since a cartesian factorization system is, by definition, reflective, and as we have pointed out above, a modality which is reflective is automatically left-exact. (See Theorem 3.4.7). But the force of this observation comes from the fact that the left class of a cartesian factorization system admits the following relatively simple characterization: Proposition 3.2.10. A factorization system (L, R) in a category with finite limits E is cartesian if and only if 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 showing that for each map u ∈ L the associated base change functor is an equivalence of categories. In fact, it turns out 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 shall say that u is a fiberwise R-equivalence if every base change of u is an R-equivalence.

4 The usefulness of the notion of fiberwise R-equivalence comes from the following proposition, which shows that 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 left class of a modality generated by a set of maps Σ may be identified as Σa, that is, the acyclic class generated by 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 is acyclic. That the fiberwise R-equivalences contain identities and are closed under composition and base change is immediate from the definition. Hence the only remaining task is to show that they are closed under colimits. It is at this point that the second use of the theory of cartesian factorization systems makes its appearance. As it happens, a rich source of cartesian factorization systems comes from the theory of Grothendieck fibrations (Section 2.4). A crucial observation, then, is that for a topos E and small indexing category K, the colimit functor colim ∶ [K, E] → E is a Grothendieck fibration, and hence gives rise to a cartesian factorization system in which the right maps are exactly the cartesian natural transformations of diagrams. This can be seen as a reformulation of the property of descent in a topos. A careful calculation in 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 our stated goal is to understand left-exact localizations, and modalities do not have a simple analog in terms of localization theory. There are at least two reasons, however, to recommend the approach taken here. A first is that, though modalities have not received a great deal of attention in the literature on category theory and homotopy theory, they do arise quite naturally from the logical perspective on topos theory, a point of view whose higher analog goes by the name Homotopy Type Theory. From the perspective of the internal language, modalities appear as operations on types similar to the operators of modal logic, which is where the concept gets its name. Indeed, many of the early properties of modalities were first worked out in in this setting [HoTT, RSS]. Furthermore, our original approach to Theorem 3.5.7 was very much inspired by ideas 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 sheafification techniques. A second motivation is that, owing to the ubiquity with which they appear in various parts of homotopy theory 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 known in 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 A1-local objects of motivic homotopy theory are part of a modality on the Nisnevich topos [MV]. Furthermore, arbitrary modalities in a topos may be 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 present paper will contribute to their adoption and study. Acknowledgments: The authors would like to thank Mike Shulman, Egbert Rijke and Simon Henry for useful discussions on the material of this paper. The first author gratefully acknowledges the support of the Air 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 of Canada through grant 70733401.

5 2 Preliminaries 2.1 Conventions and notations Throughout the paper, we use the language of higher category theory. The word category refers to (∞, 1)-cat- egory, and all constructions are assumed to be homotopy invariant. When necessary, we shall refer to an ordinary category as a 1-category and to an ordinary Grothendieck topos as a 1-topos. Furthermore, we work in a model independent style, which is to say, we do not choose an explicit combinatorial model for (∞, 1)-categories such as quasicategories, but rather give arguments which we feel are robust enough to hold in any model. We use the word space to refer generically to a homotopy type or ∞-groupoid and write S for the category of spaces. We shall denote by C(A, B) or by MapC(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 object of C. The opposite of a category C is denoted Cop and defined by the fact that Cop(B, A) ∶= C(A, B) with its category 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 the object and structure map visible when working in a slice category. If a category C has a terminal object, we denote it by 1. Every category C has a homotopy category hoC which is a 1-category with the same objects as C, but where hoC(A, B) = π0C(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 hoC. We write A ≃ B to indicate that two objects are isomorphic. We make a small exception to this terminology with regard to categories and spaces: 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 an object 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(FA,FB) is an equivalence for every A, B ∈ C. A functor F ∶ C → D is an if it is fully faithful and essentially surjective. We denote the category of functors from C to D alternatively by [C, D] or DC as seems appropriate from the context. For a small category C, we will write P(C) ∶= [Cop, 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 an isomorphism is unique in this sense. We assume that all subcategories and classes of maps in a category are defined by properties which are invariant under isomorphism, and consequently adopt the convention that all subcategories are replete. We will denote by [1] the poset {0 < 1}, regarded as a category. If C is a category, then the functor category C[1] can be described as the category of arrows of C. By construction, an object of C[1] is a map u ∶ A → B in C, and a morphism (f,g) ∶ u → u′ in C[1] is a commutative square

f A A′

u u′ g B B′ For any class of maps A in a category C, we shall write A for the full subcategory of C[1] whose objects are the maps in A. This construction permits us to succinctly describe closure properties of the class A in terms of the associated subcategory A. For example, we say that A is closed under colimits or limits if this is true of A. Given a commutative square g A B

f h CD k

6 in a category C with finite limits we refer to the canonically induced map (f,g) ∶ A → C ×D B as the cartesian gap map of the square.

Finally, for an object A of a category C with finite limits, we will write ∆(A) = (1A, 1A) ∶ A → A × A for the 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) = (1A, 1A) ∶ A → A ×B A

1A A ∆(u)

p2 A ×B A A

1Af p1  u

Au B

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.

2.2 Localizations, Reflections, and Presentability 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 denote by [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. We shall say that a functor F ∶ E → F is a localisation if it is a Σ-localisation with respect to some class of maps Σ ⊆ E. If Σ is a class of maps in a category E, then the codomain of any Σ-localization is unique up to equivalence of categories, and we denote this category generically by E[Σ−1]. 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 of maps in E, then we can ask that a cocontinuous functor F ∶ E → F invert the maps in Σ universally among Σ cocontinuous functors. If we write [E, F]cc for the category of colimit preserving functors and [E, F]cc for the 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

Σ (−) ○ F ∶ [F, G]cc → [E, G]cc

−1 is an equivalence of categories for every cocomplete category G. We will write E[Σ ]cc for the codomain of a 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 finite limit preserving functors F ∶ E → F, and is defined completely analogously. In this case, we denote the target −1 lex of such a localization by E[Σ ]cc . Remark 2.2.1. The previous notions of localization need not coincide. For example, if Σ = {Sn+1 → 1} is −1 the singleton set consisting of the terminal map from the (n + 1)-sphere to the point in S, then S[Σ ]cc is −1 lex the category of n-truncated spaces, while S[Σ ]cc is the terminal category.

7 While arbitrary localizations can be difficult to describe in general, those which arise from reflective subcategories tend to be considerably more tractable. Recall that a full subcategory E′ of a category E is said 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 right adjoint ι ∶ F → E. In this case, the essentialy image of ι coincides with the subcategory spanned by those objects 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 of all 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 map

Map(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 if the 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 respect to 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 said to be presentable if it is equivalent to a category Loc(P(K), Σ) for a small category K and a (small) set of maps 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 −1 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) = [Kop, S] be the category of presheaves on a small category K. Then the set of representable presheaves R(K) generates strongly P(K). Every presentable category E admits a set of generators. If E is a cocomplete category, then so is the category E~B for any object B ∈ E. If G ⊆ E is a class of objects, 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 cocomplete category 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 objects of G, since G strongly generates E. If ιk ∶ F (k) → X denotes the inclusion for each object k ∈ K, consider ′ ′ the functor F ∶ K → E~B defined by putting F (k) = (F (k), fιk) for every object k ∈ K. The family of ′ ′ morphisms ιk ∶ (F (k), fιk) → (X,f) is defining a colimit cone ι ∶ F → (X,f) since the family of morphisms

8 ′ ι ∶ 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), fιk) ∈ 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 the class G~B.

2.3 Colimits, Descent, and Topoi If E is a category and K is a small category, then the diagonal functor δ ∶ E → [K, E] takes an object A ∈ E to the 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 as a diagram K → E~A that we shall denote α~A. Let us denote by K ⋆ 1 the category obtained by adjoining freely a terminal object 1 to the category K. Let iK ∶ K → K ⋆ 1 be the inclusion functor. Then a functor C ∶ K ⋆ 1 → E is the same thing as a cone α ∶ F → δA, where F = C ○ iK and A = F (1). If the category E is ⋆ cocomplete, then the restriction functor i ∶ [K ⋆ 1, E] → [K, E] has a left adjoint γ = i! ∶ [K, E] → [K ⋆ 1, 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):

p2 g⋆(F ) F

g⋆(α) α δ(g) δB δA computed in the functor category. If E is moreover cocomplete, colimits are said to be universal if for any small category K, the base change g⋆(α) ∶ g⋆(F ) → δB of a colimit cone α ∶ F → δA ∶ K ⋆ 1 → E along any map 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 naturality square F (r) F (i) F (j)

u(i) u(j) G(r) G(i) G(j)

is cartesian for every morphism r ∶ i → j in K. Colimits in E are said to be effective (or van Kampen) if, for any cartesian transformation f ∶ F → G ∶ K → E of K-diagrams, the following square is cartesian for every object k ∈ K, γ(F )(k) F (k) colim(F )

f(k) colim(f) γ(G)(k) G(k) colim(G)

Definition 2.3.1 (Rezk). A cocomplete and finitely complete category is said to have descent if its colimits are 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 category and 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 either −1 −1 lex the cocomplete E → E[Σ ]cc or cocomplete and left-exact E → E[Σ ]cc localizations of E with respect to Σ. Recall from Remark 2.2.1 that the two need not coincide. But it is the latter plays a crucial role in the theory of topoi because owing to the following:

9 Theorem 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 category 2. 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 in Proposition 2.2.4, the cocotinuous localization generated by Σ has a straigtforward description in terms of local objects. Our strategy, therefore, will be to understand the left-exact localization generated by a class Σ in terms of the cocontinuous localization generated by a class Σ′ related to Σ.

2.3.1 Surjective families and Local Classes Definition 2.3.3. We shall say that a map f ∶ X → Y in a topos E is surjective, or that f is a surjection, if ⋆ the base change functor f ∶ E~Y → E~X is conservative. We shall say that a family of maps {gi ∶ Yi → Y }i∈I is surjective if the induced map ⟨gi ∶ i ∈ I⟩ ∶ + Yi → Y i∈I

is surjective. Equivalently, a family of maps {gi ∶ Yi → Y }i∈I is surjective if and only if the induced functor

E~Y → M E~Yi i∈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 π0(f) ∶ π0(X) → π0(Y ) is surjective. A pointed object (X, x) in a topos E is connected if and only if the map x ∶ 1 → X is surjective. x b) If X ∈ S is a space, then the family of maps {x ∶ 1 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 inclusion maps {ιk ∶ 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 if the following two conditions hold: 1. A is closed under base change

2. Given a map f ∶ X → Y ∈ E and a surjective family {gi ∶ Yi → Y }i∈I of Y , if the base change of f along gi 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 {gi ∶ Yi → Y }i∈I is surjective and the base change ∗ gi (f) ∶ Yi ×Y X → Yi 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 −1(y) → 1 belong to A for every y ∶ 1 → Y . In particular, a map f ∶ X → Y in S is invertible if and only the map f −1(y) → 1 is invertible for every point y ∈ Y .

10 2.4 Grothendieck Fibrations

Recall that a functor F ∶ E → B induces a functor F~A ∶ E~A → B~FA 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 the functor F~A ∶ E~A → B~FA 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 the notion 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 Grothendieck fibration if and only if it admits a factorization F = F˜ ○W ∶ E → E′ → B, where W is a categorical equivalence and F ′ is a cartesian fibration. Example 2.4.3. Some basic examples of Grothendieck fibrations: a) The unique functor E → 1 for any category E b) Any functor E → B, if B is a groupoid

c) The forgetful functor E~B → E for any object B in a category E

d) The codomain functor E[1] → 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 2.4.12 f) 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,p1) of the map G(f) ∶ GX → GF B along the map η(B) ∶ B → GF B as in the following diagram:

p2 B ×GF B GX GX

p1 G(f) B GFB η(B)

That this construction is indeed right adjoint to the functor FB 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~BG~B (X,f) → (X,f) of F~F B which corresponds to the map p2 ∶ 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~BF~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: η(A) A GF A

u GF (u) η(B) B GFB

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.

11 Proof. Since E has finite limits and φ is a reflector, we are in the situation of Lemma 2.4.4. We thus obtain, E E′ E E E′ for every B ∈ , a right adjoint ψ~B ∶ ~φB → ~B to the induced functor φ~B ∶ ~B → ~φB. By the definition of Grothendieck fibration, it therefore suffices to show that each of the ψ~B is fully faithful. For E′ f ∶ X → φB ∈ ~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 notation by omitting mention of the inclusion ψ ∶ E′ ↪ E)

φ(B ×φB X) X

f φB φB

But this map is an isomorphism since φ is a reflection which preserves finite limits.

2.4.1 Basic properties Definition 2.4.6. 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:

f○− Map(K,X) Map(K, Y )

F F F (f)○− Map(FK,FX) Map(FK,FY )

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, the two squares of the following diagram are cartesian for every object K ∈ E

f○− g○− Map(K,X) Map(K, Y ) Map(K,Z)

F F F F (f)○− F (g)○− Map(FK,FX) Map(FK,FY ) Map(FK,FZ)

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~FY 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 have an isomorhism i ∶ F~Y (E,p) = (FE,F (p)) → (B,u) as claimed. That p ∶ E → Y is cartesian is easily deduced from the fact that, for any m ∶ K → Y , one has an equivalence

MapE ((K,m), (E,p)) ≃ MapB ((FK,F (m)), (B,u)) ~Y ~F Y 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) ∶ FX → F Y , there exists a F -cartesian morphism p ∶ E → Y together with an isomorphism i ∶ F E → FX such that F (f)i = F (p).

12 By the definition of cartesian morphism, there therefore exists a unique map v ∶ X → E such that pv = f and F (v) = i−1. The map v is F -vertical, since F (v) is invertible. The proof of the uniqueness of the decomposition 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

A! E~A E

F~A F (FA)! B~FA B

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~FA for every object 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~FA)~FA(B,f) is isomorphic 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 tedious calculuation shows that the map h ∶ (X,fh) → (B,f) in the category E~A is F~A-cartesian if and only if the map h ∶ X → B in E is F -cartesian. Finally, by definition, a map h ∶ (X,fh) → (B,f) in the category E~A is F~A-vertical if and only if the map F~A(h) = F (h) ∶ (FX,F (fh)) → (FB,F (f)) in the category B~FA is invertible. But the map F (h) ∶ (FX,F (fh)) → (FB,F (f)) is invertible if and only if the map F (h) ∶ FX → FB in the category B is invertible, since the forgetful functor B~FA → B is conservative. Hence a map h ∶ (X,fh) → (B,f) in the category E~A is F~A-vertical, if and only if the map h ∶ X → B in the 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 and let η ∶ Id → GF be the unit of the adjunction F ⊣ G. Then a morphism f ∶ X → Y in E is F -cartesian if and only if the following square is cartesian:

η(X) X GFX

f GF (f) (1) η(Y ) Y GF Y

Proof. The square (1) is cartesian if and only if the following square is cartesian for every object K in E:

η(X)○− Map(K,X) Map(K,GFX)

f○− GF (f)○− (2) η(Y )○− Map(K, Y ) Map(K, GF Y )

On the other hand, we have a natural isomorphism

≃ θX ∶ Map(FK,FX) Ð→ Map(K,GFX) by adjunction, from which we deduce that the right hand square in the following diagram is (trivially) cartesian. θX Map(K,X) F Map(FK,FX) Map(K,GFX)

f○− F (f)○− GF (f)○− (3)

θY Map(K, Y ) F Map(FK,FY ) Map(K, GF Y )

13 It follows that the square (2) is cartesian if and only if the left hand square of diagram (3) is cartesian, which proves 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 XUu

f g (4) Yv V 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 cartesian square (4) are F -cartesian. This stronger result will be proved below as Lemma 3.2.5 after the introduction of 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 square of functors commutes V! E~V E

F~V F (F V )! B~F V B where the horizontal arrows are the forgetful functors. Note that the square (4) which we would like to prove cartesian is the image by the forgetful functor V! ∶ E~V → E of the following square

(X,vf)u (U,g)

f g v (Y, v) (V, 1V )

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, a square 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, 1V ) is terminal in the category E~V and the functor F~V ∶ E~V → B~F V is a reflector 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 a fully faithful right adjoint G ∶ B → E, since it is a reflector, and the essential image E′ ⊆ E of the functor G is a reflective subcategory E′ ⊆ E with reflector R ∶ E → E′ equivalent to the functor F ∶ E → B. Hence we may suppose 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 object is local with respect to every map in E and we may identify E′ = Loc(E, Σ), where Σ is the class of maps ′ inverted by the reflector R ∶ E → E . Next, let us show that if a map pZ ∶ Z → 1 in E is R-cartesian, then ′ Z ∈ E . Note that the following square is cartesian by Lemma 2.4.9, since pZ is R-cartesian:

η(Z) ZRZ

pZ R(pZ ) (5) η(1) 1 R1 But the map η(1) is invertible, since 1 ∈ E. Hence the map η(Z) is invertible, since the square (5) is cartesian so that we have Z ∈ E′. It follows that the square XUu f (6) Y 1

14 belongs to the subcategory E′ since every map in the square is R-cartesian by hypothesis, and R-cartesian maps compose by Lemma 2.4.7. Hence the square (6) is isomorphic to its image by the functor R ∶ E → E′, since the endofunctor RSE′ ∶ E′ → E′ induced by R is isomorphic to the identity functor E′ → E′. But the square (6) is cartesian in the subcategory E′, since it is cartesian in E and the inclusion E′ ↪ E is fully faithful. Thus the image of the square (6) by the functor R ∶ E → E′ is cartesian and completes the proof.

2.4.2 The fibration of diagrams A fundamental example of Grothendieck fibration arises from the category of diagrams in a topos, as we now 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, cartesian with 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] sending and object X ∈ E to the constant K-diagram at X. Since the category [K, E] has finite limits, we are in the situation of Lemma 2.4.4. We have, therefore, a right adjoint δ~B ∶ E~ colim(B) → [K, E]~B to the induced functor 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

p2 δ~B(X) δ(X)

p1 δ(f) B δ colim(B) η(B)

The counit of the adjunction colim~B ⊣ δ~B is the map colim(δ~B(X)) → X that corresponds to the map p2 ∶ δ~B(X) → δ(X) via the adjunction colim ⊣ δ. The cone p2 ∶ δ~B(X) → δ(X) is a colimit cone since colimits are 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 the functor colim~B is a reflector. It follows by definition that the functor colim ∶ [K, E] → E is a Grothendieck fibration. 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 the adjunction colim~B ⊣ δ~B is invertible. By construction, η(A, u) is the cartesian gap map of the following square: γ(A) A δ colim(A)

u δ colim(u) (7) γ(B) B δ colim(B)

Hence the map η(A, u) is invertible if and only if the square (7) is cartesian. But it is easily seen by descent in 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 3 Higher Sheaves and Left-Exact Localizations 3.1 Factorization Systems Definition 3.1.1. Let u ∶ A → B and f ∶ X → Y be two maps in a category E. We say that u is left orthogonal to f, or that f is right orthogonal to u, if every commutative square

Ax X u f y B Y 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 cartesian

Map(u,X) Map(B,X) Map(A, X)

Map(B,f) Map(A,f) Map(u,Y ) Map(B, 1) Map(A, 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 , and write A ⊥ f, if the map A → 1 is left orthogonal 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 an object A ∈ E is left orthogonal to an object X ∈ E, and write A ⊥ X, if the map A → 1 is left orthogonal the map X → 1. Lemma 3.1.2. Let E be a category with a terminal object 1. Then an object X ∈ E is right orthogonal to a map 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 → 1 if and only if the following square is cartesian

Map(u,X) Map(B,X) Map(A, X)

Map(B,p) Map(A,p) Map(u,1) Map(B, 1) Map(A, 1)

But the map Map(u, 1) is invertible, since Map(B, 1) = 1 = Map(A, 1). Hence the square is cartesian if and only if the map Map(u,X) is invertible. This shows that u ⊥ p if and only if the object X is local with respect 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 for every 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[1] whose objects 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

16 The class L of a factorization system S = (L, R) is said to be the left class of the factorization system and the class R to be the right class. A factorization f = pu ∶ X → E → Y with u ∈ L and p ∈ R is called an 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.

f X Y

L(f) R(f) S(f)

Moreover, the operations f ↦ L(f), f ↦ R(f) and f ↦ S(f) are functorial. When the category E has a terminal object, the we shall write YXYS for S(X → 1). We shall see in Proposition 3.1.16 that the endo-functor Y − YS ∶ 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 factorization systems. b) A map between two spaces f ∶ X → Y is said to be injective if the square

X X f (8) X Y f

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 the pair (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 of surjective 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 map and u ∶ J → Y an injective map, f X Y

p u J 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 AK the class of natural transformations 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 (LK, RK) in the category [K, E] for every small category K. Proof. [HTT, Corollary 5.2.8.18]

17 We record here for future reference a number of standard facts about the closure properties of the right and left classes of a factorization system. Recall that if A is a class of maps in a category E, then A denotes the full subcategory of E[1] whose objects are the maps in A. We shall say that the class A is closed under colimits is the full subcategory A ⊆ E[1] is closed under colimits. The closure of A under finite colimits, or finite 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 composition 2. The class L is closed under cobase change 3. The class R is closed under base change 4. 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 colimits 7. The class R is closed under limits 8. The intersection L ∩ R is the class of isomorphisms 9. 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 if the following conditions hold: 1. The class L contains the isomorphisms and is closed under composition 2. 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 condition that a saturated class be stable by cobase change. The following lemma, however, shows that this additional condition 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 contains the isomorphisms and is closed under pushouts in E[1]. Then the class Q is closed under cobase change. Proof. If the following square is a pushout in the category E

f A A′

u u′ g B B′

then the following square is also a pushout in the category E[1],

(f,f) 1A 1A′

′ (1A,u) (1A′ ,u ) (f,g) u u′

′ We have 1A, 1A′ ∈ Q, since the class Q contains the isomorphisms. Thus, u ∈ Q ⇒ u ∈ Q, since the subcategory Q is closed under pushouts.

18 Lemma 3.1.11. If A is a class of maps in a cocomplete category E, then the class ⊥A is saturated. The left class 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 the first, 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 in a smallest saturated class Σs. We shall say that Σs is the saturated class generated by Σ. If L ⊆ E is a saturated class in E then we say L is of small generation if L = Σs for a set of maps Σ ⊆ E. Lemma 3.1.13. If Σ is a class of maps in a cocomplete category E, then (Σs)⊥ = Σ⊥. Proof. We have (Σs)⊥ ⊆ Σ⊥, since Σ ⊆ Σs. Conversely, the class ⊥(Σ⊥) is saturated by Lemma 3.1.11. Thus Σs ⊆ ⊥(Σ⊥), since Σ ⊆ ⊥(Σ⊥). It follows that Σs ⊥ Σ⊥ and hence that Σ⊥ ⊆ (Σs)⊥. Proposition 3.1.14. Let Σ be a set of maps in a presentable category E. If Σs ⊆ E is the saturated class generated by Σ, then the pair (Σs, Σ⊥) is a factorization system. Proof. The saturated class Σs is of small generation, since Σ is a set of maps. Hence the pair (Σs, (Σs)⊥) is a factorization system by [HTT, Proposition 5.5.5.7.]. We have (Σs)⊥ = Σ⊥ by Lemma 3.1.13. Hence the pair (Σs, Σ⊥) = (Σs, (Σs)⊥) is a factorization system. We shall say that the factorization system (Σs, Σ⊥) of Proposition 3.1.14 is generated by the set of maps Σ ⊆ E.

3.1.1 Factorization systems and reflective subcategories If S = (L, R) is a factorization system in a category E, then for every object B ∈ E, let us denote by R(B) the full subcategory of E~B whose objects are the maps p ∶ E → B in R. For every X = (X,f) ∈ E~B application of the functor S produces an object (S(f), R(f)) ∈ R(B).

L(f) X S(f)

R(f) f B Moreover, in view of the commutativity of the above triangle, the map L(f) ∶ X → S(f) determines a morphism 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 every object (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 the reflection 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

Xh Y

f g B there exists a unique map h′ ∶ S(f) → Y such that gh′ = R(f) and h′L(f) = h.

L(f) ′ X S(f) h Y

R(f) f g

B

19 Such a map h′ may be obtained as the unique diagonal filler of the following square:

Xh Y L(f) g R(f) S(f) B

It remains to prove the last statement of the proposition. If a map h ∶ (X,f) → (Y,g) in E~B exhibits the reflection of (X,f) into the subcategory R(B), let us show that h ∈ L and g ∈ R. We saw above that the map L(f) ∶ (X,f) → (YXYS, R(f)) in E~B exhibits the reflection of (X,f) into the subcategory R(B). It follows that the map h ∶ (X,f) → (Y,g) of E~B is isomorphic to the map L(f) ∶ (X,f) → (YXYS, R(f)) and hence that h ∈ L. Similarly, g ∈ R. The special case where B = 1 ∈ E is the terminal object will occur frequently. For every object X ∈ E, we shall denote the unique map X → 1 by pX ∶ X → 1. We shall write YXYS for the object S(pX ) and write L(X) for the map L(pX ) ∶ X → YXYS.

L(X) X YXYS

pX 1 Proposition 3.1.16. Let S = (L, R) be a factorization system in a category E with a terminal object 1. Then the subcategory R(1) is reflective. Moreover,

1. For every object X ∈ E the map L(X) ∶ X → YXYS exhibits the reflection of the object X into the subcategory R(1) 2. A map r ∶ X → X′ in E exhibits the reflection of X into the subcategory R(1) if and only if X′ ∈ R(1) and r ∈ L 3. An object X ∈ E belongs to R(1) if and only it is L-local 4. Every map in R(1) belongs to R

5. The functor Y − YS ∶ E → R(1) inverts every map in L

Proof. 1. Follows from Proposition 3.1.15 since L(X) ∶= L(pX ) ∶ X → YXYS. 2. Follows from Proposition 3.1.15.

⊥ 3. By definition, an object X ∈ E belongs to R(1) if and only if the map pX ∶ X → 1 belongs to R = L . ⊥ But the map pX ∶ X → 1 belongs to 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(1) then the maps pX ∶ X → 1 and pY ∶ Y → 1 belongs to R. It follows that f ∈ R, since pY f = pX 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 YfYS ∶ YXYS → YY YS is invertible. The following square commutes by naturality L(X) X YXYS

f YfYS L(Y ) Y YY YS

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 YfYSL(X) = L(Y )f ∈ L as well. It follows that the map

20 YfYS belongs to L, since the class L is right cancellable by Proposition 3.1.7. But we have YfYS ∈ R by (4). Thus, YfYS ∈ L ∩ R and this shows that YfYS 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(1) = Loc(E, Σ) and the category R(1) is presentable. Proof. An object X ∈ E belongs to R(1) if and only if the map X → 1 belongs to R. But R = Σ⊥ by Proposition 3.1.14. Hence X → 1 belongs to 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(1) = Loc(E, Σ). The category Loc(E, Σ) is presentable, since the category E is presentable and Σ is a set.

If u ∶ A → B is a map in a category E, then the pushforward functor u! ∶ E~A → E~B is defined by putting u!(X,f) = (X,uf) for every map f ∶ X → A. If S = (L, R) is a factorization system in E, then the pushforward 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. L(uf) X S(uf)

f R(uf) Au B It follows from the definition that the following square commutes

u! E~A E~B

u R(A) ♯ R(B)

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,p1) 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 equivalence

≃ ⋆ MapB(u!(X), Y ) Ð→ MapA(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) ∶ MapB (u♯(X), Y ) Ð→ MapB(u!(X), Y )

⋆ is invertible. The adjunction u♯ ⊣ u follows.

21 3.2 Cartesian Factorization Systems The theory of cartesian factorization systems in 1-categories is developped in [RT], [My]. For cartesian factorization systems in ∞-categories, see [Lan]. Recall that a class of maps A in a category E is said to have the 3-for-2 property if every commutative triangle Y f g

XZh having two sides in A in fact has all three sides in A. Equivalently, A is closed under composition and both left 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 factorization system (L~B, R~B) in the category E~B for every object B ∈ E.

3.2.1 Cartesian Factorization Systems and Grothendieck Fibrations Proposition 3.2.3. 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 following diagram Map(B,X) −○u Map(A, X)

f○− f○−

Map(B, Y ) −○u Map(A, Y )

Map(FB,FX) Map(FA,FX) −○F (u) F (f)○− F (f)○− −○F (u) Map(FB,FY ) Map(F A, F Y ) where the vertical arrows of the cube are the maps induced by the functor F . The left hand face and right hand faces are cartesian since the map f is F -cartesian by hypothesis. On the other hand, the bottom face of the cube is trivially cartesian, since the map F (u) is invertible. It follows that the top face of the cube is 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 shows that 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, since a map u ∈ E belongs to L if and only if the map F (u) is invertible. Hence it only remains then to check that the 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 → FB is invertible, since u belongs to L. We now consider the composite F (u)−1 ○ F (q) ∶ FB′ → 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)−1 ○ 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 that

22 q ○ u′ = u ○ p and F (u′) = i. Moreover, clearly u′ ∈ L since F (u) = i and i is invertible by assumption. We claim that the square p A′ A

u′ u q B′ B is cartesian, which will complete the proof. For this, it suffices to show that the top face of the following cube is cartesian for every object K ∈ E.

p○− Map(K, A′) Map(K, A)

u○− u′○− q○− Map(K,B′) Map(K,B)

Map(FK,FA′) Map(FK,FA)

F (u)○− F (u′)○− Map(FK,FB′) Map(FK,FB) 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 follows that 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 Lφ of maps inverted by φ is the left class of a cartesian factorization system (Lφ, Rφ). If η ∶ Id → φ is the unit of the reflector, then a map f ∶ X → Y in E belongs to Rφ if and only if the following square is cartesian η(X) X φ(X)

f φ(f) Y φ(Y ). η(Y )

′ Moreover, Rφ(1) = E . Proof. The functor φ ∶ E → E′ ⊆ E is a Grothendieck fibration by Proposition 2.4.5 and hence we obtain a cartesian factorization system from Proposition 3.2.3. It remains, therefore, only to justify the description of the φ-horizontal maps (the identification of Rφ(1) we leave to the reader). By definition, a map f ∶ X → Y is φ-horizontal if and only if the square

f○− Map(K,X) Map(K, Y )

φ φ φ(f)○− Map(φK, φX) Map(φK, φY )

23 is cartesian for every K ∈ E. Since φ is a reflector, on the other hand, the above square factors as

f○− Map(K,X) Map(K, Y )

−○ηX −○ηY φ(f)○− Map(K,φX) Map(K, φY )

≃ ≃ φ(f)○− Map(φK, φX) Map(φK, φY )

with the lower vertical maps equivalences as indictated. It follows that the outer square is cartesian if and only 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 the functor 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 a cartesian square f X A v u (9) g Y B is a cartesian square. If L ⊆ E is the class of F -vertical maps and R ⊆ E is the class of F -horizontal maps, then the 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

p q X E A v w u (10) L(g) R(g) Y S(g) B

The map w ∶ E → S(g) belongs to R, since the map u ∶ A → B belongs to R and the class R is closed under 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 and right squares are. From this it follows that p ∈ L since we have seen that the factorization system (L, R) is cartesian and we have both L(g) ∈ L and w ∈ R. But then both F (p) and F (L(g)) are invertible so that the image of the left square by F is trivially cartesian.

3.2.2 Fibering subcategories We have seen above that factorization systems give rise to reflective subcategories. When the factorization system 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 a Grothendieck fibration. Example 3.2.7. Let E be a category with finite limits and let φ ∶ E → E′ ⊆ E be a left-exact reflector. Then the subcategory E′ ⊆ E is fibering.

24 Lemma 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 change of a reflecting map r ∶ Y → Y ′ along any map u ∶ X′ → Y ′ in E′ exists and is itself a reflecting map. Up to isomorphism, 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

p X Y η(X) η(Y ) R(p) RX RY

Hence the map η(X) ∶ X → RX is a base change of the map η(Y ) along the map R(p). It follows that the map i ○ η(X) ∶ X → X′ is a base change of the map η(Y ) along the map u = R(p) ○ i−1 ∶ 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′. E E′ (⇐) Conversely, by Definition 2.4.1, we are to show that the functor R~A ∶ ~A → ~RA induced by the functor R is a reflector for every object A ∈ E. First of all, we observe that this functor admits a right adjoint A E′ E E′ R ∶ ~RA → ~A. Indeed, given some (Y,g) ∈ ~RA, the map η(A) admits a base change along g ∶ Y → RA ′ A since the subcategory E is fibering, and we may therefore define R (Y,g) = (A ×RA Y,p1).

p2 A ×RA Y Y

p1 g A RA η(A)

But since p2 is the base change of a reflecting map by construction, it is also a reflecting map and hence we A have R(A ×RA Y ) = Y . But then R~AR (Y,g) = (Y,g) which shows that the counit of the adjunction is an isomorphism. Consequently, R~A is a reflector as required. Proposition 3.2.9. Let E be a category with a terminal object 1. If (L, R) is a cartesian factorization system in E, then the subcategory R(1) is fibering. Proof. Recall from Proposition 3.1.16 that a map y ∶ Y → Y ′ in E reflects the object Y into R(1) if and only Y ′ ∈ R(1) and y ∈ L. Every map v ∶ X′ → Y ′ in R(1) belongs to R by Proposition 3.1.16. Hence the base change of a reflecting map y ∶ Y → Y ′ along any map v ∶ X′ → Y ′ in R(1) exists and belongs to L, since the factorization system (L, R) is cartesian.

Xu Y x y (11)

′ ′ X v Y

The map x ∶ X → X′ in the cartesian square (11) reflects the object X into R(1), since X′ ∈ R(1) and x ∈ L. This shows that the subcategory R(1) is fibering.

3.2.3 A characterization of cartesian factorization systems As described in the introduction, the following characterization of cartesian factorization systems in terms of their associated base change functors will be a key tool in the proof of Theorem 3.6.8 below.

25 Proposition 3.2.10. A factorization system (L, R) in a category with finite limits E is cartesian if and only if 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))

L(uf) X S(uf)

f R(uf) (12) Au B

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) of the following square L(uf) X S(uf)

f R(uf)

Au B and hence fits into the following diagram where we put η(X,f) ∶= (f, L(uf)).

η(X,f) p2 X A ×B S(uf) S(uf)

p1 R(uf) f

Au B

Now, as the map p2 is a base change of u along R(uf) by construction, and the factorization system (L, R) is cartesian, we have p2 ∈ L. It follows then that η(X,f) ∈ L, since the class L has the 3-for-2 property and p2η(X,f) = L(uf) belongs to L. The rest of the argument follows from Proposition 3.1.7. Indeed, we have p1 ∈ R since it is a base change of a map in R. But then η(X,f) ∈ R, since p1η(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,p1) is defined by the following cartesian square

p2 A ×B Y Y

p1 g

Au B

Hence we have p2 ∈ L, since u ∈ L and g ∈ R and (L, R) is cartesian. It follows that L(up1) = p2 and ⋆ R(up1) = g. Thus, u♯(A ×B Y,p1) = (Y,g) which shows that the counit of the adjunction u♯ ⊣ u is invertible as 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 equivalence of categories, so is its left adjoint u♯ ∶ R(A) → R(B). Now given some u and g as above, it follows from the essential surjectivity of u♯ that there exists (X,f) ∈ R(A) such that u♯(X,f) = (Y,g). That is, we have a commutative square Xh Y f g

Au B

26 with h ∶= L(uf) and g = R(uf). Futhermore, as pointed out in the first part of the proof, the unit of the ⋆ adjunction u♯ ⊣ u is exactly the cartesian gap map of this square. It follows then that since this adjunction is 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 show that 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

L(u) A S(u)

vu vR(u) C

from which we see that the composite vR(u) ∶ S(u) → C belongs to L, since the class L has the right cancellation property by Proposition 3.1.7 and we have vu ∈ L and L(u) ∈ L. It follows that v♯(S(u), R(u)) = (C, 1C ). 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, 1C ) 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 an equivalence, a condition which makes no mention of finite limits. As we will not have occasion to use the result at this level of generality, and as the proof is considerably longer, we omit it here.

3.3 Modalities and Acyclic Classes 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 a map 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 (Connn(E), Trunn(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 cartesian factorization system (Lφ, Rφ) 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 fiberwise left 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 that we have u ñ f for every u ∈ A and f ∈ B. We shall denote by Añ (resp. ñA) the class of maps in E that are fiberwise right orthogonal (resp. fiberwise left orthogonal) to every map in A. We have

A ⊆ ñB ⇔ A ñ B ⇔ Añ ⊇ B

27 Lemma 3.3.4. Let E be a category with finite limits. Then a factorization system (L, R) in E is a modality if and only if L ñ R, in which case R = Lñ and L = ñR. 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 (LK, RK) on the category of diagrams [K, E] for any small category K. In each case, the latter is a modality as soon as the original factorization system (L, R) is, a fact which is easily checked. 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 (LK, RK) 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 and descends along any surjective family of maps. The following proposition says that modalities provide us with a 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 p1 in the following pullback square belongs to L,

p2 A ×B A A

p1 u

Au B

since p1∆(u) = 1A and the class L has the right cancellation property by Proposition 3.1.7. Thus, u ∈ L by descent (Proposition 3.3.7), since u ∶ A → B is surjective.

3.3.1 Acyclic Classes Definition 3.3.9. Let E be a topos. We shall say that a class of maps L ⊆ E is acyclic if the following conditions hold: 1. The class L contains the isomorphisms and is closed under composition 2. The class L is closed under colimits 3. The class L is closed under base changes By definition, a class L is acyclic if and only if it is saturated and closed under base changes. An acyclic class 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 of maps 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 −1(y) → 1 belongs to L for every y ∶ 1 → Y . That is, 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.

28 Proof. 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 of the class L. It remains to show that the subcategory L′ ⊆ E[1] is closed under colimits. If K is a small category, then a diagram K → E[1] is the same thing as a natural transformation u ∶ A → B between two diagrams A, B ∶ K → E. If the map u(k) ∶ A(k) → B(k) belongs to L′ for every object k ∈ K, let us show that the map colim(u) ∶ colim A → colim B belongs to L′. We have the following commutative square of natural transformations γ(A) A δ colim(A)

u δ colim(u) (13) γ(B) B δ colim(B) 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 closed under colimits. It remains to show that every base change colim(u) is in L. For this, we have to show that in every pullback square Dw C g colim(u) colim(A) colim(B)

the map w belongs to L. By pulling back the square (13) along the map δ(g) ∶ δC → δ colim(B) we obtain the following cube of natural transformations, where every vertical face is a cartesian square.

′ A′ u B′ γ(A′) γ(B′)

δ(w) δD δC

u A B δ(g) γ(A) γ(B)

δ colim(A) δ colim(B) δ colim(u)

The natural transformations γ(A′) ∶ A′ → δD and γ(B′) ∶ B′ → δC are colimit cones, since colimit are universal in a topos. It follows that w = colim(u′). The map u′(k) ∶ A′(k) → B′(k) is a base change of the map 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[1] is closed under colimits. This shows that colim(u) ∈ L′. Hence the subcategory L′ ⊆ E[1] 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 an acylic class, let us show that L′′ ⊆ L′. If u ∶ A → B is a map in L′′, then L′′ contains every base change u′ of the 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 Proposition 3.3.13. If A is a class of maps in a topos E, then the class ñA is acyclic. Proof. The class ⊥A is saturated by Lemma 3.1.11. By Lemma 3.3.12 the saturated class ⊥A contains a largest 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)′ = ñA. Hence the class ñA 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 acyclic class Σa ⊆ E. We shall say that Σa is the acyclic class generated by Σ. We shall say that an acyclic class L ⊆ E is of small generation if L = Σa for a set of maps Σ ⊆ E. Lemma 3.3.15. If Σ is a class of maps in a topos E, then (Σa)ñ = Σñ. Proof. We have (Σa)ñ ⊆ Σñ, since Σ ⊆ Σa. Conversely, the class ñ(Σñ) is acyclic by Proposition 3.3.13. Thus Σa ⊆ ñ(Σñ), since Σ ⊆ ñ(Σñ). It follows that Σa ñ Σñ and hence that Σñ ⊆ (Σa)ñ.

♭ [1] 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,

p2 X ×B A A

u♭(f) u f X B

♭ [1] 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 = 1. [1] Consider the diagonal functor δ ∶ E → E (which takes an object X ∈ E to the map 1X ∶ X → X). If ♭ ♭ [1] pA ∶ A → 1, then for every X ∈ E, we have (pA) (X) = δ(X) × pA. Hence the functor (pA) ∶ E → E is the [1] [1] [1] composite of the functor δ ∶ E → E followed by the functor (−) × pA ∶ E → E . The functor δ is left [1] [1] [1] adjoint to the domain functor dom ∶ E → E. Moreover, the functor (−) × pA ∶ E → E has a right adjoint [1] [1] [1] [pA, −] ∶ E → E since the category E is cartesian closed. It follows that the functor g ↦ dom([pA,g]) ♭ is right adjoint to the functor pA. This proves the lemma in the case B = 1. In the general case, observe first ♭ [1] that the functor (A, u) ∶ E~B → (E~B ) defined by the object (A, u) of E~B has a right adjoint by the first [1] [1] ♭ ♭ part of the proof. Moreover, if FB ∶ (E~B) → E is the forgetful functor, then we have u = FB ○ (A, u) . ⋆ [1] [1] The forgetful functor FB 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 = FB ○ (A, u) has a right 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-base change, then Σa = Σs and Σñ = Σ⊥. Proof. Obviously, Σs ⊆ Σa. Let us show that Σa ⊆ Σs. The saturated class L ∶= Σs contains a largest acyclic class 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 is ♭ [1] strongly 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 be the full subcategory of E[1] spanned by the maps in L. The subcategory L ⊆ E[1] is closed under colimits, since the class L is saturated and a saturated class is closed under colimits. Hence the full subcategory ♭ −1 ♭ ♭ (u ) (L) ⊆ E~B is closed under colimits, since the functor u is cocontinuous. We have u (G, g) ∈ L for every object (G, g) ∈ E~B, since every G-base change of the map u ∶ A → B belongs to L. In other words,

30 ♭ −1 ♭ −1 G~B ⊆ (u ) (L). Thus, E~B ⊆ (u ) (L), since the class G~B strongly generates E~B by Lemma 2.2.5. This shows 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 that Σ ⊆ L′. It follows that Σa ⊆ L′, since the class L′ is acyclic. Thus, Σa ⊆ Σs, since Σa ⊆ L′ ⊆ L = Σs. The equality Σa = Σs is proved. By Lemma 3.1.13 we have (Σs)⊥ = Σ⊥ and by Lemma 3.3.15 we have (Σa)ñ = Σñ. Thus, Σñ = Σ⊥. Theorem 3.3.19. Let Σ be a set of maps in a topos E. If Σa ⊆ E is the acyclic class generated by Σ, then the pair (Σa, Σñ) is a modality. Proof. The topos E admits a set of generators G since it is presentable. Let us denote by Λ the closure of Σ over G-base changes (Definition 3.3.17). The class Λ is a set, since the classes G and Σ are sets. Hence the pair (Λs, Λ⊥) is a factorization system by Proposition 3.1.14, since Λ is a set. But Λa = Λs and Λñ = Λ⊥ by Lemma 3.3.18, since Λ is closed under G-base change. Hence the pair (Λa, Λñ) is a factorization system, since the pair (Λs, Λ⊥) is a factorization system. We have Λa ñ (Λa)ñ ⊆ Λñ, since Λ ⊆ Λa. Hence the factorization system (Λa, Λñ) is a modality by Lemma 3.3.4. Let us show that (Σa, Σñ) = (Λa, Λñ). We have Σa ⊆ Λa, since Σ ⊆ Λ. Moreover, we have Λ ⊆ Σa, since Σa is closed under base change; thus, Λa ⊆ Σa. This shows that Σa = Λa. Let us show that Σñ = Λñ. We have Σñ = (Σa)ñ by Lemma 3.3.15. But we have (Σa)ñ = (Λa)ñ, since Σa = Λa. Thus, Σñ = (Λa)ñ = Λñ, since the pair (Λa, Λñ) is a modality. This shows that (Σa, Σñ) = (Λa, Λñ). Hence the pair (Σa, Σñ) is a modality. Definition 3.3.20. We shall say that the modality (Σa, Σñ) of Theorem 3.3.19 is generated by the set of maps 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 it is 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 subcategory of E spanned by Σ-modal objects. It follows from Lemma 3.3.15 that we have Mod(E, Σ) = Mod(E, Σa). Lemma 3.3.22. Let (L, R) be a modality generated by a set of maps Σ in a topos E. Then R(1) = Mod(E, Σ) and the category R(1) is presentable. Proof. By Definition 3.3.20, we have (L, R) = (Σa, Σñ). The topos E admits a set of generators G, since a topos is presentable. Let us denote by Λ the closure of Σ over G-base changes (Definition 3.3.17). The class Λ is a set, since the classes G and Σ are sets. We saw in the proof of Theorem 3.3.19 that (Σa, Σñ) = (Λs, Λ⊥). Hence the factorization system (L, R) is generated by the set Λ ⊆ E. It follows from Lemma 3.1.17 that R(1) = Loc(E, Λ) and that the category R(1) is presentable. By Lemma 3.1.2, an object X belongs to ⊥ ⊥ ñ R(1) = Loc(E, Λ) if and only if the map pX ∶ X → 1 belongs to Λ . But we have Λ = Σ . Thus, an object ñ X belong the R(1) if and only if the map pX ∶ X → 1 belongs to Σ . Thus, an object X belongs to R(1) if and only if it is Σ-local.

3.4 Left-Exact Modalities Among the modalities (L, R) in a topos E, we may distinguish those for which the subcategory R(1) is again a topos. As we have seen above, this will be the case as soon as the associated reflector preserves finite limits. 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[1]. Remark 3.4.2. It can be shown that a factorisation system (L, R) in a category with finite limits is a left-exact modality if and only if the functor S ∶ E[1] → E preserves finite limits.

31 Example 3.4.3. Let E be a category with finite limits and let φ ∶ E → E′ ⊆ E be a left-exact reflector. We saw in Proposition 3.2.4 that the class of maps Lφ ⊆ E inverted by the functor φ is the left class of a cartesian factorization system Sφ ∶= (Lφ, Rφ). We shall see in Proposition 3.4.8 that the factorization system Sφ is in fact 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 are given in [RSS]). We will content ourselves with those given by Theorem 3.4.7, which are most relevant to the current setup. The proof of the theorem will require a couple of preparatory lemmas concerning classes of 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 full subcategory Q ⊆ E[1] whose objects are the maps in Q contains the isomorphisms of E (as objects) and is closed 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 the category E[1] is cartesian. (1A,v) u vu

(u,1B ) (u,1C )

1B v (1B ,v)

We have 1B ∈ Q, since Q contains the isomorphisms. Thus, if v and vu belongs to Q, then u ∈ Q since the subcategory Q ⊆ E[1] 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 the isomorphisms and is closed under base changes. Write L(B) for the full subcategory of E~B whose objects are 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[1]. It suffices therefore to show that the subcategory L is closed under fiber products. Consider, therefore, a cartesian square in the category E[1] g z v f q (14) p u w If u,v,w ∈ L, let us show that w ∈ L. The codomain functor cod ∶ E[1] → E is a Grothendieck fibration, since the category E has finite limits. Hence the category E[1] 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 pulling back the factorisation p = H(p)V(p) ∶ u → T (p) → v along the map q ∶ v → w, the cartesian square (14) is decomposed into two cartesian squares:

z v′ v

q (15) V(p) H(p) u T (p) w

The morphism of H(p) ∶ T (p) → p of E[1] is a cartesian square in E. Hence the map T (p) of E is a base change of the map w. Thus, T (p) ∈ L, since w ∈ L and the class L is closed under base changes. The

32 morphism v′ → v in the diagram (15) belongs to the class H, since the right class of a factorisation system is 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 lemma in the case of the left hand square of the diagram (15), since u,T (p), v′ ∈ L. Equivalently, we may suppose that 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 into two cartesian squares: z v V(q) u′ T (q) (16)

H(q) p u w The argument above can be applied to show that u′,T (p), v ∈ L. Hence it suffices to prove the lemma in the case of the top square of the diagram (16), But the morphism u′ → T (q) in the diagram (16) is a base change of 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 lemma for the original square (14) with the extra hypothesis that the morphisms u → w and v → w belongs to V. In which case the four maps cod(p), cod(q), cod(f) and cod(g) of the category E are invertible. Hence we may suppose that cod(z) = cod(u) = cod(v) = cod(w) = B for some object B ∈ E and that the square (14) belongs [1] to 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 hence that L is closed under fiber products. The class L contains the terminal object of E[1], since this terminal object is an isomrphism in E and L contains the isomorphisms. This completes the proof that L is closed under 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-exact 2. The class L has the 3-for-2 property

3. 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 cartesian

5. The reflector Y − YB ∶ 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. Hence it suffices to show that the class L is left cancellable. Since the modality (L, R) is left-exact, the class L is closed under pullbacks in E[1] 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 p1 ∶ A ×B A → A, since the latter is a base change of the former. Thus, ∆(u) ∶ A → A ×B A belongs to L by 3-for-2, since p1∆(u) = 1A 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. We shall use the following diagram of cartesian squares:

(u,1A) p2 A B ×C A A

a u ( ) B×C u (b) u

∆(v) p2 B B ×C B B

p1 (c) v

Bv C

33 The square (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, 1A) ∶ A → B ×C B belongs to L, since the square (a) is cartesian and the class L is closed under base change. Similarly, the projection p1 ∶ 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 = p1(u, 1A) 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 stable by 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 Y − Y ∶ E → R(1) preserves finite limits. We shall prove that the functor Y − Y takes a cartesian square in the category E

f X A v u (17) g Y B

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

p q X E A v w u (18) L(g) R(g) Y S(g) B

The functor Y − Y ∶ E → R(1) 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 Y − Y takes the right hand square of diagram (18) to a cartesian square. Let us show that the functor Y−Y takes the left hand square to a cartesian square. The map p ∶ X → E belongs to L, since the map L(g) belongs to L and the class L is closed under base change. Hence the maps YpY and YL(g)Y are invertible by Proposition 3.1.16. It follows that the functor Y−Y ∶ E → R(1) takes the left hand square of diagram (18) to a cartesian square. This shows that the functor Y − Y ∶ E → R(1) takes the square (17) to a cartesian square. Hence the functor Y − Y ∶ E → R(1) preserves finite limits, since it preserves terminal objects. More generally, let us show that the reflector Y − YB ∶ E~B → R(B) preserves finite limits for every object B ∈ E. The factorization system S~B ∶= (L~B , R~B) is a modality by Lemma 3.3.5, since (L, R) is a modality by hypothesis. Moreover, the factorization system (L~B , R~B) is cartesian by Lemma 3.2.2, since the factorization system (L, R) is cartesian by hypothesis. It then follows by the first part of the argument that the reflector Y − YB ∶ 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 show that 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) if and only if the object Y(A, u)YB = (S(u), R(u)) of the category R(B) is terminal. But the full subcategory of terminal 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 Y − YB ∶ 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 the ′ class Lφ of maps inverted by φ is the left class of a left-exact modality Sφ ∶= (Lφ, Rφ) and Rφ(1) = E . If η ∶ Id → φ is the unit of the reflector, then a map f ∶ X → Y in E belongs to Rφ if and only if the following

34 square is cartesian η(X) X φ(X)

f φ(f) (19) Y φ(Y ). η(Y )

Conversely, if S = (L, R) is a left-exact modality in E and φ is the reflector Y − YS ∶ E → R(1), then S = Sφ. ′ The construction φ ↦ Sφ 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 class of φ-vertical maps and R ⊆ E is the class of φ-cartesian maps, then the pair (L, R) is a cartesian factorization ′ system by Proposition 3.2.3. By definition, we have L = Lφ. The functor φ ∶ E → E is left adjoint to the inclusion functor E′ ⊆ E. It then follows by Lemma 2.4.9 that a map f ∶ X → Y in E is φ-cartesian if and only ′ if the square (19) is cartesian. Let us show that Rφ(1) = E . For every object X ∈ E, the map pX ∶ X → 1 belongs to Rφ if and only if the following square is cartesian

η(X) X φ(X)

pX φ(pX ) 1 1 ′ ′ if and only if the map η(X) is invertible, if and only if X ∈ E . Thus, Rφ(1) = E . Conversely, if S = (L, R) is a left-exact modality in E and φ is the reflector Y − YS ∶ E → R(1), let us show that S = Sφ. The modality (L, R) is a cartesian factorization system by Theorem 3.4.7. It then follows from ⊥ ⊥ Proposition 3.2.9 that L = Lφ. Thus, (L, R) = (Lφ, Rφ) since R = L and Rφ = Lφ by Proposition 3.1.7. ′ Hence construction φ ↦ Sφ is a one to one correpondance between the left-exact reflectors φ ∶ E → E ⊆ E and the left-exact modalities S = (L, R) in E.

3.5 Fiberwise R-equivalences In this section, we give a criterion under which the modality generated by a set of maps Σ is left-exact. The theorem 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 shall say 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 the following 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 the ⋆ ⋆ functor u ∶ R(B) → R(A) if fully faithful if and only if the counit of the adjunction u♯ ⊣ u is invertible. By ⋆ definition, for every map g ∶ Y → B ∈ R we have u (Y,g) = (A ×B Y,p1).

p2 A ×B Y Y

p1 g Au B

35 ⋆ If u belongs to L, let us show that the canonical map u♯u (Y,g) → (Y,g) is invertible. The map p2 belongs to L, since u belongs to L and L is closed under base change. But g belongs to R by hypothesis. Thus, L(up1) = p2 and R(up1) = g. This shows that u♯(A ×B Y,p1) = (Y,g) and hence that the canonical ⋆ ⋆ map u♯u (Y,g) → (Y,g) is the identity. We have proved that the counit of the adjunction u♯ ⊣ u is ⋆ invertible. Conversely, if the the counit u♯u (Y,g) → (Y,g) is invertible for every (Y,g) ∈ R(B), then it is ⋆ in particular in the special case where (Y,g) = (B, 1B). In this case, we have u (Y,g) = (A, 1A) and hence u♯(A, 1A) = (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. To this end, consider the following cubical diagram:

∗L ∗ u (uf) ∗ u (X) u (u♯(X))

L(uf) ∗ u (f) X u♯(X)

u∗R(uf) f

p2 R A ×B A A (uf)

u p1

Au B

f L(uf) Viewing the maps A ←Ð X ÐÐÐ→ u♯(X) as a span in the category E~B, we obtain the back face of the cube ∗ by applying the functor u ∶ E~B → E~A. It follows immediately that the left, right, top and bottom faces of 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

∗ ∗ (p2)♯(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 we ∗ have a canonical isomorphism ∆(u)♯(∆(u) (Y )) = Y for any Y → A ×B A. We now simply calculate:

X = (p2)♯(∆(u)♯(X)) since p2 ○ ∆(u) = 1A ∗ ∗ = (p2)♯(∆(u)♯((∆(u)) ((p1) (X)))) since p1 ○ ∆(u) = 1A ∗ ∗ = (p2)♯((p1) (X)) since ∆(u) is fully faithful ∗ = (p2)♯(u (X)) since the left face is cartesian ∗ = u (u♯(X)) from above This 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 the ⋆ functor u ∶ R(B) → R(A) is fully faithful. Let us show that ∆(u) ∈ L. The projection p1 ∶ A ×B A → B is a base change of the map u ∶ A → B, since the following square is cartesian.

p2 A ×B A A

p1 u

Au B

36 Hence the map p1 ∶ A ×B A → B is an R-equivalence, since u is a fiberwise R-equivalence. Hence the functor ⋆ ⋆ ⋆ p1 ∶ R(A) → R(A×B A) is an equivalence of categories. But we have ∆(u) p1 = id, since p1∆(u) = 1A. Hence ⋆ ⋆ the functor ∆(u) ∶ R(A ×B A) → R(A) is an equivalence of categories, since the functor p1 is an equivalence of 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. We shall 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 remains to 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. The fact that it is closed under isomorphisms, composition and base change is immediate from the definition, so that 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

Xv Y p q (20) Au B be a cartesian square in the category [K, E]. Suppose that the horizontal maps of the square belong to the class LK and that the vertical maps belong to RK. If the natural transformation p ∶ X → A is cartesian, then so 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(r) X(i) X(j)

v(j) v(i) Y (r) p(i) Y (i) Y (j) p(j) (21) q(i) A(r) A(i) A(j) q(j)

u(j) u(i) B(r) B(i) B(j)

The left and the right faces of the cube are cartesian, since the square (20) is cartesian. Let us show that the 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 that the composite of the left hand face with the front face is cartesian, since the cube commutes. Hence the boundary square of the following diagram is cartesian.

v(i) Y (r) X(i) Y (i) Y (j)

p(i) q(i) q(j) (22) u(i) B(r) A(i) B(i) B(j)

37 Let us show that right hand face of the diagram is cartesian. For this, it suffices to show that its cartesian gap map g ∶ Y (i) → B(i) ×B(j) Y (j) is invertible. The vertical maps of the diagram (22) belong to R, K since the vertical maps of the square (20) belong to R . Hence the projection p1 ∶ 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. Hence the 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 LK. 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 prove that g is invertible by showing that the map u(i)⋆(g) is invertible. But u(i)⋆(g) is the cartesian gap map of the boundary square of the diagram (22), since the left hand square of this diagram is cartesian. But the cartesian gap map of the boundary square is invertible, since the boundary square is cartesian. It follows that u(i)⋆(g) is invertible and hence g is invertible. We have proved that the front face of the cube (21) is cartesian. 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 is acyclic. 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 the subcategory W ⊆ E[1] is closed under colimits. If K is a small category, then a diagram K → E[1] is the same 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

γ(A) A δ colim(A)

u δ colim(u) (23) γ(B) B δ colim(B) 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[1] 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 p q (24) colim(u) colim(A) colim(B)

For this we shall show that the square (24) is the colimit over k ∈ K of a square of natural transformations in the category [K, E], X′ v Y ′

p′ q′ Au B The whole procedure depends on a step by step construction of the following cube, where k is a variable

38 object of K: v(k) X′(k) Y ′(k)

γ(X′)(k) γ(Y ′)(k)

colim(v) p′(k) X Y

q′(k) p u(k) A(k) B(k) q

γ(A)(k) γ(B)(k) colim(A) colim(B) colim(u) The bottom face of the cube is obtained from the square (23). The left hand face of the cube is obtained by pulling back the map p ∶ X → colim(A) along the canonical map γ(A)(k) ∶ A(k) → colim(A),

γ(X′)(k) X′(k) X

p′(k) p (25) γ(A)(k) A(k) colim(A)

This defines a diagram X′ ∶ K → E together with a natural transformation p′ ∶ X′ → A. Let δX ∶ K → E be the 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 (LK, RK) in the category of K-diagrams [K, E] by Lemma 3.3.6. The natural transformation p′ ∶ X′ → A belongs to RK, 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 LK. It follows by Proposition 3.5.2 that the natural transformation u ∶ A → B is a K ′ ′ ′ ′ K ′ ′ K ′ R -equivalence. Let us put (Y , q ) = u♯(X ,p ), v = L (up ) and q = R (up ). The resulting square

X′ v Y ′

p′ q′ Au B is the square (24). The square is cartesian, since u ∶ A → B is a RK-equivalence. The horizontal arrows of square belong to the class LK and the vertical arrows belong to RK. Moreover, the natural transformation q′ ∶ Y ′ → B is cartesian by Lemma 3.5.5, since the natural transformation p′ ∶ X′ → A is cartesian. Hence the following square is cartesian by Lemma 3.2.5, since the colimit functor colim ∶ [K, E] → E is a Grothendieck fibration by Proposition 2.4.12.

colim(v) colim(X′) colim(Y ′)

colim(p′) colim(q′) (26) colim(u) colim(A) colim(B)

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, since

39 the 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:

γ(Y ′)(k) Y ′(k) colim(Y ′)

q′(k) colim(q′) γ(B)(k) B(k) colim(B)

But we have q′(k) ∈ R for every object k ∈ K, since q′ ∈ RK. Thus, q = colim(q′) ∈ R, since the class R is local by Proposition 3.3.7. Hence the object (Y, q) belongs to R(colim(B)). This shows that the functor colim(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. It remains to show that the map colim(u) is a fiberwise R-equivalence. For this, we have to show that in every pullback square Dw C g colim(u) colim(A) colim(B) the map w is an R-equivalence. By pulling back the square (23) along the map δ(g) ∶ δC → δ colim(B) we obtain the following cube of natural transformations, where every vertical face is a cartesian square.

′ A′ u B′ γ(B′) γ(A′) δ(w) δD δC

u A B δ(g) γ(B) γ(A) δ colim(A) δ colim(B) δ colim(u) The natural transformations γ(A′) ∶ A′ → δD and γ(B′) ∶ B′ → δC are colimit cones, since a base change functor in a topos preserves colimits. Thus, w = colim(u′). The map u′(k) ∶ A′(k) → B′(k) is a base change of 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 a fiberwise 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 the modality (L, R) is left-exact. Proof. We again denote by W ⊆ E the class of fiberwise R-equivalences. In view of the fact that we have ∆(Σ) ⊆ L, it follows from Proposition 3.5.2 that we have Σ ⊆ W. As the class W is acyclic by Proposition 3.5.6, we therefore have L ⊆ W. It follows that the functor u⋆ ∶ R(B) → R(A) is an equivalence of categories for every map u ∶ A → B in L and therefore that the factorization system (L, R) is cartesian by Proposition 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 every isomorphism is in L by Proposition 3.1.7.

40 3.6 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 following conditions hold: 1. The class L contains the isomorphisms and is closed under composition 2. The class L is closed under colimits 3. 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 inverted by 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 cancellable by 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 Σc ⊆ E the congruence generated by Σ. We shall say that a congruence L ⊆ E is of small generation if L = Σc for a set of maps Σ ⊆ 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 also closed 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 saturated class L is of small generation by [HTT, Proposition 5.5.4.16.]. Thus, L = Σss for a set of maps Σ ⊆ E. Let us show that L = Σc. The class Σc is strongly saturated, since a congruence is strongly saturated by Lemma 3.6.3. Thus, L = Σss ⊆ Σc, since Σ ⊆ Σc. But the class L is a congruence, since the functor φ ∶ E → E′ is left exact and cocontinuous. Thus Σc ⊆ L, since Σ ⊆ L. This shows that L = Σc. For Σ a set of maps in a topos E, we write

∆∞(Σ) = {∆iu S u ∈ Σ,i ≥ 0} 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 Σc = ∆∞(Σ)a. Proof. Let us put Λ ∶= ∆∞(Σ). The pair (Λa, Λñ) is a modality by Theorem 3.3.19. Furthermore, the modality (Λa, Λñ) is left-exact by Theorem 3.5.7 since we have ∆(Λ) ⊆ Λ ⊆ Λa. Hence the class Λa is a congruence. Thus, Σc ⊆ Λa, since Σ ⊆ Λa. Conversely, we have Λ ⊆ Σc, since the congruence Σc is closed under finite limits. Hence we have Λa ⊆ Σc, since Σc 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 map Map(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 maps in Σ. The following lemma shows that it suffices verify this condition for a set of maps.

41 Lemma 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 be the set of G-base changes of the maps in ∆∞(Σ). Let us show that Λa = ∆∞(Σ)a. We have Λ ⊆ ∆∞(Σ)a, since every map in Λ is a base change of a map in ∆∞(Σ). Thus, Λa ⊆ ∆∞(Σ)a. Conversely, let us show that ∆∞(Σ) ⊆ Λa. The set Λ is closed under G-base changes by transitivity of base changes. Thus, Λ⊥ = Λñ by Lemma 3.3.18. Moreover, the pair (Λa, Λñ) is a modality by Theorem 3.3.19. Hence the class Λa is local by 3.3.7. Let us show that ∆∞(Σ) ⊆ Λa. If u ∶ A → B is a map in ∆∞(Σ), then its codomain B is the colimit of a diagram F ∶ K → E of objects F (k) ∈ G, since G is a set of generators. The collection of inclusions ⋆ ιk ∶ F (k) → B is surjective by Example 2.3.5. The map ιk(u) is the G-base change of the map u ∶ A → B ⋆ along the map ιk ∶ F (k) → B. Hence we have ιk(u) ∈ Λ for every object k ∈ K. It follows that we have u ∈ Λa, since Λ ⊆ Λa and the class Λa is local. Thus, ∆∞(Σ) ⊆ Λa and hence ∆∞(Σ)a ⊆ Λa. The equality Λa = ∆∞(Σ)a is proved. An object X ∈ E is a Σ-sheaf if and only if it is modal with respect to the set of maps ∆∞(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 Λñ = Λ⊥. As an example, let P(K) = [Kop, 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 set of 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 local with 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 the reflector ρ ∶ E → Sh(E, Σ) is left-exact. In particular, Sh(E, Σ) is a topos. Furthermore, the reflector ρ inverts the maps in Σ universally among cocontinuous and left-exact functors. In other words, we may identify the −1 lex reflection ρ ∶ E → Sh(E, Σ) with the cocontinuous and left-exact Σ-localization E → E[Σ ]cc . Proof. Let us put Λ ∶= ∆∞(Σ). By definition, an object X is a Σ-sheaf if and only if it is modal with respect to Λ. In other words, we have Sh(E, Σ) = Mod(E, Λ). Now, the pair (Λa, Λñ) is a modality by Theorem 3.3.19 and therefore the subcategroy Sh(E, Σ) = Mod(E, Λ) = R(1) is reflective by Lemma 3.3.22. Furthermore, the modality (Λa, Λñ) is left-exact by Theorem 3.5.7 since we have ∆(Λ) ⊆ Λ ⊆ Λa. It follows that the reflector ρ = Y − YS ∶ E → R(1) = Sh(E, Σ) is left-exact by Theorem 3.4.7 and hence that the category Sh(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.

ρ E Sh(E, Σ)

ψ φ E′

To see this, note that the class of isomorphisms J ⊆ E′ is acyclic and hence also the class A ∶= φ−1J since the functor φ preserves colimits and finite limits. We have Σ ⊆ A since the functor φ inverts the maps in 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 Λa ⊆ A, since the class A is acyclic. This shows that the functor φ ∶ E → E′ inverts every map in Λa. Hence there exists a unique cocontinuous functor ψ ∶ Loc(E, Λa) → 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 left exact 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,

42 however, provide us with concrete methods of presenting topoi by means of generators and relations, and thus lead us to propose a modification of the notion of higher site. Specifically, we lift the requirement that the localization generated by Σ be left exact, since the definition of Σ-sheaf provides us with a means of recognizing the local objects for the left-exact localization which Σ generates. Our notion of site also extends that 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)-base changes 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 = Σc for a set of maps Σ ⊆ P(K). But we have Σc = ∆∞(Σ)a by Proposition 3.6.5. Thus, a presheaf X ∈ P(K) is local with respect to 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).

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