Kato-Nakayama spaces, infinite root stacks, and the profinite homotopy type of log schemes
David Carchedi, Sarah Scherotzke, Nicolò Sibilla, Mattia Talpo
aa r X i v : . [ m a t h . AG ] N ov KATO-NAKAYAMA SPACES, INFINITE ROOT STACKS, AND THEPROFINITE HOMOTOPY TYPE OF LOG SCHEMES
DAVID CARCHEDI, SARAH SCHEROTZKE, NICOL `O SIBILLA, AND MATTIA TALPO
Abstract.
For a log scheme locally of finite type over C , a natural candidate for its profinitehomotopy type is the profinite completion of its Kato-Nakayama space [26]. Alternatively,one may consider the profinite homotopy type of the underlying topological stack of itsinfinite root stack [48]. Finally, for a log scheme not necessarily over C , another naturalcandidate is the profinite ´etale homotopy type of its infinite root stack. We prove that,for a fine saturated log scheme locally of finite type over C , these three notions agree. Inparticular, we construct a comparison map from the Kato-Nakayama space to the underlyingtopological stack of the infinite root stack, and prove that it induces an equivalence onprofinite completions. In light of these results, we define the profinite homotopy type of ageneral fine saturated log scheme as the profinite ´etale homotopy type of its infinite rootstack. Contents
1. Introduction 12. Profinite homotopy types 63. The homotopy type of topological stacks 164. Construction of the map 275. The topology of log schemes 346. The equivalence 397. The profinite homotopy type of a log scheme 44Appendix A. 45References 551.
Introduction
Log schemes are an enlargement of the category of schemes due to Fontaine, Illusie andKato [24]. The resulting variant of algebraic geometry, “logarithmic geometry”, has appli-cations in a variety of contexts ranging from moduli theory to arithmetic and enumerativegeometry (see [2] for a recent survey).In the past years there have been several attempts to capture the “log” aspect of theseobjects and translate it into a more familiar terrain. In the complex analytic case, Kato
Mathematics Subject Classification.
MSC Primary: 14F35, 55P60; Secondary: 55U35.
Key words and phrases. log scheme, Kato-Nakayama space, root stack, profinite spaces, infinity category,´etale homotopy type, topological stack. and Nakayama introduced in [26] a topological space X log (where X is a log analytic space),which may be interpreted as the “underlying topological space” of X , and over which, in somecases, one can write a comparison between logarithmic de Rham cohomology and ordinarysingular cohomology. In a different direction, for a log scheme X, Kato introduced two sites,the Kummer-flat site X Kfl and the Kummer-´etale site X Ket , that are analogous to the smallfppf and ´etale site of a scheme, and were used later by Hagihara and Nizio l [17, 33] to studythe K-theory of log schemes.Recently in [48], the fourth author together with Vistoli introduced and studied a thirdincarnation of the “log aspect” of a log structure, namely the infinite root stack ∞ √ X , andused it to reinterpret Kato’s Kummer sites and link them to parabolic sheaves on X . Thisstack is defined as the limit of an inverse system of algebraic stacks ∞ √ X = lim ←− n n √ X ,parameterizing n -th roots of the log structure of X .The infinite root stack can be thought of as an “algebraic incarnation” of the Kato-Nakayama space: if X is a log scheme locally of finite type over C , both X log and ∞ √ X havea map to X . The fiber of X log → X an over a point x ∈ X an is homeomorphic to ( S ) r ,where r is the rank of the log structure at x . For all n, the reduced fiber of n √ X → X overthe corresponding closed point of X is equivalent to the classifying stack B ( Z /n Z ) r (for thesame r ). Regarding the infinite root stack not as the limit lim ←− n n √ X, but instead as thediagram of stacks n n √ X, i.e. as a pro-object or “formal limit,” yields then that the reduced fiber of ∞ √ X → X is thediagram of stacks, n B ( Z /n Z ) r , which regarded as a pro-object is simply B b Z r ≃ d B Z r , the profinite completion of ( S ) r . In this paper we formalize this analogy and prove a comparison result between the profinitecompletions of X log and ∞ √ X for a fine saturated log scheme X locally of finite type over C . Furthermore, we put this result in a wider circle of ideas, centered around the concept ofthe profinite homotopy type of a log scheme.Our approach relies in a crucial way on a careful reworking of the foundations of the theoryof topological stacks and profinite completions within the framework of ∞ -categories [30].This allows us to have greater technical control than earlier and more limited treatments,and plays an important role in the proof of our main result. In the second half on thepaper we construct a comparison map between X log and ∞ √ X and show that it is induces anequivalence between their profinite completions. The proof involves an analysis of the localgeometry of log schemes, and a local-to-global argument which reduces the statement to alocal computation. Next, we review the main ideas in the paper in greater detail.1.1. Topological stacks and profinite completions of homotopy types.
The firstingredient that we need in order to compare X log and ∞ √ X is the notion of a topological stack [34] associated with an algebraic stack. This is an extension of the analytification functordefined on schemes and algebraic spaces, that equips algebraic stacks with a topologicalcounterpart, and allows one for example to talk about their homotopy type. Given analgebraic stack X locally of finite type over C , let us denote by X top its “underlying topological N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES stack”. This formalism allows us to carry over ∞ √ X to the topological world, where X log lives.The second ingredient we need is a functorial way of associating to a topological stack itshomotopy type. Although this is in principle accomplished in [35] and [36], the constructionis a bit complicated and it is difficult to notice the nice formal properties this functor hasfrom the construction. We instead construct a functor Π ∞ associating to a topologicalstack X its fundamental ∞ -groupoid . The source of this functor is a suitable ∞ -categoryof higher stacks on topological spaces, and the target is the ∞ -category S of spaces. Usingthe language and machinery of ∞ -categories makes the construction and functoriality of Π ∞ entirely transparent; it is the unique colimit preserving functor which sends each space T toits weak homotopy type.The third ingredient we need is a way of associating to a space its profinite completion.Combining this with the functor Π ∞ gives a way of associating to a topological stack aprofinite homotopy type. The notion of profinite completion of homotopy types is originallydue to Artin and Mazur [4]. Profinite homotopy types have since played many importantroles in mathematics, perhaps most famously in relation to the Adams conjecture fromalgebraic topology [41, 47, 13]. A more modern exposition using model categories is given in[39, 40, 22]; however the notion of profinite completion is a bit complicated in this framework.Finally, Lurie briefly introduces an ∞ -categorical model for profinite homotopy types in [29],which has recently been shown to be equivalent to Quick’s model in [5] (and also to a specialcase of Isaksen’s). The advantage of Lurie’s framework is that the definition of profinitespaces and the notion of profinite completion become very simple. A π -finite space is aspace X with finitely many connected components, and finitely many homotopy groups, allof whom are finite, and a profinite space is simply a pro-object in the ∞ -category of π -finitespaces. The profinite completion functor c ( · ) : S → Prof ( S )from the ∞ -category of spaces to the ∞ -category of profinite spaces preserves colimits,and composing this functor with Π ∞ gives a colimit preserving functor b Π ∞ which assigns atopological stack its profinite homotopy type. This property is used in an essential way inthe proof of our main theorem. Using this machinery, we are able to derive some non-trivialproperties of profinite spaces that are used in a crucial way to prove our main result; inparticular we show that profinite spaces can be glued along hypercovers (Lemma 6.1).1.2. The comparison map and the equivalence of profinite completions.
Our mainresult states:
Theorem (see Theorem 6.4) . Let X be a fine saturated log scheme locally of finite type over C . Then there is a canonical map of pro-topological stacks Φ X : X log → ∞ √ X top that induces an equivalence upon profinite completion b Π ∞ ( X log ) ∼ −−−−−−−→ b Π ∞ (cid:16) ∞ √ X top (cid:17) . DAVID CARCHEDI, SARAH SCHEROTZKE, NICOL `O SIBILLA, AND MATTIA TALPO
This theorem makes precise the idea that the infinite root stack is an algebraic incarnationof the Kato-Nakayama space, and that it completely captures the “profinite homotopy type”(`a la Artin-Mazur) of the corresponding log scheme.The construction of the comparison map Φ X is first performed ´etale locally on X wherethere is a global chart for the log structure, and then globalized by descent. The localconstruction uses the quotient stack description of the root stacks, that reduces the problemof finding a map to constructing a (topological) torsor on X log with an equivariant map toa certain space.This permits the construction of Φ X as a canonical morphism of pro-topological stacksover X an : X log Φ X / / π log " " ❊❊❊❊❊❊❊❊ ∞ √ X topπ ∞ z z ✈✈✈✈✈✈✈✈✈ X an . The jump patterns of the fibers of π log and π ∞ reflect the way in which the rank of thelog structure varies over X an . More formally, the log structure defines a canonical stratifi-cation on X an called the “rank stratification”, which makes X log and ∞ √ X top into stratifiedfibrations. After profinite completion, the fibers of π log and π ∞ on each stratum becomeequivalent: indeed they are equivalent respectively to real tori of dimension n , and to the(pro-)classifying stacks B b Z n . The fact that the fibers of π log and π ∞ are profinite homotopyequivalent was in fact our initial intuition as to why the main result should be true. Ex-tracting from this fiber-wise statement a proof that Φ X induces an equivalence of profinitehomotopy types requires a local-to-global argument that makes full use of the ∞ -categoricalframework developed in the first half of the paper.The Kato-Nakayama space models the topology of log schemes, but its applicability islimited to schemes over the complex numbers. Our results suggest that the infinite root stackencodes all the topological information of log schemes (or at least its profinite completion)in a way that is exempt from this limitation. More precisely, if X is a log scheme locally offinite type over C , there are three natural candidates for its “profinite homotopy type”: theprofinite completion of the Kato-Nakayama space X log , the profinite ´etale homotopy type of ∞ √ X and the profinite completion of the (pro-)topological stack ∞ √ X top . Theorem 6.4 andTheorem 7.2 (proved in [9]) imply that these three constructions give the same result. Thisjustifies the definition of the profinite homotopy type for a log scheme X , even outside ofthe complex case, as the profinite ´etale homotopy type of its infinite root stack ∞ √ X .Another possible approach to this would be to define the homotopy type of a log scheme viaKato’s Kummer-´etale topos (see [25]). As proved in [48, Section 6.2], this topos is equivalentto an appropriately defined small ´etale topos of the infinite root stack. It is not immediate,however, to link the resulting profinite homotopy time and the one that we define in thepresent paper. We plan to address this point in future work.We believe that our results hold in the framework of log analytic spaces as well. Eventhough root stacks of those have not been considered anywhere yet, the construction andresults about them that we use in the present paper should carry through without difficulty,using some notion of “analytic stacks” instead of algebraic ones. N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES In recent unpublished work, Howell and Vologodsky give a definition of the motive of alog schemes inside Voevodsky’s triangulated category of motives. Based on our results weexpect that infinite root stacks should provide an alternative encoding of the motive of logschemes, or a profinite approximation of it. It is an interesting question to explore possibleconnections between these two viewpoints.
Description of content.
The paper is structured as follows.In the first two sections we develop the framework necessary to associate profinite ho-motopy types to (pro-)algebraic and topological stacks. Along the way, in Section 3.4 weprove an interesting result (Theorem 3.25) which expresses the homotopy type of the Kato-Nakayama space of a log scheme as the classifying space of a natural category.As a first step towards the main theorem, we construct in Section 4 (Proposition 4.1) acanonical map of pro-topological stacks(1) Φ X : X log → ∞ √ X top by exploiting the local quotient stack presentations of the root stacks n √ X , and gluing theresulting maps.Section 5 contains results about the topology of the Kato-Nakayama space and the topo-logical infinite root stack that we use in an essential way in the proof of our main result.In Section 6, we give the proof of Theorem 6.4: we show that the canonical map (1)induces an equivalence after profinite completion. The proof is based on a local-to-globalanalysis: we use a suitable hypercover U • of X an constructed in Section 5 to reduce thequestion to the restriction of the map Φ X to each element of this hypercover. We thenuse the results about the topology of the Kato-Nakayama space and the topological infiniteroot stack proven in the same section to reduce to showing that the map induces a profinitehomotopy equivalence along fibers. This concludes the proof.Finally, in Section 7 we make some remarks about the definition of the profinite homotopytype of a general log scheme.In Appendix A, we gather definitions and facts that we use throughout the paper about logschemes, the analytification functor, the Kato-Nakayama space, root stacks, and topologicalstacks. In particular, in (A.6), we carefully construct the “rank stratification” of X (and X an ), over which the characteristic monoid M of the log structure is locally constant. Acknowledgements.
All of the authors would like to thank their respective home institu-tions for their support.We are also happy to thank Kai Behrend, Thomas Goodwillie, Marc Hoyois, Jacob Lurie,Thomas Nikolaus, Behrang Noohi, Gereon Quick, Angelo Vistoli, and Kirsten Wickelgrenfor useful conversations.We are grateful to the anonymous referee for a careful reading and useful comments, inparticular for pointing out the short proof of Proposition 4.4.
Notations and conventions.
We will always work over a field k , which will almost alwaysbe the complex numbers C . In particular all our log schemes will be fine and saturated, andlocally of finite type over C , unless otherwise stated. DAVID CARCHEDI, SARAH SCHEROTZKE, NICOL `O SIBILLA, AND MATTIA TALPO If P is a monoid we denote by P gp the associated group. Our monoids will typicallybe integral, finitely generated, saturated and sharp (hence torsion-free). A monoid P withthese properties has a distinguished “generating set”, consisting of all its indecomposableelements. This gives a presentation of any such monoid P through generators and relations.If F is a sheaf of sets on the small ´etale site of a scheme, its “stalks” will always be stalkson geometric points.By an ∞ -category, we mean a quasicategory or inner-Kan complex. These are a modelfor ( ∞ , C and D objects of an ∞ -category C , we will denote by Hom C ( C, D ) the space of morphismsfrom C to D in C , rather than using the notation Map C ( C, D ) , in order to highlight theanalogy with classical category theory. A very brief heuristic introduction to ∞ -categoriescan be found in Appendix A of [10]. See also [16].2. Profinite homotopy types
In this section we will introduce the ∞ -categorical model for profinite spaces that we willuse in this article. This ∞ -category is introduced in [29, Section 3.6]; a profinite space willsuccinctly be a pro-object in the ∞ -category of π -finite spaces. This notion is equivalentto the notion of profinite space introduced by Quick in [39, 40] (see [5]), but the machineryand language of ∞ -categories is much more convenient to work with. Most importantly, thenotion of profinite completion becomes completely transparent in this set up, and it is left-adjoint to the canonical inclusion of profinite spaces into pro-spaces, and hence in particularpreserves all colimits. We use this fact in an essential way in the proof of our main result,and we do not know how to prove the analogous fact about profinite completion in any otherformalism.We start first by reviewing the notion of ind-objects and pro-objects.We will interchangeably use the notation S and Gpd ∞ for the ∞ -category of spaces, andthe ∞ -category of ∞ -groupoids. These two ∞ -categories are one and the same, and we willuse the different notations solely to emphasize in what way we are viewing the objects.Recall that for D a small category, the category of ind-objects is essentially the categoryobtained from D by freely adjoining formal filtered colimits. This construction carries overfor ∞ -categories. Moreover, if D is an essentially small ∞ -category, the ∞ -category of ind-objects in D , Ind ( D ) , admits a canonical functor j : D → Ind ( D )satisfying the following universal property:For every ∞ -category E which admits small filtered colimits, composition with j inducesan equivalence of ∞ -categoriesFun filt . (Ind ( D ) , E ) → Fun ( D , E ) , where Fun filt . (Ind ( D ) , E ) denotes the ∞ -category of all functors Ind ( D ) → E which pre-serve filtered colimits. N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES A more concrete description of the ∞ -category Ind ( D ) is as follows. First, recall thefollowing proposition: Proposition 2.1 ([30, Corollary 5.3.5.4]) . Denote by
Psh ∞ ( D ) the ∞ -category of ∞ -presheaves on D , that is, the functor category Fun ( D op , Gpd ∞ ) . Let D be an essentially small ∞ -category and let F : D op → Gpd ∞ be an ∞ -presheaf. Thenthe following conditions are equivalent: i) The associated right fibration Z D F → D classified by F has R D F a filtered ∞ -category. ii) There exists a small filtered ∞ -category J and a functor f : J → D such that F is the colimit of the composite J f −→ D y ֒ → Psh ∞ ( D ) (where y denotes the Yoneda embedding).and if D has finite colimits, i ) and ii ) are equivalent to: iii) F is left exact (i.e. preserves finite limits). The ∞ -category Ind ( D ) may be described as the full subcategory of Psh ∞ ( D ) satisfyingthe equivalent conditions i ) and ii ) (or iii ) if D has finite colimits). In particular, this impliesthat j is full and faithful, since it is a restriction of the Yoneda embedding. In a nutshellPsh ∞ ( D ) is the ∞ -category obtained from D by freely adjoining formal colimits, and ii )above states that Ind ( D ) is the full subcategory thereof on those formal colimits of objectsin D which are filtered colimits.The notion of a pro-object is dual to that of an ind-object; it is a formal cofiltered limit.By definition, the ∞ -category of pro-objects of an essentially small ∞ -category D isPro ( D ) .. = Ind ( D op ) op . If D has small limits, we see that Pro ( D ) can be described as the full subcategory ofFun ( D , Gpd ∞ ) op on those functors F : D → Gpd ∞ such that F preserves finite limits. Since this definition makes sense even when D is notessentially small, we make the following definition, due to Lurie: Definition 2.2. If E is any accessible ∞ -category with finite limits, then we define the ∞ -category of pro-objects of E , Pro ( E ) , to be the full subcategory of Fun ( E , Gpd ∞ ) op onthose functors F : E → Gpd ∞ which are accessible and preserve finite limits. DAVID CARCHEDI, SARAH SCHEROTZKE, NICOL `O SIBILLA, AND MATTIA TALPO
Remark 2.3. If E is any accessible ∞ -category and E is an object of E , then the functorHom ( E, · ) : E → Gpd ∞ co-represented by E is accessible and preserves all limits. This induces a fully faithful functor E j ֒ → Pro ( E ) . The functor j satisfies the following universal property:If D is any ∞ -category admitting small cofiltered limits, then composition with j inducesan equivalence of ∞ -categories(2) Fun co − filt . (Pro ( E ) , D ) → Fun ( E , D ) , where Fun co − filt . (Pro ( E ) , D ) is the full subcategory of Fun (Pro ( E ) , D ) spanned by thosefunctors which preserve small cofiltered limits, see [29, Proposition 3.1.6]. Remark 2.4. If C is any (not necessarily accessible) ∞ -category, there always exists an ∞ -category Pro ( C ) satisfying the universal property (2). This is a special case of [30,Proposition 5.3.6.2]. Remark 2.5.
Let E be any accessible ∞ -category which is not necessarily essentially small.Let U be the Grothendieck universe of small sets and let V be a Grothendieck universe suchthat U ∈ V , so that we may regard V as the Grothendieck universe of large sets. Let d Gpd ∞ denote the ∞ -category of ∞ -groupoids in the universe V . By the proof of [29, Proposition3.1.6], it follows that the essential image of the compositionPro ( E ) ֒ → Fun ( E , Gpd ∞ ) op ֒ → Fun (cid:16) E , d Gpd ∞ (cid:17) op consists of those functors F : E → d Gpd ∞ for which there exists a small filtered ∞ -category J and a functor f : J → E op such that F is the colimit of the composite J f −→ E op ֒ → Fun (cid:16) E , d Gpd ∞ (cid:17) . Remark 2.6.
In light of Remark 2.5, any object X of Pro ( E ) , for E an accessible ∞ -category, can be written as a cofiltered limit of a diagram of the form F : I → E j ֒ → Pro ( E ) , or in more informal notation, X = lim ←− i ∈ I X i . Unwinding the definitions, we see that if Y = lim ←− j ∈ J Y j is another such object of Pro ( E ) , thenthe usual formula for the morphism space holds:Hom Pro( E ) ( X, Y ) ≃ lim ←− j ∈ J colim −−−→ i ∈ I Hom E ( X i , Y j ) . N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES Now suppose that E has a terminal object 1 . ThenHom
Pro( E ) ( X, j (1)) ≃ colim −−−→ i ∈ I Hom E ( X i , . Notice that each space Hom E ( X i ,
1) is contractible since 1 is terminal, and since ( − S by [30, Corol-lary 5.5.7.4], it follows that Hom Pro( E ) ( X, j (1)) itself is a contractible space, and hence weconclude that j (1) is a terminal object. Example 2.7.
Let E = S be the ∞ -category of spaces. Then the ∞ -category of pro-spaces , Pro ( S ) , can be identified with the opposite category of functors F : S → S suchthat F is accessible and left exact. Notice that any space X gives rise to a pro-spaceHom ( X, · ) : S → S which moreover preserves all limits. Moreover if F : S → S is any functor which preservesall limits, then by the Adjoint Functor theorem for ∞ -categories ([30, Corollary 5.5.2.9]), F must have a right adjoint G , and is moreover accessible by Proposition 5.4.7.7 of op. cit.This then implies that Hom ( G ( ∗ ) , X ) ≃ Hom ( ∗ , F ( X )) ≃ F ( X ) . Hence F ≃ j ( G ( ∗ )) . We conclude that the essential image of j : S ֒ → Pro ( S )is precisely those ∞ -functors S → S which preserve all small limits. Proposition 2.8.
The functor T : Pro ( S ) Hom( j ( ∗ ) , · ) −−−−−−−−−−−−−→ S is right adjoint to the canonical inclusion j : S → Pro ( S ) . Proof.
By Remark 2.5, Pro ( S ) op may be identified with a subcategory of Fun (cid:16) S , d Gpd ∞ (cid:17) of large ∞ -co-presheaves, and since limits commute with limits, this subcategory is stableunder small limits. Note that this implies that Pro ( S ) is cocomplete. Since the Yonedaembedding into large ∞ -presheaves S op y ֒ → d Psh ∞ ( S )preserves small limits, it follows that j : S ֒ → Pro ( S )preserves small colimits. Since S ≃ Psh ∞ (1) , where 1 is the terminal ∞ -category, andsince Pro ( S ) is cocomplete, one has by [30, Theorem 5.1.5.6] that j ≃ Lan y ( t ) where y is the Yoneda embedding 1 → S and t : 1 → Pro ( S ) is the functor picking out the object j ( ∗ ) . It follows immediately from the Yoneda lemma that Hom ( j ( ∗ ) , · ) is right adjoint toLan y ( t ) . (cid:3) Remark 2.9.
Let P : S → S be a pro-space. By [30, Proposition 5.4.6.6], since P isaccessible, it follows that the associated left fibration Z S P is accessible, and hence has a small cofinal subcategory r : C P ֒ → Z S P, and P may be identified with the limit of the composite C P r ֒ → Z S P π P −−−−−−−→ S j ֒ → Pro ( S ) . We claim that T ( P ) ≃ lim ←− π P ◦ r. Indeed: T ( P ) = Hom ( j ( ∗ ) , P ) ≃ Hom (cid:0) j ( ∗ ) , lim ←− j ◦ π P ◦ r (cid:1) ≃ lim ←− Hom ( j ( ∗ ) , j ◦ π P ◦ r ) ≃ lim ←− Hom ( ∗ , π P ◦ r ) ≃ lim ←− π P ◦ r. By the same proof, if one has P presented as a cofiltered limit P = lim ←− j ( X α ) of spaces,then T ( P ) ≃ lim ←− X α . In fact, this holds more generally by the following proposition:
Proposition 2.10.
Let C be an accessible ∞ -category which admits small filtered limits.Then the canonical inclusion j : C ֒ → Pro ( C ) has a right adjoint T and if F : I → C is a cofiltered diagram corresponding an object in Pro ( C ) , then T ( F ) = lim ←− F. Proof.
By Remark 2.5, composition with j : C ֒ → Pro ( C )induces an equivalence of ∞ -categoriesFun co − filt . (Pro ( C ) , C ) → Fun ( C , C ) , so we can find a functor T : Pro ( C ) → C and an equivalence η : id C ∼ −→ T ◦ j. Let Z be an arbitrary object of Pro ( C ) , then we can write Z = lim ←− i ∈ I j ( X i ) . First note thatsince η is an equivalence and T preserves cofiltered limits (by definition), we have that forsuch a Z, T ( Z ) ≃ lim ←− i ∈ I X i . N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES This shows that T has the desired properties on pro-objects. Let us now show that T is aright adjoint to j . Let C be an object of C , then we haveHom C ( D, T ( Z )) ≃ Hom C D, lim ←− i ∈ I X i ! ≃ lim ←− i ∈ I Hom C ( D, X i ) , and since j is fully faithful, we have for each i Hom C ( D, X i ) ≃ Hom
Prof( C ) ( j ( D ) , j ( X i )) . It follows then thatHom C ( D, T ( Z )) ≃ lim ←− i ∈ I Hom
Prof( C ) ( j ( D ) , j ( X i )) ≃ Hom
Prof( C ) j ( D ) , lim ←− i ∈ I j ( X i ) ! = Hom Prof( C ) ( j ( D ) , Z ) . (cid:3) Definition 2.11.
A space X in S is π -finite if all of its homotopy groups are finite, it hasonly finitely many non-trivial homotopy groups, and finitely many connected components. Definition 2.12.
Let S fc denote the full subcategory of the ∞ -category S on the π -finitespaces. S fc is essentially small and idempotent complete (and hence accessible). The ∞ -category of profinite spaces is defined to be the ∞ -categoryProf ( S ) .. = Pro (cid:0) S fc (cid:1) . Proposition 2.13.
Let V be a π -finite space. Note that V is n -truncated for some n, sinceit has only finitely many homotopy groups. The associated profinite space j ( V ) is also n -truncated.Proof. Let X = lim ←− i ∈ I X i be a profinite space. Then by Remark 2.6, we have thatHom Prof( S ) ( X, j ( V )) ≃ colim −−−→ i ∈ I Hom S fc ( X i , V ) . Each space Hom S fc ( X i , V ) is n -truncated since V is, and n -truncated spaces are stableunder filtered colimits by [30, Corollary 5.5.7.4], so it follows that Hom Prof( S ) ( X, j ( V )) isalso n -truncated. (cid:3) Remark 2.14.
The assignment C Pro ( C ) is functorial among accessible ∞ -categorieswith finite limits. Given a functor f : C → D , the composite C f −→ D j ֒ → Pro ( D ) corresponds to an object of the ∞ -category Fun ( C , Pro ( D )) , which by Remark 2.3 is equiv-alent to the ∞ -category Fun co − filt . (Pro ( C ) , Pro ( D )) . Hence, one gets an induced functorPro ( f ) : Pro ( C ) → Pro ( D )which preserves cofiltered limits. Moreover, Pro ( f ) is fully faithful if f is. If f happens tobe accessible and left exact, then there is an induced functor in the opposite direction, givenby f ∗ : Pro ( D ) → Pro ( C ) (cid:16) D F −→ Gpd ∞ (cid:17) (cid:16) C f −→ D F −→ Gpd ∞ (cid:17) , and f ∗ is left adjoint to Pro ( f ) . See [29, Remark 3.1.7] (but note there is a typo, since f ∗ is in fact a left adjoint, not a right adjoint). Example 2.15.
The canonical inclusion i : S fc ֒ → S induces a fully faithful embeddingPro ( i ) : Prof ( S ) ֒ → Pro ( S )of profinite spaces into pro-spaces. Moreover, i is accessible and preserves finite limits, hencethe above functor has a left adjoint i ∗ : Pro ( S ) → Prof ( S ) . This functor sends a pro-space P to its profinite completion . Definition 2.16.
We denote by c ( · ) the composite S j ֒ → Pro ( S ) i ∗ −→ Prof ( S )and call it the profinite completion functor . Concretely, if X is a space in S , then b X corresponds to the composite S fc i ֒ → S Hom( X, · ) −−−−−−−−−−−−−→ S . This functor has a right adjoint given by the compositeProf ( S ) Pro( i ) ֒ −−→ Pro ( S ) T −→ S . We will denote this right adjoint simply by U . Remark 2.17.
We will sometimes abuse notation and denote the profinite completion of apro-space Y also by b Y rather than i ∗ Y, when no confusion will arise.2.1. The relationship with profinite groups.
In this subsection, we will touch brieflyupon the relationship between profinite groups and profinite spaces. Recall the notion ofprofinite completion of a group. A profinite group is a pro-object in the category of finitegroups. Equivalently, a profinite group is a group object in profinite sets, see [29, Proposition3.2.12].Denote by i : F inGp ֒ → Gp the fully faithful inclusion of the category of finite groups intothe category of groups. The composite Gp ֒ → Pro ( Gp ) i ∗ −−−−−−−→ Pro (
F inGp ) ≃ Gp (Pro ( F inSet )) N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES is the functor assigning a group its profinite completion. We also denote this functor by c ( · )when no confusion will arise. Recall that the profinite completion of a group has a classicalconcrete description as follows: Let G be a group, then its profinite completion is the limitlim ←− N j ( G/N ) , where N ranges over all the finite index normal subgroups of G. Similarly,denote by i ab : F inAbGp ֒ → AbGp the fully faithful inclusion of the category of finite abeliangroups into the category of abelian groups. By the analogous construction to the above,there is an induced profinite completion functor c ( · ) ab : AbGp → Pro (
F inAbGp ) . It can be described classically by the same formula as in the non-abelian case. If φ : AbGp ֒ → Gp is the canonical inclusion of abelian groups into groups, it follows that the following diagramcommutes up to canonical natural equivalence: AbGp c ( · ) ab / / φ (cid:15) (cid:15) Pro (
F inAbGp ) Pro( φ ) (cid:15) (cid:15) Gp c ( · ) / / Pro (
F inGp ) . By [29, Proposition 3.2.14], there is a canonical equivalence of categoriesPro (
F inAbGp ) ≃ AbGp (Pro (
F inSet )) , between the category of pro-objects in finite abelian groups and the category of abelian groupobjects in profinite sets. In particular, finite coproducts (direct sums) in Pro ( F inAbGp )coincide with finite products. Since c ( · ) ab is a left adjoint, it preserves direct sums, and byRemark 2.14, Pro ( φ ) is a right adjoint (since φ preserves finite limits), so Pro ( φ ) preservesproducts. It follows that the composite c ( · ) ◦ φ : AbGp → Pro (
F inGp )preserves finite products.
Corollary 2.18.
Let k be a non-negative integer. Then there is a canonical isomorphism ofprofinite groups c Z k ∼ = b Z k . We now note a recent result which compares the ∞ -categorical model for profinite spacesjust presented with the model categorical approach developed by Quick in [39, 40]: Theorem 2.19 ([5, Corollary 7.4.6]) . The ∞ -category associated to the model category pre-sented in [39, 40] is equivalent to Prof ( S ) . The details of Quick’s model category need not concern us here, but we cite the abovetheorem in order to freely use results of [39, 40] about profinite spaces.
Proposition 2.20.
Let k be a non-negative integer. There is a canonical equivalence ofprofinite spaces \ B ( Z k ) ≃ B (cid:16)b Z k (cid:17) . Proof.
Since Z k is a finitely generated free abelian group, it is good in the sense of Serre in[44]. It follows from [40, Proposition 3.6] and Theorem 2.19 that the canonical map \ B ( Z k ) → B (cid:16)c Z k (cid:17) is an equivalence of profinite spaces. The result now follows from Corollary 2.18. (cid:3) The following lemma will be used in an essential way several times in this paper:
Lemma 2.21.
Let f : ∆ → C be a cosimplicial diagram and suppose that C is an ( n + 1 , -category, i.e. an ∞ -category whose mapping spaces are all n -truncated. Then, provided bothlimits exist, the canonical map lim ←− f → lim ←− ( f | ∆ ≤ n ) is an equivalence.Proof. Let C be an arbitrary ( ∞ , n + 1)-category. Notice that for any diagram f : ∆ → C , and any object C of C , we haveHom C, lim ←− k ∈ ∆ f ( k ) ! ≃ lim ←− k ∈ ∆ Hom (
C, f ( k )) , and since C is an ( ∞ , n + 1)-category, each Hom ( C, f ( k )) is an n -truncated space. Thereforethe general case follows from the case when C is the full subcategory S ≤ n of S on the n -truncated spaces. By [30, Theorem 4.2.4.1], to prove the lemma for the special case C = S ≤ n , it suffices to prove the corresponding statement about homotopy limits in the Quillenmodel structure on the category of compactly generated spaces CG , since the associated ∞ -category is S . Suppose that X • : ∆ → CG is a cosimplicial space which is fibrant with respect to the projective model structure onFun (∆ , CG ) (with respect to the Quillen model structure on CG ), i.e., the diagram X • consists entirely of Serre fibrations. Then the homotopy limit of X • may be computedas Tot ( X ) , and moreover, Tot ( X ) can be written as the (homotopy) limit of a tower offibrations . . . → Tot ( X ) k → Tot ( X ) k − → . . . → Tot ( X ) → Tot ( X ) = X, where each Tot ( X ) k is a model for the homotopy limit of X | ∆ ≤ k . Moreover, the (homotopy)fiber of each map Tot ( X ) k → Tot ( X ) k − is homotopy equivalent to the k -fold loop space Ω k (cid:0) M k ( X • ) (cid:1) , where N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES M k X • = lim ←− [ k +1] ։ [ j ] j ≤ k X j is the k -th matching object of X • (see e.g. the introduction of [31]).Now, let us assume that each X k is n -truncated. Then as X • is fibrant, the diagraminvolved in the limit above consists entirely of fibrations, so the limit is a homotopy limit,hence each matching object is also n -truncated (since n -truncated objects are stable underlimits in S by [30, Proposition 5.5.6.5]). It follows then that each homotopy fiberTot ( X ) k → Tot ( X ) k − is weakly contractible for k > n, and hence the natural mapholim ←−−−−−−− X • = Tot ( X ) → Tot ( X ) n = holim ←−−−−−−− X • | ∆ ≤ n is a weak homotopy equivalence. (cid:3) Proposition 2.22.
Let lim ←− i ∈ I G i be a pro-object in the category of finite groups, or equivalently,a group object in Pro (
FinSet ) . Consider the profinite space B lim ←− i G i ! .. = colim −−−→ " . . . lim ←− i G i × lim ←− i G i →→→ lim ←− i G i ⇒ ∗ , where the colimit is computed in Prof ( S ) , and ∗ denotes the terminal profinite space. Inmore detail, the diagram whose colimit is being taken is the simplicial diagram which is theˇCech nerve of the unique map lim ←− i G i → ∗ in Prof ( S ) . Consider for each i the object in S B ( G i ) .. = colim −−−→ [ . . . G i × G i →→→ G i ⇒ ∗ ] , i.e. the colimit in S of the ˇCech nerve of G i . Then these spaces assemble into a profinitespace lim ←− i B ( G i ) , and we have a canonical equivalence B lim ←− i G i ! ≃ lim ←− i B ( G i ) in Prof ( S ) . Proof.
It suffices to prove that for each π -finite space V, we have an equivalenceHom Prof( S ) B lim ←− i G i ! , j ( V ) ! ≃ Hom
Prof( S ) lim ←− i B ( G i ) , j ( V ) ! . Recall that by Proposition 2.13 j ( V ) is n -truncated for some n. As such, we have
Hom
Prof( S ) B lim ←− i G i ! , j ( V ) ! ≃ Hom
Prof( S ) colim −−−→ ∆ op N lim ←− i G i ! , j ( V ) ! ≃ lim ←− ∆ Hom
Prof( S ) N lim ←− i G i ! , j ( V ) ! ≃ lim ←− ∆ ≤ n Hom
Prof( S ) N lim ←− i G i ! , j ( V ) ! , that last equivalence following from Lemma 2.21. Expanding this out we get lim ←− ∆ ≤ n Hom
Prof( S ) (1 , j ( V )) ⇒ Hom
Prof( S ) lim ←− i G i , j ( V ) ! →→→ Hom
Prof( S ) lim ←− i G i ! , j ( V ) . . . Hom
Prof( S ) lim ←− i G i ! n , j ( V ) ! which is equivalent to lim ←− ∆ ≤ n " Hom S ( ∗ , V ) ⇒ colim −−−→ i Hom S ( G i , j ( V )) →→→ colim −−−→ i Hom S ( G i , j ( V )) . . . colim −−−→ i Hom S ( G ni , j ( V )) and since by [30, Proposition 5.3.3.3] finite limits commute with filtered colimits in S , we get colim −−−→ i lim ←− ∆ ≤ n [Hom S ( ∗ , V ) ⇒ Hom S ( G i , V ) →→→ Hom S ( G i , j ( V )) . . . Hom S ( G ni , V )] . Now since j ( V ) is n -truncated by Proposition 2.13, it follows from Lemma 2.21 that we canrewrite this as colim −−−→ i lim ←− ∆ [Hom S ( ∗ , V ) ⇒ Hom S ( G i , V ) →→→ Hom S ( G i , j ( V )) . . . Hom S ( G ni , V ) . . . ] which is equivalent tocolim −−−→ i Hom S colim −−−→ ∆ op N ( G i ) , V ! ≃ colim −−−→ i Hom S ( B ( G i ) , V ) ≃ Hom
Prof( S ) lim ←− i B ( G i ) , j ( V ) ! . (cid:3) The homotopy type of topological stacks
In this section we use the formalism of ∞ -categories to produce two important construc-tions necessary for our paper. Firstly, we extend the construction of analytification, whichsends a complex variety to its underlying topological space with the complex analytic topol-ogy, to a colimit-preserving functor( · ) top : Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) → H ypSh ∞ ( Top C )from the ∞ -category of ∞ -sheaves over the ´etale site of affine schemes of finite type over C to the ∞ -category of hypersheaves over an appropriate topological site. This functor, inparticular, sends an Artin stack locally of finite type over C to its underlying topological N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES stack in the sense of Noohi [34]. Using this functor, one associates to the infinite root stack ∞ √ X of a log-scheme a pro-topological stack ∞ √ X top . In Section 4, we produce a map(3) X log → ∞ √ X top from the Kato-Nakayama space to the underlying (pro-)topological stack of the infinite rootstack. The main result of the paper is that this map is a profinite homotopy equivalence,but to make sense of such a statement, one first needs to associate to each of these objectsa (pro-)homotopy type, in a functorial way. To achieve this, the second construction weproduce is a colimit-preserving functorΠ ∞ : H ypSh ∞ ( Top C ) → S which sends every topological space X to its underlying homotopy type, and sends everytopological stack to its homotopy type in the sense of Noohi [35]. Using this constructionand the map (3), one has an induced map in Pro ( S )Π ∞ ( X log ) → Π ∞ (cid:16) ∞ √ X top (cid:17) which we prove in Section 6 to become an equivalence after applying the profinite completionfunctor, i.e. the map (3) is a profinite homotopy equivalence.3.1. The underlying topological stack of an algebraic stack.
Let
Top be the categoryof topological spaces and let
Top s C denote the full subcategory of Top of all contractible andlocally contractible spaces which are homeomorphic to a subspace of R n for some n. Notethat
Top s C is essentially small. Denote by Top C the following subcategory of topologicalspaces: • A topological space T is in Top C if T has an open cover ( U α ֒ → T ) α such that each U α is an object of Top s C . We use the subscript C to highlight the fact that Top C will serve as the target of theanalytification functor from the category of algebraic spaces over C . Note that the objectsof Top C are closed under taking open subspaces. As such, it makes sense to equip Top C with the Grothendieck topology generated by open covers. Denote by H ypSh ∞ ( Top C ) the ∞ -topos of hypersheaves on Top C , i.e. the hypercompletion of the ∞ -topos of ∞ -sheaves.There is also a natural structure of a Grothendieck site on Top s C as follows: • Let T be a space in Top s C . A covering family of T consists of an open cover ( U α ֒ → T )such that each U α is in Top s C .Note that every open cover of T can be refined by such a cover. We denote the associated ∞ -topos of hypersheaves by H ypSh ∞ ( Top s C ) . By the Comparison Lemma of [1] III, we havethat restriction along the canonical inclusion
Top s C ֒ → Top C induces an equivalence between their respective categories of sheaves of sets. It then followsfrom [23, Theorem 5] and [30, Proposition 6.5.2.14 ] that this lifts to an equivalence H ypSh ∞ ( Top C ) ∼ −→ H ypSh ∞ ( Top s C ) , and in particular, H ypSh ∞ ( Top C ) is an ∞ -topos (which is not a priori clear for sites whichare not essentially small).Denote by Aff LFT C the category of affine schemes of finite type over C . Note that it is asmall category with finite limits. Denote by( · ) an : Aff LFT C → Top the functor associating to such an affine scheme its space of C -points, equipped with theanalytic topology. The above functor preserves finite limits, and is the restriction of afunctor defined for all algebraic spaces locally of finite type over C , see [49, p. 12]. Note alsothat if V is a scheme which is separated and locally of finite type, then V an is locally (overany affine) a triangulated space by [28], so in particular V an is locally contractible. Alsoobserve that V an is locally cut-out of C n by polynomials, so it follows that V an is in Top C . Consequently ( · ) an restricts to a functor( · ) an : Aff LFT C → Top C , which preserves finite limits.Note that the category Aff LFT C can be equipped with the Grothendieck topology generatedby ´etale covering families. Denote the associated ∞ -topos of ∞ -sheaves on this site bySh ∞ (cid:0) Aff LFT C , ´et (cid:1) . The following theorem is an extension of [34, Proposition 20.2]:
Theorem 3.1.
The functor ( · ) an : Aff LFT C → Top C lifts to a left exact colimit preserving functor ( · ) top : Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) → H ypSh ∞ ( Top C ) . Proof.
Note that the image under ( · ) an of an ´etale map is a local homeomorphism. Alsonote that if S → T is a local homeomorphism and T is in Top C , so is S. Furthermore, since the inclusion of anyopen subspace of a topological space is a local homeomorphism, and since any cover by localhomeomorphisms can be refined by a cover by open subspaces, it follows that open coversand local homeomorphisms generate the same Grothendieck topology on
Top C . It followsthat any ∞ -sheaf on Top C , so in particular any hypersheaf, satisfies descent with respect tocovers by local homeomorphisms. The result now follows from [30, Proposition 6.2.3.20]. (cid:3) Remark 3.2.
Denote by Y the Yoneda embedding Y : Top C ֒ → H ypSh ∞ ( Top C )and denote by y the Yoneda embedding y : Aff LFT C ֒ → Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) . Explicitly, ( · ) top is the left Kan extension of Y ◦ ( · ) an along y, Lan y [ Y ◦ ( · ) an ] : Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) → H ypSh ∞ ( Top C ) , N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES or more concretely, it is the unique colimit preserving functor such that for a representable y ( X ) , i.e. an affine scheme, y ( X ) top ∼ = Y ( X an ) . By the proof of Theorem 3.1, we see that given any hypersheaf F on Top C , the functor F ◦ ( · ) an is an ∞ -sheaf on (cid:0) Aff LFT C , ´et (cid:1) , i.e. we have a well-defined functor( · ) ∗ an : H ypSh ∞ ( Top C ) → Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) . Proposition 3.3.
The functor ( · ) top is left adjoint to ( · ) ∗ an .Proof. Since ( · ) top is colimit preserving, it follows from [30, Corollary 5.5.2.9] that it has aright adjoint. Let us denote the right adjoint by R. By the Yoneda lemma, we have that if F is a hypersheaf F on Top C , then R ( F ) is the ∞ -sheaf on (cid:0) Aff LFT C , ´et (cid:1) such that if X isan affine scheme, R ( F ) ( X ) ≃ Hom ( y ( X ) , R ( F )) ≃ Hom (cid:16) ( y ( X )) top , F (cid:17) ≃ Hom ( Y ( X an ) , F ) ≃ F ( X an ) . (cid:3) Remark 3.4.
The adjoint pair ( · ) top ⊣ ( · ) ∗ an assemble into a geometric morphism of ∞ -topoi f : H ypSh ∞ ( Top C ) → Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) , with direct image functor f ∗ = ( · ) ∗ an and inverse image functor f ∗ = ( · ) top . Lemma 3.5.
Let
AlgSp
LFT C denote the category of algebraic spaces locally of finite type over C . Equip
AlgSp
LFT C with the ´etale topology. Then restriction along the canonical inclusion Aff LFT C ֒ → AlgSp
LFT C induces an equivalence of ∞ -categories Sh ∞ (cid:0) AlgSp
LFT C , ´et (cid:1) ∼ −→ Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) . Proof.
The inclusion satisfies the conditions of the Comparison Lemma of [1] III, so we havean induced equivalence Sh (cid:0)
AlgSp
LFT C , ´et (cid:1) ∼ −→ Sh (cid:0) Aff LFT C , ´et (cid:1) between sheaves of sets. Since both sites have finite limits, the result now follows from [30,Proposition 6.4.5.4]. (cid:3) Proposition 3.6.
Let X be any algebraic space locally of finite type over C . Then X top ∼ = X an . Proof.
Let U be the Grothendieck universe of small sets and let V be the Grothendieckuniverse of large sets with U ∈ V . Denote by d Gpd ∞ the ∞ -category of large ∞ -groupoids,and denote by \H ypSh ∞ ( Top C ) the ∞ -category of hypersheaves on Top C with values in d Gpd ∞ , and similarly let d Sh ∞ (cid:0) AlgSp
LFT C , ´et (cid:1) denote the ∞ -category of sheaves on the ´etalesite of algebraic spaces with values in d Gpd ∞ . Then by same proof as Theorem 3.1, by leftKan extension there is a V -small colimit preserving functor L : d Sh ∞ (cid:0) AlgSp
LFT C , ´et (cid:1) → \H ypSh ∞ ( Top C )such that for all representable sheaves y ( P ) on (cid:0) AlgSp
LFT C , ´et (cid:1) ,L ( y ( P )) ∼ = Y ( P an ) . By [30, Remark 6.3.5.17], both inclusions H ypSh ∞ ( Top C ) ֒ → \H ypSh ∞ ( Top C )and Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) ֒ → d Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) preserve U -small colimits. Hence both compositesSh ∞ (cid:0) Aff LFT C , ´et (cid:1) ֒ → d Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) ≃ d Sh ∞ (cid:0) AlgSp
LFT C , ´et (cid:1) L −→ \H ypSh ∞ ( Top C )and Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) ( · ) top −−−−−−−→ H ypSh ∞ ( Top C ) ֒ → \H ypSh ∞ ( Top C )are U -small colimit preserving, and agree up to equivalence on every representable y ( X ), for X an affine scheme. It follows from [30, Theorem 5.1.5.6] that both compositions must infact be equivalent. However, the inclusionSh ∞ (cid:0) Aff LFT C , ´et (cid:1) ֒ → d Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) ≃ d Sh ∞ (cid:0) AlgSp
LFT C , ´et (cid:1) carries an algebraic space P to its representable sheaf y ( P ). The result now follows. (cid:3) The following lemma follows immediately from the fact that ( · ) top preserves finite limits: Lemma 3.7.
Let G be a groupoid object in sheaves of sets on the ´etale site (cid:0) Aff LFT C , ´et (cid:1) .Then applying ( · ) top level-wise produces a groupoid object in sheaves of sets on Top C , denotedby G top . Moreover, if the original groupoid G is a groupoid object in algebraic spaces, then G top is degree-wise representable, i.e. a topological groupoid. Proposition 3.8.
Let G be a groupoid object in sheaves of sets on the ´etale site (cid:0) Aff LFT C , ´et (cid:1) . Denote by [ G ] its associated stack of torsors, and denote by [ G top ] the stack of groupoids on Top C associated to G top , i.e. stack on Top C of principal G top -bundles. Then [ G ] top ≃ [ G top ] . N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES Proof.
The stack [ G ] is the stackification of the presheaf of groupoids ˜ y ( G ) which sends anaffine scheme X to the groupoid G ( X ) . Denote by N ( G ) the simplicial presheaf which is thenerve of this presheaf of groupoids. Consider the diagram∆ op N ( G ) −−−−−−−→ Psh (cid:0) Aff LFT C , Set (cid:1) ֒ → Psh (cid:0) Aff LFT C , Gpd ∞ (cid:1) . We claim that the colimit of the above functor is ˜ y ( G ) . Since colimits are computed object-wise, it suffices to show that if H is any discrete groupoid, then N ( H ) is the homotopycolimit of the diagram ∆ op N ( H ) −−−−−−−→ Set ֒ → Set ∆ op , which follows easily from the well-known fact that the homotopy colimit of a simplicialdiagram of simplicial sets can be computed by taking the diagonal. It follows then that [ G ]is the colimit of the diagram∆ op N ( G ) −−−−−−−→ Sh (cid:0) Aff LFT C , ´et (cid:1) ֒ → Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) , since ∞ -sheafification preserves colimits, as it is a left adjoint. By the same argument, wehave that [ G top ] is the colimit of the diagram∆ op N ( G top ) −−−−−−−→ Sh (
Top C ) ֒ → Sh ∞ ( Top C ) . Notice that for all n we have N ( G top ) n = ( N ( G ) n ) top . The result now follows from the fact that ( · ) top preserves colimits. (cid:3) Definition 3.9. A topological stack is a stack on Top C of the form [ G ] for G a groupoidobject in Top C . Denote the associated (2 , TopSt . Remark 3.10.
In the literature, typically there is no restriction on a topological stack tocome from a topological groupoid which is locally contractible, and such a stack is representedby its functor of points on the Grothendieck site of all topological spaces. However, the (2 , , Corollary 3.11.
The functor ( · ) top : Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) → H ypSh ∞ ( Top C ) restricts to a left exact functor ( · ) top : A lgSt LF T C → TopSt from Artin stacks locally of finite type over C to topological stacks. Up to the identification mentioned in Remark 3.10, the construction in the above corollaryagrees with that of Noohi in [34, Section 20].
The fundamental infinity-groupoid of a stack.
The following proposition will allow us to talk about homotopy types of topological stacks.
Proposition 3.12.
There is a colimit preserving functor L : H ypSh ∞ ( Top s C ) → S sending every representable sheaf y ( T ) for T a space in Top s C to its weak homotopy type.Proof. The proof is essentially the same as [10, Proposition 3.1]. By Lemma 3.1 in op. cit.,there is a functor
Top s C ֒ → Top h −→ S assigning to each space T its associated weak homotopy type. Denote this functor by π .Since Top s C is essentially small, by left Kan extension there is a colimit preserving functorLan Y π : Psh ∞ ( Top s C ) → S sending every representable presheaf Y ( T ) to the underlying weak homotopy type of T . Itfollows from the Yoneda lemma that this functor has a right adjoint R π which sends an ∞ -groupoid Z to the ∞ -presheaf R π ( Z ) : T Hom ( π ( T ) , Z ) . We claim that R π ( Z ) is a hypersheaf. To see this, it suffices to observe that if U • : ∆ op → Top s C /T is a hypercover of T with respect to the coverage of contractible open coverings, then thecolimit of the composite ∆ op U • −−−−−−−→ Top s C /T → Top s C π −→ S is π ( T ) , which follows from [12, Theorem 1.3]. We thus have that Lan y π and R π restrict toan adjunction L ⊣ ∆between H ypSh ∞ ( Top s C ) and S , so in particular, L preserves colimits. (cid:3) Corollary 3.13.
Let G be an ∞ -groupoid. Denote by ∆ ( G ) the constant presheaf on Top s C . Then ∆ ( G ) is a hypersheaf.Proof. Following the proof of the above theorem, we have that R π ( G ) is a hypersheaf. More-over, for each space T in Top s C , we have that R π ( G ) ( T ) ≃ Hom ( Y ( T ) , R π ( G )) ≃ Hom (cid:0) L ( Y ( T )) , G (cid:1) ≃ Hom ( ∗ , G ) ≃ G , since each such T is in fact contractible. (cid:3) N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES Remark 3.14.
The ∞ -category S of spaces is the terminal ∞ -topos. In particular, if C is any ∞ -category equipped with a Grothendieck topology, then the unique geometricmorphism Sh ∞ ( C ) → S has as direct image functor the global sections functorΓ : Sh ∞ ( C ) → S defined by Γ ( F ) = Hom (1 , F ) , which is the same as F (1) if C has a terminal object. Theinverse image functor is given by ∆ : S → Sh ∞ ( C )and it sends an ∞ -groupoid G to the sheafification of the constant presheaf with value G .Similarly, the unique geometric morphism H ypSh ∞ ( C ) → S has its direct image functor Γ given by the same construction as for ∞ -sheaves, and theinverse image functor ∆ assigns an ∞ -groupoid G the hypersheafification of the constantpresheaf with values G . In either case we have ∆ ⊣ Γ . In particular, Corollary 3.13 impliesthat for the ∞ -topos H ypSh ∞ ( Top s C ) we have a triple of adjunctions: L ⊣ ∆ ⊣ Γ . Although we will not prove it here, there is in fact a further right adjoint to Γ , coDisc ⊢ Γand moreover the quadruple L ⊣ ∆ ⊣ Γ ⊣ coDisc exhibits H ypSh ∞ ( Top s C ) as a cohesive ∞ -topos in the sense of [43]. Proposition 3.15.
The composite H ypSh ∞ ( Top C ) ∼ −→ H ypSh ∞ ( Top s C ) L −→ S is colimit preserving and sends a representable sheaf Y ( X ) , for X in Top C , to its underlyingweak homotopy type.Proof. By [10, Lemma 3.1], there is a functor
Top C ֒ → Top h −→ S assigning to each space X its associated weak homotopy type. Denote this functor by Π.By exactly the same proof as Proposition 3.12, by using that Top C is V -small, with V theGrothendieck universe of large sets, we construct a colimit preserving functor L : \H ypSh ∞ ( Top C ) → b S , where b S is the ∞ -category of large spaces (or large ∞ -groupoids), which sends every rep-resentable sheaf Y ( X ) to its underlying weak homotopy type. The rest of the proof isanalogous to that of Proposition 3.6. (cid:3) Definition 3.16.
We denote the composite from Proposition 3.15 byΠ ∞ : H ypSh ∞ ( Top C ) → S . For F a hypersheaf on Top C , we call Π ∞ ( F ) its fundamental ∞ -groupoid . Remark 3.17.
In light of Remark 3.14, we have that Π ∞ ⊣ ∆ ⊣ Γ , where Γ is globalsections, and ∆ assigns an ∞ -groupoid G the hypersheafification of the constant presheafwith value G . In particular, we have a formula for ∆ ( G ) , namely, if X is a space in Top C , ∆ ( G ) ( X ) ≃ Hom (Π ∞ ( X ) , G ) , that is, the space of maps from the homotopy type of X to G . The following proposition may be seen as an extension of the results of [35]:
Proposition 3.18.
Let G be a groupoid object in Top C and denote by [ G ] denote the asso-ciated stack of groupoids on Top C , i.e. the stack of principal G -bundles. Then Π ∞ ([ G ]) hasthe same weak homotopy type as B G = || N ( G ) || - the fat geometric realization of the simplicial space arising as the topologically enrichednerve of G . Proof.
We know that [ G ] is the colimit in H ypSh ∞ ( Top C ) of the diagram∆ op N ( G ) −−−−−−−→ Top C Y ֒ → H ypSh ∞ ( Top C )(as in the proof of Proposition 3.8). The result now follows from Proposition 3.15 and [10,Lemma 3.3] (cid:3) Lemma 3.19.
Let F be a hypersheaf on Top s C . Then L ( F ) is the colimit of F, i.e. thecolimit of the diagram F : ( Top s C ) op → S . Proof.
By the proof of Proposition 3.12, L factors as the composition H ypSh ∞ ( Top s C ) ֒ → Psh ∞ ( Top s C ) Lan Y π −−−−−−−−−−−−−→ S = Gpd ∞ . Note however that every space in
Top s C is contractible, so the canonical morphism π → t, to the terminal functor t : Top s C → Gpd ∞ (i.e. the functor with constant value the one point set), is an equivalence, and hence Lan Y π isleft adjoint to the constant functor t ∗ which sends an ∞ -groupoid G to the constant presheafwith value G . Since Psh ∞ ( Top s C ) = Fun (( Top s C ) op , Gpd ∞ ) , the result now follows from theuniversal property of colim −−−→ ( · ) . (cid:3) Corollary 3.20.
Let F be a hypersheaf on Top C . Then Π ∞ ( F ) is the colimit of F | Top s C . N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES The profinite homotopy type of a (pro-)stack.
Let us define the profinite versionof the homotopy type of a stack.
Definition 3.21.
We denote the composite H ypSh ∞ ( Top C ) Π ∞ −−−−−−−→ S c ( · ) −−−−−−−→ Prof ( S )by b Π ∞ . For F a hypersheaf on Top C , we call b Π ∞ ( F ) its profinite fundamental ∞ -groupoid or simply its profinite homotopy type .Let us extend the constructions of this section to pro-objects. Note that the functor( · ) top : Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) → H ypSh ∞ ( Top C )extends to a functor on pro-objects, which by abuse of notation, we will denote by the samesymbol ( · ) top : Pro (cid:0) Sh ∞ (cid:0) Aff LFT C , ´et (cid:1)(cid:1) → Pro ( H ypSh ∞ ( Top C )) . We now describe how to define the profinite homotopy type of a pro-object in H ypSh ∞ ( Top C ).First, we may extend the profinite fundamental ∞ -groupoid functor on H ypSh ∞ ( Top C ) topro-objects. This can be achieved easily by the universal property of Pro ( H ypSh ∞ ( Top C )) . Indeed, consider the functor b Π ∞ : H ypSh ∞ ( Top C ) → Prof ( S )and denote its unique cofiltered limit preserving extension, by abuse of notation, again by b Π ∞ : Pro ( H ypSh ∞ ( Top C )) → Prof ( S ) . Unwinding the definitions, we see that if lim ←− i ∈ I Y i is a pro-object in hypersheaves on Top C , then its profinite homotopy type is b Π ∞ lim ←− i ∈ I Y i ! = lim ←− i ∈ I b Π ∞ (cid:0) Y i (cid:1) . The homotopy type of Kato-Nakayama spaces.
In this subsection, we will givea formula expressing the homotopy type of the Kato-Nakayama space of a log scheme interms of algebro-geometric data. We first start by reviewing a functorial approach to Kato-Nakayama spaces which is due to Kato, Illusie and Nakayama. Let (
X, M, α ) be a log scheme,and let X an be the analytification of X , which is an object of Top C .Consider the slice category Top C /X an . If (cid:16) T p → X an (cid:17) is an object in Top C /X an , one canpullback M to T and take the section-wise group completion. In this way we obtain a sheafof abelian groups on T that we denote p ∗ M gp . Note that p ∗ M gp contains p ∗ O × X as a sub-sheafof abelian groups.Let G be any abelian topological group. If T is a topological space, we denote G T thesheaf on T of continuous maps to G equipped with the group structure coming from additionin G . Note that we have f ∗ ( G S ) = G T . Definition 3.22.
We denote by F log the presheaf of sets on Top C /X an that is defined onobjects by the following assignment: (cid:16) T p → X an (cid:17) (cid:26) morphisms of sheaves s : p ∗ M gp → S T such that s ( f ) = f | f | for f ∈ O × X (cid:27) . Theorem 3.23 ([20, Section 1.2]) . The presheaf F log is represented by X log . Since X log is an object of Top C , the functor F log completely determines X log . Moreover,we can use this functorial description to give an expression for the homotopy type of X log , as we will now show. Definition 3.24.
Denote by C KN ( X ) the following category: the objects consist of triples( T, p, s ) where • T is a topological space in Top s C , • p is a continuous map p : T → X an , • and s is a morphism of sheaves of abelian groups s : p ∗ M gp → S T such that s ( f ) = f | f | for f ∈ O × X .The morphisms ( T, p, s ) → ( S, q, r ) are continuous maps f : T → S such that f ∗ ( r ) = s. Theorem 3.25.
Let X be a log scheme. The weak homotopy type of the Kato-Nakayamaspace is that of B C KN ( X ) . Proof.
The reader may notice that C KN ( X ) is nothing but the Grothendieck construction Z Top s C (cid:0) F log | Top s C /X an (cid:1) . Notice also that
Top s C /X an → Top s C is the Grothendieck construction of Y ( X an ) | Top s C (where Y denotes the Yoneda embedding)i.e. the corresponding fibered category. Now, there is a canonical equivalence of categoriesSh ( Top s C /X an ) ≃ Sh (
Top s C ) /Y ( X an ) | Top s C , and it follows that R Top s C (cid:0) F log | Top s C /X an (cid:1) is equivalent to the Grothendieck construction of Y ( X log ) . By Proposition 3.15, we have that Π ∞ ( Y ( X log )) is the weak homotopy type of X log . The result now follows from Corollary 3.20 and [10, Corollary 3.2]. (cid:3)
Corollary 3.26.
Let X be a log scheme. The profinite homotopy type of its Kato-Nakayamaspace X log is that of the profinite completion of B C KN ( X ) . N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES Construction of the map
In all that follows X will be a fine and saturated log scheme over C that is locally of finitetype. See Appendix A for a condensed introduction to the main concepts and notations thatwe will use in this section. Our goal is to prove the following proposition: Proposition 4.1.
There is a canonical morphism of pro-topological stacks Φ X : X log → ( ∞ √ X ) top . Later (Section 6) we will show that this map induces a weak equivalence of profinitehomotopy types. The proof of Proposition 4.1 will take up the rest of this section.Our strategy will be to construct the morphism Φ X ´etale locally on X , where the logstructure has a Kato chart, and then to show that the locally defined morphisms glue togetherto give a global one. Step 1 (local case): first let us assume that X → Spec C [ P ] is a Kato chart for X ,where P is a fine saturated sharp monoid. In this case everything is very explicit: asexplained in Section A.4 in the appendix, there is an isomorphism n √ X ≃ [ X n /µ n ( P )],where X n = X × Spec C [ P ] Spec C [ n P ], the group µ n ( P ) is the Cartier dual of the cokernel of P gp → n P gp , and the action on X n is induced by the natural one on Spec C [ n P ].By following Noohi’s construction (see A.19) we see that n √ X top is canonically isomorphicto the quotient [( X n ) an /µ n ( P ) an ], where µ n ( P ) an ∼ = ( Z /n ) r . Note that the finite morphismSpec C [ n P ] → Spec C [ P ] is ´etale on the open torus Spec C [ P gp ] ⊆ Spec C [ P ], and ramifiedexactly on the complement.Now let us construct a morphism of topological stacks X log → n √ X top . By the quotientstack description of the target, this is equivalent to giving a µ n ( P ) an -torsor (i.e. principalbundle) on X log , together with a µ n ( P ) an -equivariant map to ( X n ) an .Let us look at a couple of examples first. Example 4.2.
Let X be the standard log point Spec C with log structure given by N ⊕ C × → C sending ( n, a ) to 0 n · a . Then X log ∼ = S , and n √ X top ≃ B( Z /n ). In this case the morphism S → B( Z /n ) corresponds to the ( Z /n )-torsor S → S defined by z z n . Example 4.3.
Let X be A with the divisorial log structure at the origin. Then X log ∼ = R ≥ × S and n √ X top ≃ [ C / ( Z /n )], where the morphism [ C / ( Z /n )] → ( A ) an = C is inducedby z z n , and ( Z /n ) acts by roots of unity.In this case the map R ≥ × S → [ C / ( Z /n )] corresponds to the ( Z /n )-torsor R ≥ × S → R ≥ × S defined by ( a, b ) ( a n , b n ) and the equivariant map R ≥ × S → C given by( a, b ) a · b .Note that the map R ≥ × S → R ≥ × S coincides with z z n outside of the “origin” { } × S , and this is ´etale even on the algebraic side. Over the “origin”, it is precisely thepresence of the S introduced by the Kato-Nakayama construction that allows the map tobe a ( Z /n )-torsor. This is what happens in general (see also [26, Lemma 2.2]). Proposition 4.4.
Consider the map φ log : ( X n ) log → X log induced by the morphism of logschemes φ : X n → X . The map φ log is a µ n ( P ) an -torsor, and the projection ( X n ) log → ( X n ) an is a µ n ( P ) an -equivariant map. Note (Definition A.9) that if P is a monoid, C ( P ) will denote the log analytic space(Spec C [ P ]) an with the induced natural log structure. Proof.
The action of µ n ( P ) on Spec C [ n P ] induces an action on X n , and the map X n → X is invariant. Consequently we have an induced action of µ n ( P ) an on ( X n ) log , and the map φ log : ( X n ) log → X log is invariant.Moreover, since taking ( · ) log commutes with strict base change (see Proposition A.12),we have a cartesian diagram ( X n ) log / / φ log (cid:15) (cid:15) C ( n P ) logφ P,log (cid:15) (cid:15) X log / / C ( P ) log and because the action of µ n ( P ) an on ( X n ) log comes from the one on C ( n P ) log , it suffices toprove the statement for the right-hand column.Similarly, in order to verify that ( X n ) log → ( X n ) an is µ n ( P ) an -equivariant we are reducedto checking that C ( n P ) log → C ( n P ) is µ n ( P ) an -equivariant.Now note that µ n ( P ) an is precisely the kernel of the map Hom( n P, S ) → Hom(
P, S ),So that the action of µ n ( P ) an on Hom( n P, S ) is free and transitive. It is also not hardto check that there are local sections (note that Hom( P, S ) = Hom( P gp , S ) ∼ = ( S ) k non-canonically), so that the map is a µ n ( P ) an -torsor.Furthermore, φ P,log : C ( n P ) log → C ( P ) log is the restriction map Hom( n P, R ≥ × S ) → Hom( P, R ≥ × S ), and this is the product of the two maps Hom( n P, R ≥ ) → Hom( P, R ≥ )(which is a homeomorphism) and Hom( n P, S ) → Hom(
P, S ). The action of µ n ( P ) an istrivial on the factor Hom( n P, R ≥ ) and the one given by the aforementioned inclusion asa subgroup, on the factor Hom( n P, S ). Consequently, φ P,log is a µ n ( P ) an -torsor for thenatural action, as required.The map C ( n P ) log → C ( n P ) coincides with the map Hom( n P, R ≥ × S ) → Hom( n P, C )induced by the natural map R ≥ × S → C , and therefore it is manifestly Hom( n P, S )-equivariant, and in particular µ n ( P ) an -equivariant. (cid:3) The previous proposition gives a morphism of pro-topological stacks Φ n,P : X log → n √ X top .It is clear from the construction that if n | m , then the diagram X log Φ m,P / / Φ n,P ●●●●●●●● m √ X top (cid:15) (cid:15) n √ X top is (2-)commutative, so we obtain a morphism (Φ X ) P : X log → ( ∞ √ X ) top of pro-topologicalstacks. Step 2 (compatibility of the local constructions): let us extend this local constructionto a global one. The idea is of course to use descent and glue the local constructions, andintuitively, one would expect that these local maps patch together to define a global one
N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES without incident. However, writing down all the necessary 2-categorical coherences getspretty technical quickly, and it is much cleaner to use the machinery of ∞ -categories.We will need some preliminary lemmas and constructions. Lemma 4.5.
Let X be a fine saturated log scheme over a field k with two Kato charts X → Spec k [ P ] and X → Spec k [ Q ] for the log structure. Then for every geometric point x of X , after passing to an ´etale neighborhood of x , there is a third chart X → Spec k [ R ] withmaps of monoids P → R and Q → R inducing a commutative diagram Spec k [ P ] X / / / / / / Spec k [ R ] ♣♣♣♣♣♣♣♣♣♣♣ & & ◆◆◆◆◆◆◆◆◆◆◆ Spec k [ Q ] . Proof.
We can take R = M x . There is a chart with monoid R in an ´etale neighborhood of x by A.6, and we have maps P → R and Q → R that induce a commutative diagram as inthe statement, possibly after further localization. (cid:3) Now let us define a category I of ´etale open subsets of X with a global chart: objects aretriples ( φ : U → X, P, f ) where φ : U → X is ´etale, P is a fine saturated sharp monoid and f : U → Spec C [ P ] is a chart for the log structure on U (pulled back via φ ).A morphism ( φ : U → X, P, f ) → ( ψ : V → X, Q, g ) is given by a (necessarily ´etale) map U → V over X and a morphism Q → P , such that the diagram U f / / (cid:15) (cid:15) Spec C [ P ] (cid:15) (cid:15) V g / / Spec C [ Q ]is commutative.We have two (lax) functors ( · ) log and ( n √ · ) top : I → TopSt /X an , as follows:for each A = ( φ : U → X, P, f ) ∈ I we get, via strict pullback through the chart morphism,a local model for the Kato-Nakayama space X Alog (over U ) and one for the n -th root stack n √ X Atop . We set A log = X Alog and n √ A top = n √ X Atop . The maps to X an are given by thecomposites of the projections to U an and the local homeomorphism U an → X an . The actionof these two functors on morphisms is clear.The construction in the local case (i.e. Step 1 above) gives an assignment, for each A ∈ I ,of a morphism of topological stacks α nA : A log → n √ A top . Lemma 4.6.
The family ( α nA ) gives a lax natural transformation α n : ( · ) log ⇒ ( n √ · ) top , in the sense of [15, Appendix A] . Proof.
By translating the definition, in the present case this means the following: if a : A =( φ : U → X, P, f ) → ( ψ : V → X, Q, g ) = B is a morphism in I , then the diagram A log α nA / / (cid:15) (cid:15) n √ A top (cid:15) (cid:15) α n ( a ) x (cid:0) ①①①①①①①① B log α nB / / n √ B top α n ( a ) satisfy a compatibility condition.This follows from the fact that the morphism a = ( U → V, Q → P ) gives a commutativediagram ( U n ) log / / y y sssssss (cid:15) (cid:15) ( U n ) an y y ttttttt ( V n ) log / / (cid:15) (cid:15) ( V n ) an X Alog z z ✉✉✉✉✉✉✉✉ X Blog between the two objects corresponding to the functors α nA and α nB . This gives a canonicalnatural transformation that makes the diagram X Alog (cid:15) (cid:15) α nA / / [( U n ) an /µ n ( P ) an ] = n √ X Atop (cid:15) (cid:15) α n ( a ) p x ✐✐✐✐✐✐✐✐✐✐ X Blog α nB / / [( V n ) an /µ n ( Q ) an ] = n √ X Btop . N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES Now if C = ( η : W → X, R, h ) is a third object of I with a morphism b : B → C in I ,then the fact that the diagram ( U n ) log ( U n ) an ( V n ) log ( V n ) an ( W n ) log ( W n ) an X Alog X Blog X Clog commutes implies that the composite of the two 2-cells α n ( b ) and α n ( a ) is equal to α n ( b ◦ a ). (cid:3) By composition with the natural functor
TopSt /X an ֒ → H ypSh ∞ ( Top C ) /X an to hypersheaves on Top C (see Section 3) and by abuse of notation we get a natural trans-formation of functors of ∞ -categories: I H ypSh ∞ ( Top C ) /X an . ( · ) log + + n √ · top α n (cid:11) (cid:19) Step 3 (the global case): we will now use the natural transformation α n above to constructa global map Φ X : X log → n √ X. We will first need a crucial lemma:
Lemma 4.7.
Let ι : I → H ypSh ∞ ( Top C ) be the functor sending a triple ( φ : U → X, P, f ) to U an . Then the canonical map colim −−−→ ι → X an is an equivalence. Before proving the above lemma, we will show how we may use this lemma to producethe global morphism we seek. The key idea is the following basic fact about ∞ -topoi: Proposition 4.8 ( colimits are universal ) . Let colim −−−→ i ∈ I A i → B be a morphism in an ∞ -topos E , and let C → B be another morphism. Then the canonical map colim −−−→ i ∈ I ( C × B A i ) → C × B colim −−−→ i ∈ I A i is an equivalence. The above fact is standard and is an immediate consequence of the fact that any ∞ -toposis locally Cartesian closed.Let us now see how we may complete the construction. Suppose we know that the canonicalmap colim −−−→ ι → X an is an equivalence. We can write this informally ascolim −−−→ ( φ : U → X,P,f ) U an ∼ −→ X an . Consider the morphism X log → X an . Then since colimits are universal we have that thefollowing is a pullback diagram:colim −−−→ ( φ : U → X,P,f ) U an × X an X log / / (cid:15) (cid:15) X log (cid:15) (cid:15) colim −−−→ ( φ : U → X,P,f ) U an ∼ / / X an . It follows that top map colim −−−→ ( φ : U → X,P,f ) U an × X an X log → X log is also an equivalence. However, notice that we have a canonical identification U an × X an X log ∼ = U log , hence X log ≃ colim −−−→ ( φ : U → X,P,f ) U log = colim −−−→ ( · ) log . By a completely analogous argument, one sees that n √ X top ≃ colim −−−→ ( φ : U → X,P,f ) n √ U top = colim −−−→ n √ · top . For each n , the global map is then defined to becolim −−−→ α n : colim −−−→ ( · ) log → colim −−−→ n √ · top . Just as in the local case, one easily sees that the mapscolim −−−→ α n : X log → n √ X top assemble into a morphism of pro-objectsΦ X : X log → ∞ √ X top . It is immediate from the construction that this map agrees locally with the map constructedin Step 1. In the next sections we will prove that Φ X induces an equivalence of profinitespaces.To finish the proof of the existence of the above map, it suffices to prove Lemma 4.7.Without further ado, we present the proof below. N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES Proof of Lemma 4.7.
Equip I with the following Grothendieck topology: A collection ofmorphisms (( φ i : U i → X, P i , f i ) → ( φ : U → X, P, f )) i will be a covering family if the induced family( U i → U ) i is an ´etale covering family. Note that there is a canonical morphism of sites F : I → X ´ et from I to the small ´etale site of X. Moreover, by Lemma 4.5, one easily checks that F satisfies the conditions of the Comparison Lemma of [27, p. 151], so the induced geometricmorphism of topoi Sh ( I ) → Sh ( X ´ et )is an equivalence. It then follows from [23, Theorem 5] and [30, Proposition 6.5.2.14] thatthe induced geometric morphism between the respective ∞ -topoi of hypersheaves H ypSh ∞ ( I ) → H ypSh ∞ ( X ´ et )is an equivalence. By Remark 3.4, the analytification functor is the inverse image part of ageometric morphism f : H ypSh ∞ ( Top C ) → Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) . By [30, Proposition 6.5.2.13], there is an induced geometric morphism˜ f : H ypSh ∞ ( Top C ) → H ypSh ∞ (cid:0) Aff LFT C , ´et (cid:1) . By left Kan extension of the canonical functor X ´ et → H ypSh ∞ (cid:0) Aff LFT C , ´et (cid:1) /X which sends each ´etale open U → X to itself, one produces a colimit preserving functor ω : H ypSh ∞ ( X ´ et ) → H ypSh ∞ (cid:0) Aff LFT C , ´et (cid:1) /X. Consider the composite H ypSh ∞ ( I ) ≃ H ypSh ∞ ( X ´ et ) ω −→ H ypSh ∞ (cid:0) Aff LFT C , ´et (cid:1) /X → H ypSh ∞ (cid:0) Aff LFT C , ´et (cid:1) ˜ f ∗ −→ H ypSh ∞ ( Top C ) , where H ypSh ∞ (cid:0) Aff LFT C , ´et (cid:1) /X → H ypSh ∞ (cid:0) Aff LFT C , ´et (cid:1) is the canonical projection. Denotethe composite by Θ . The functor Θ is colimit preserving as it is the composite of colimitpreserving functors, and unwinding definitions, one sees that the composite I y −→ H ypSh ∞ ( I ) Θ −→ H ypSh ∞ ( Top C )is canonically equivalent to ι. It follows that there is a canonical equivalencecolim −−−→ ι ≃ Θ (cid:16) colim −−−→ y (cid:17) . But y is strongly generating, so by the proof of [8, Lemma 5.3.5], the colimit of y is theterminal object. Unwinding the definitions, one sees that the terminal object gets sent to X an by Θ . This completes the proof. (cid:3) The topology of log schemes
This section contains preliminaries about some topological properties of fine saturated logschemes locally of finite type over C , the Kato-Nakayama space and the root stacks.5.1. Stratified fibrations.
The following proposition is a consequence of the material inSection A.6.Recall that if X is a fine saturated log scheme locally of finite typer over C there is astratification R = { R n } n ∈ N of X , the rank stratification (Definition A.25), given by R n = { x ∈ X | rank Z M gp x ≥ n } . Proposition 5.1.
The Kato-Nakayama space X log , the topological root stacks m √ X top andthe topological infinite root stack ∞ √ X top are stratified fibrations over X an with respect tothe stratification R , i.e. they are fibrations over the strata ( S n ) an = ( R n \ R n +1 ) an of thestratification R an .Proof. All constructions are compatible with arbitrary base change along strict morphisms,so X log | ( S n ) an ∼ = ( S n ) log and m √ X | S n ≃ m p S n where m can be ∞ , and S n has the log structure pulled back from X . It suffices then toshow that the two maps ( S n ) log → ( S n ) an and ( m √ S n ) top → ( S n ) an are fibrations over S n .Let us cover ( S n ) an with open subsets over which the sheaf M is constant, and recall thatby definition of S n it will have rank n . We can choose such opens in order to have a cartesiandiagram ( S n ) log / / (cid:15) (cid:15) (Spec k [ P ]) log (cid:15) (cid:15) ( S n ) an / / (Spec k [ P ]) an over each of them, where the bottom horizontal arrow sends everything to the vertex v P (as in the proof of A.27). It follows that ( S n ) log ∼ = ( S ) n × ( S n ) an , and that the map( S n ) log → ( S n ) an is identified with the projection. The factor ( S ) n is the fiber of the map(Spec k [ P ]) log → (Spec k [ P ]) an over the point v P .The analogous diagram ( m √ S n ) top / / (cid:15) (cid:15) m p Spec k [ P ] top (cid:15) (cid:15) ( S n ) an / / (Spec k [ P ]) an shows the same conclusion for root stacks. In this case we get an isomorphism ( m √ S n ) top ≃ X × ( S n ) an , where X is the fiber of the map m p Spec k [ P ] top → (Spec k [ P ]) an over the vertex v P . (cid:3) N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES We need a similar (local) statement for groupoid presentations of the root stacks.Take x ∈ X , and an open ´etale neighborhood U → X of x where there is a global chart U → Spec C [ P ] for the log structure, where P is fine, saturated and sharp. Then we have aquotient stack presentation for the topological n -th root stack n √ U top ≃ ( n √ X | U ) top for every n (see the discussion preceding Proposition A.18). Let us denote by G ( n ) the simplicialtopological space associated with this quotient presentation. There are compatible maps G ( m ) → G ( n ) whenever n | m , and the whole system gives a (simplicial) presentation forthe topological infinite root stack ∞ √ U top .Explicitly, the simplicial space G ( n ) is obtained from the action of µ n ( P ) on the scheme U n = U × Spec C [ P ] Spec C [ n P ] (see the local description of the root stacks in A.4), so that G ( n ) k ∼ = ( U n × µ n ( P ) × · · · × µ n ( P )) an where there are k copies of µ n ( P ), and the map G ( n ) k → U an is the composite of theprojection to ( U n ) an followed by the map ( U n ) an → U an . Proposition 5.2.
Every x ∈ U an has arbitrarily small neighborhoods over which for every n and k the map G ( n ) k → U an is a product over U an ∩ ( S r ) an , where x ∈ ( S r ) an In particular, for every n and k the topological space G ( n ) k is a stratified fibration over U an . Proof.
Note first of all that since the map U → Spec C [ P ] is strict, the rank stratification ofSpec C [ P ] with its natural log structure is pulled back to the rank stratification of U , in theobvious sense.Moreover from the cartesian diagram( U n ) an / / (cid:15) (cid:15) C ( n P ) (cid:15) (cid:15) U an / / C ( P )and from the fact that G ( n ) k → U an is the projection G ( n ) k ∼ = ( U n × µ n ( P ) ×· · ·× µ n ( P )) an → ( U n ) an followed by ( U n ) an → U an , we see that it suffices to prove that the map π : C ( n P ) =(Spec C [ n P ]) an → C ( P ) = (Spec C [ P ]) an is a stratified fibration. The proof will show thatfor a stratum S we can find an open subset V ⊆ C ( P ) such that the map π is a productover V ∩ S for every n .Let us pick φ ∈ C ( P ) = Hom( P, C ), and call p , . . . , p l the (finitely many) indecomposableelements of P (cfr. [37, Proposition 2.1.2]). Assume (by reordering) that the first h of thoseget sent to 0 by φ , and the last ones are sent to non-zero complex numbers. Call r the rankof the group associated to the quotient P / h p i | i = h + 1 , . . . , l i (i.e. the rank of the log structureof C ( P ) at φ ).The stratum of the rank stratification of C ( P ) to which φ belongs will be then S r , theset of points of C ( P ) where the log structure has rank exactly equal to r . It’s clear that φ actually belongs to the open subset S φ of S r consisting of the morphisms ψ ∈ Hom( P, C )such that ψ ( p i ) = 0 for 1 ≤ i ≤ h and ψ ( p i ) = 0 for h < i ≤ l . Note also that the same condition on images of indecomposables of n P will determine asubset S ′ φ ⊆ C ( n P ) = Hom( n P, C ) (of those morphisms such that the image of p i n is zeroexactly if 1 ≤ i ≤ h ), that a moment’s reflection will show to be exactly the preimage π − ( S φ ). Let us check that we can choose a neighborhood of φ in C ( P ) over which therestriction of π : π − ( S φ ) → S φ is a product.For each i = h + 1 , . . . , l let us choose a small open disk D i around φ ( p i ) in C that doesnot contain the origin, and for i = 1 , . . . , h let D i be a small open disk around the origin.These define an open neighborhood W of φ in C ( P ), made up of those functions ψ such that ψ ( p i ) ∈ D i for every i .Let us also choose an n -th root n p φ ( p i ) of the non-zero complex number φ ( p i ) for i = h + 1 , . . . , l . There are a finite number of such choices, and there is a subset of those choicesfor which the homomorphism n P → C given by p i n n p φ ( p i ) is well-defined (note that thisassignment might not give a well-defined homomorphism due to the relations among theindecomposable elements of the monoid P ). Let us call A this set of “good” choices.Any element of A determines for each i = h + 1 , . . . , l an n -th root function n √− i definedon the small disk D i . Let us define a map W ∩ S φ → π − ( W ∩ S φ ) ⊆ Hom( n P, C ) bysending ψ to the morphism defined by p i n n p ψ ( p i ) i . This is a section of the projection π − ( W ∩ S φ ) → W ∩ S φ , and one can check that this induces a homeomorphism W ∩ S φ × A ∼ = π − ( W ∩ S φ ), where A is seen as a discrete set. We leave the details to the reader.These arguments are uniform in n ∈ N , so the open subset W that we identified will workfor any n . (cid:3) A system of open neighborhoods for X an . In this subsection we will prove thefollowing crucial lemma:
Lemma 5.3.
For all x ∈ X an there exists a fundamental system of contractible analytic openneighborhoods U x of x with global charts f : U → (Spec C [ P ]) an for U ∈ U x , such that: (1) the map f sends x into the vertex of (Spec C [ P ]) an (i.e. the maximal ideal generatedby all non-zero elements of P ), and (2) each of the following maps is a weak homotopy equivalence: ( X log ) x → X log | U and ( G ( n ) i ) x → ( G ( n ) i ) | U where { G ( n ) } n ∈ N is the family of topological groupoid presentations for the topological n -th root stack coming from the chart f , as in Proposition 5.2. First of all we review some standard facts on triangulations and open covers. Let M be atopological space equipped with a triangulation T . Denote V T the set of vertices of T . If f is a simplex of T , we denote s ( f ) the union of the relative interiors of the simplices of T thatcontain f . We call s ( f ) the star of f . Note that s ( f ) is a contractible open subset of M . If v is a vertex of T , we set U v .. = s ( v ). The star of a simplex f is naturally stratified by thesimplices containing f : the strata are the relative interiors of the simplices containing f . N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES We say that a subspace of R n is a cone if it is invariant under the action of R > by rescaling.We say that a cone is linear if it can be expressed as an intersection of linear spaces andlinear half-spaces. Lemma 5.4.
Let v be in V T . Then there exists an N ∈ N such that U v can be embeddedas a linear cone in R N . Further we can choose this embedding in such a way that, for allsimplices f containing v , s ( f ) ⊂ U v is mapped to a linear subcone.Proof. Let v . . . v N be the one dimensional simplices that contain v , and let e , . . . , e N bethe standard basis of R N . If I is a subset of { , . . . , N } we denote O I .. = { Σ i ∈ I α i e i | α i ≥ } ⊂ R N . Every simplex ∆ containing v determines a subset I ∆ of { , . . . , N } in the following way: i belongs to I ∆ if and only if ∆ contains v i . We obtain an embedding of U v into R N byconsidering a piecewise linear homeomorphism U v ≃ [ v ∈ ∆ O I ∆ . This embedding has all the properties claimed by the lemma. (cid:3)
Lemma 5.5.
Let x be in M , and let f be the lowest dimensional simplex such that x belongsto f . Then there exists a system of open neighborhoods U x of x such that all U in U x havethe following properties: (1) U is contractible. (2) U does not intersect simplices of T that do not contain f .Proof. Let v be a vertex incident to f . By Lemma 5.4 the open neighborhood U v can beembedded as a linear cone in R n in such a way that s ( f ) ⊂ U v is a linear subcone. Equip R n with a Euclidean metric. Then U x can be obtained by intersecting s ( f ) with a system ofopen neighborhoods given by open balls in R n centered at x . (cid:3) Next we turn to the log scheme X . Let x be in X an . Since we are interested in constructinga system of open neighborhoods for x we can assume, by ´etale localizing around x , that • X is affine, and • that we have a global chart f : X → Spec C [ P ], where P = M x (see A.6), whichsends x to the vertex of Spec C [ P ].The fact that X is affine is key in order to produce triangulations, which we do in lemma5.6. By Lemma A.23 the log structure determines a stratification R X of X . Lemma 5.6.
There exists a triangulation T X of X that refines R X .Proof. The existence of triangulations refining stratifications of affine schemes goes back toLojasiewicz [28]. See also Shiota’s work [45] for a more recent reference. (cid:3)
By Lemma 5.5, the triangulation T X gives us a system of open neighborhoods U x of x in X an satisfying the two properties stated there. We claim that U x has all the propertiesrequired by lemma 5.3. Note that, since we assumed without loss of generality that X is affine and that has a global chart to Spec C [ P ] sending x to the vertex of Spec C [ P ], we onlyneed to prove that property (3) holds. We do this next.The following lemma was proved in [42]. Lemma 5.7 (Lemma 3.25 [42]) . Let W and W be locally compact and locally contractibleHausdorff spaces. Let p : W → W be a continuous map, and let K ⊂ W be a closeddeformation retract. Suppose that the restriction p − ( W − K ) → W − K is homeomorphicto the projection from a product F × ( W − K ) → W − K . Then K .. = p − ( K ) is adeformation retract of W . We will actually need a slight variant of Lemma 5.7. Assume that W − K decomposesas a finite disjoint union of m components, that we denote ( W − K ) i , W − K = i = m [ i =1 ( W − K ) i . Then the claim still holds if the restriction p − ( W − K ) → W − K is homeomorphic tothe projection from a disjoint union of products i = m [ i =1 F i × ( W − K ) i → i = m [ i =1 ( W − K ) i . This stronger statement is proved exactly as Lemma 5.7: and in fact, follows from it throughan induction on the number of connected components of W − K .We conclude the proof of Lemma 5.3 by showing that the following proposition holds. Proposition 5.8.
For all U in U x each of the following maps is a weak homotopy equivalence: ( X log ) x → X log | U , ( G ( n ) i ) x → ( G ( n ) i ) | U where { G ( n ) } n ∈ N is the family of topological groupoid presentations for the topological n -throot stack coming from the chart f .Proof. The proof is the same for both X log and G ( n ) i . The argument relies exclusively on thefact that X log and G ( n ) i give stratified fibrations on X an with respect to the stratification R X . To avoid repetition, we prove the statement only for X log but the argument remainsvalid if we substitute G ( n ) i in all occurences of X log .Let f be the lowest dimensional simplex of T X such that x lies on f . Recall from the proofof Lemma 5.5 that, in order to define U x , we pick a vertex v of the triangulation T X that isincident to f . By construction, U is an open subset of U v . Thus U carries a stratificationwhich is obtained by restricting to it the stratification on U v by the simplices containing v .For all k ∈ N , denote U k ⊂ U the k -skeleton of U : that is, U k is the union of the strata ofdimension less than or equal to k . Note that if k < dim ( f ), U k is empty, and is contractibleif dim ( f ) ≤ k . Further if dim ( f ) ≤ k ≤ k ′ , U k is a strong deformation retract of U k ′ . Indeedboth U k and U k ′ are CW complexes (up to compactifying), and any contractible subcomplexof a contractible CW complex is a strong deformation retract, see e.g. [32, Lemma 1.6].We prove next that if dim ( f ) ≤ k −
1, the map X log | U k − → X log | U k N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES is a deformation retract. Note that U k − U k − is equal to the disjoint union of k -dimensionalstrata. That is U k − U k − can be written as a disjoint union of m components U k − U k − = i = m [ i =1 ( U k − U k − ) i . The restriction of the map X log | U → X an | U to each stratum of U is a principal bundle.Indeed, the stratification on U is finer that the restriction to U of R X . Further, it is atrivializable principal bundle, since the strata are paracompact Hausdorff and contractible.Thus the restriction X log | U k − U k − → U k − U k − is homeomorphic to a projection from a disjoint union of products X log | U k − U k − ≃ i = m [ i =1 F i × ( U k − U k − ) i → i = m [ i =1 ( U k − U k − ) i . We have showed that the map U k − → U k is a deformation retract. We apply Lemma 5.7, orrather the variant that was discussed immediately after the statement of Lemma 5.7, (notethat X log is locally compact Hausdorff and locally contractible by Proposition A.13), anddeduce that the map X log | U k − → X log | U k is also a deformation retract, as we claimed.There exists a N ∈ N such that U N = U . By applying recursively the retractions that wehave constructed in the previous paragraph, we obtain a deformation retract X log | U dim ( f ) → X log | U . By property (2) of Lemma 5.5, U dim ( f ) is connected. Further it is contractible and para-compact, and thus X log | U dim ( f ) is homeomorphic to a product F × U dim ( f ) . This implies thatthere are homotopy equivalences( X log ) | x ≃ F × { x } ∼ ֒ −→ F × U dim ( f ) ≃ X log | U dim ( f ) and this concludes the proof. (cid:3) The equivalence
Finally, in this section we will prove the main result of this paper, namely that there is anequivalence b Π ∞ (Φ X ) : b Π ∞ ( X log ) → b Π ∞ (cid:16) ∞ √ X top (cid:17) of profinite spaces, where b Π ∞ is the “profinite homotopy type” functor defined in 3.3, andΦ X is the morphism of pro-topological stacks constructed in Section 4.The main idea is to use the basis of open subsets constructed in Lemma 5.3 to producea suitable hypercover of X an and to use this to reduce to checking that one has a profinitehomotopy equivalence along fibers. First, we will need a few more technical lemmas.The following lemma makes precise in what way one can glue profinite spaces togetherusing hypercovers: Lemma 6.1.
Let X be a hypersheaf in H ypSh ∞ ( Top C ) . Let I be a cofiltered ∞ -categoryand let f • : I → H ypSh ∞ ( Top C ) / X be an I -indexed pro-system with associated pro-object lim ←− i ∈ I ( f i : Y i → X ) . Let U • : ∆ op → H ypSh ∞ ( Top C ) / X be a hypercover of X . For each i, denote by f ∗ i U • the pullback of the hypercover U • to ahypercover of Y i . Consider the underlying pro-object lim ←− i ∈ I Y i in H ypSh ∞ ( Top C ) . Then thereis a canonical equivalence of profinite spaces b Π ∞ lim ←− i ∈ I Y i ! ≃ colim −−−→ n ∈ ∆ op " b Π ∞ lim ←− i ∈ I f ∗ i U n ! , where b Π ∞ : Pro ( H ypSh ∞ ( Top C )) → Prof ( S ) is the functor constructed in Section 3.3.Proof. It suffices to show that for every π -finite space V, there is a canonical equivalenceHom Prof( S ) colim −−−→ n ∈ ∆ op " b Π ∞ lim ←− i ∈ I f ∗ i U n ! , j ( V ) ! ≃ Hom
Prof( S ) b Π ∞ lim ←− i ∈ I Y i ! , j ( V ) ! which is natural in V. We have thatHom
Prof( S ) colim −−−→ n ∈ ∆ op " b Π ∞ lim ←− i ∈ I f ∗ i U n ! , j ( V ) ! ≃ lim ←− n ∈ ∆ " colim −−−→ i ∈ I op Hom S (Π ∞ f ∗ i U n , V ) . Notice that V is k -truncated for some k , and hence so is j ( V ) by Proposition 2.13. Sincefiltered colimits of k -truncated spaces are k -truncated, it follows that for all n, colim −−−→ i ∈ I op Hom S (Π ∞ f ∗ i U n , V )is k -truncated. By Lemma 2.21, it then follows thatHom Prof( S ) colim −−−→ n ∈ ∆ op " b Π ∞ lim ←− i ∈ I f ∗ i U n ! , j ( V ) ! ≃ lim ←− n ∈ ∆ ≤ k " colim −−−→ i ∈ I op Hom S (Π ∞ f ∗ i U n , V ) . By using that filtered colimits commute with finite limits, we then have that this is in turnequivalent to colim −−−→ i ∈ I op " lim ←− n ∈ ∆ ≤ k Hom S (Π ∞ f ∗ i U n , V ) . Again by Lemma 2.21 this is equivalent tocolim −−−→ i ∈ I op " lim ←− n ∈ ∆ Hom S (Π ∞ f ∗ i U n , V ) . N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES Finally, we have the following string of natural equivalences:colim −−−→ i ∈ I op " lim ←− n ∈ ∆ Hom S (Π ∞ f ∗ i U n , V ) ≃ colim −−−→ i ∈ I op Hom S colim −−−→ n ∈ ∆ op Π ∞ f ∗ i U n , V ! ≃ colim −−−→ i ∈ I op Hom S Π ∞ colim −−−→ n ∈ ∆ op f ∗ i U n , V ! ≃ colim −−−→ i ∈ I op Hom S (cid:0) Π ∞ Y i , V (cid:1) ≃ Hom
Prof( S ) b Π ∞ lim ←− i ∈ I Y i ! , j ( V ) ! . (cid:3) Let X be a log scheme. Denote by U the basis of contractible open subsets of X an givenby Lemma 5.3. Lemma 6.2.
There is a hypercover U • : ∆ op → Top C /X an such that for all n, the map U n → X an is isomorphic to the coproduct of inclusions of openneighborhoods in the basis U , and all the structure maps are local homeomorphisms.Proof. Using standard techniques, since U is a basis for the topology of X an we can constructa split hypercover satisfying the above by induction (cfr. [11]). (cid:3) Remark 6.3.
The image under the Yoneda embedding of the hypercover of topologicalspaces U • just constructed is a hypercover of Y ( X an ) in the ∞ -topos H ypSh ∞ ( Top C ) . Wewill abuse notation by identifying the two.We now prove our main result:
Theorem 6.4.
Let X be a fine saturated log scheme locally of finite type over C . The inducedmap b Π ∞ (Φ X ) : b Π ∞ ( X log ) ∼ −−−−−−−→ b Π ∞ (cid:16) ∞ √ X top (cid:17) is an equivalence of profinite spaces.Proof. Consider now the hypercover U • of X an just constructed. Then each U n = ` α V α where each V α is in U . Let us restrict to one such V = V α . Since V is in U , there exists an x ∈ V such that ( X log ) x → X log | V is a weak homotopy equivalence, and such that there isa Kato chart U → Spec C [ P ], with U → X ´etale, such that U an → X an admits a section σ over V, and with the property that the composite V σ −→ U an → (Spec C [ P ]) an carries x to the vertex point of the toric variety Spec C [ P ] . Let us fix this x , and call it the center of V . Suppose that the monoid P has rank k, then the log structure at x also has rank k . Moreover, the fiber of the map V n .. = V × (Spec C [ P ]) an (cid:18) Spec C (cid:20) n P (cid:21)(cid:19) an → V over x consists of a single point (see [21, Lemma 1.2]).Let us fix an n , then we have that n √ X top | V ≃ h ( Z /n Z ) k ⋉ V n i = h V n / ( Z /n Z ) k i . Hence our groupoid presentation G ( n ) for n √ X top | V guaranteed by Proposition 5.2 is thetopological action groupoid ( Z /n Z ) k ⋉ V n . This groupoid admits a continuous functor to V (viewing V as a topological groupoid with only identity arrows) which on objects is simplythe canonical map V n → V. Similarly, regard the one-point space ∗ also as a topologicalgroupoid, and consider the canonical map ∗ → V picking out x. Since V and ∗ have no non-identity arrows, the lax fibered product of topo-logical groupoids ∗ × (2 , V (cid:16) ( Z /n Z ) k ⋉ V n (cid:17) is equivalent to the strict fibered product ∗ × V (cid:16) ( Z /n Z ) k ⋉ V n (cid:17) which is canonically equivalent to the action groupoid( Z /n Z ) k ⋉ ( V n ) x , where ( V n ) x is the fiber over V n → V. Since this fiber consists of a single point, we concludethat the lax fibered product may be identified with ( Z /n Z ) k , where we are identifying thegroup ( Z /n Z ) k with its associated 1-object groupoid.Consider the continuous functor of topological groupoids( Z /n Z ) k ≃ ∗ × (2 , V (cid:16) ( Z /n Z ) k ⋉ V n (cid:17) → ( Z /n Z ) k ⋉ V n . This induces a map of simplicial topological spaces between their simplicially enriched nerves N (cid:16) ( Z /n Z ) k (cid:17) → N (cid:16) ( Z /n Z ) k ⋉ V n (cid:17) . By Lemma 5.3, this map is degree-wise a weak homotopy equivalence. It follows fromProposition 3.18 and [10, Lemma 3.2] that the induced map B (cid:16) ( Z /n Z ) k (cid:17) ≃ Π ∞ (cid:16) ( Z /n Z ) k ⋉ ∗ (cid:17) → Π ∞ (cid:16)h ( Z /n Z ) k ⋉ V i(cid:17) ≃ Π ∞ (cid:16) n √ X top | V (cid:17) is an equivalence in S . Since the topological groupoid presentations for n √ X top constructedin Section 5.1 are compatible with the natural maps m √ X top → n √ X top when n | m , it followsthat we have a natural identification of b Π ∞ (cid:16) ∞ √ X top | V (cid:17) ≃ lim ←− n B (cid:16) ( Z /n Z ) k (cid:17) N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES in Prof ( S ) . Consider the pro-system of finite groups n ( Z /n Z ) k . This is the k th Cartesian power of the pro-system n ( Z /n Z ) , which is simply b Z . By Proposition 2.22, it follows that b Π ∞ (cid:16) ∞ √ X top | V (cid:17) ≃ B (cid:16)b Z k (cid:17) , and hence by Proposition 2.20, we have that b Π ∞ (cid:16) ∞ √ X top | V (cid:17) ≃ B (cid:16)c Z k (cid:17) . We also have that ( X log ) x ∼ = (cid:0) S (cid:1) k . It follows that Π ∞ ( X log | V ) ≃ Π ∞ (cid:0) S (cid:1) k ≃ B (cid:0) Z k (cid:1) , and so b Π ∞ ( X log | V ) ≃ \ B ( Z k ) . Since Z k is a finitely generated free abelian group, it is good in the sense of Serre in [44]. Itfollows from [40, Proposition 3.6] and Theorem 2.19 that the canonical map \ B ( Z k ) → B (cid:16)c Z k (cid:17) is an equivalence of profinite spaces, hence b Π ∞ ( X log | V ) ≃ B (cid:16)c Z k (cid:17) . It now follows that b Π ∞ (cid:16) ∞ √ X top | V (cid:17) ≃ b Π ∞ ( X log | V ) , which is a local version of our statement.Now let us globalize using the hypercover U • . For each n , denote by q n the natural map q n : n √ X top → X an . Since b Π ∞ preserves colimits, it follows that the induced mapcolim −−−→ l ∈ ∆ op b Π ∞ ◦ τ ∗ U l → colim −−−→ l ∈ ∆ op b Π ∞ ◦ lim ←− n q ∗ n U l ! is an equivalence of profinite spaces, where τ is the canonical map τ : X log → X an . However,colim −−−→ l ∈ ∆ op b Π ∞ ◦ τ ∗ U l ≃ b Π ∞ colim −−−→ l ∈ ∆ op τ ∗ U l ! ≃ b Π ∞ ( X log ) , since τ ∗ U • is a hypercover of X log . Finally, by Lemma 6.1,colim −−−→ l ∈ ∆ op b Π ∞ ◦ lim ←− n q ∗ n U l ! ≃ b Π ∞ lim ←− n n √ X top ! = b Π ∞ (cid:16) ∞ √ X top (cid:17) . (cid:3) The profinite homotopy type of a log scheme
We conclude this paper by defining the profinite homotopy type of an arbitrary log schemeover a ground ring k, by using the notion of ´etale homotopy type.´Etale homotopy theory, as originally introduced by Artin and Mazur in [4], is a way ofassociating to a suitably nice scheme a pro-homotopy type. In this seminal work they proveda generalized Riemann existence theorem: Theorem 7.1 ([4, Theorem 12.9]) . Let X be scheme of finite type over C , then the profinitecompletion of the ´etale homotopy type of X agrees with the profinite completion of X an . In light of the above theorem, the ´etale homotopy type of a complex scheme of finite typegives a way of accessing homotopical information about its analytic topology by using onlyalgebro-geometric information, and for a setting where the analytic topology is not available,such as a scheme over an arbitrary base, the profinite completion of its ´etale homotopy typeserves as a suitable replacement.In the original work of Artin and Mazur, for X a locally Noetherian scheme, one associatesa pro-object in the homotopy category of spaces Ho ( S ) . This definition was later refined byFriedlander in [14] to produce a pro-object in the category
Set ∆ op of simplicial sets, anda generalized Riemann existence theorem is also proven in this context. In recent work ofLurie [29], the ´etale homotopy type of an arbitrary higher Deligne-Mumford stack is definedby using shape theory to produce an object in the ∞ -category Pro ( S ) (in fact the definitionin op. cit. is for spectral Deligne-Mumford stacks - analogues of Deligne-Mumford stacksfor algebraic geometry over E ∞ -rings), and Hoyois has recently proven that up to profinitecompletion, this definition agrees with that of Friedlander for a classical locally Noetherianscheme in [19]. See also recent work of Barnea, Harpaz, and Horel in [5].In recent work of the first author [9], the ´etale homotopy type of an arbitrary higher stackon the ´etale site of affine k -schemes is defined, and is shown to agree with the definition ofLurie when restricted to higher Deligne-Mumford stacks. In particular, there is shown to bea functor b Π ´ et ∞ : Sh ∞ (cid:0) Aff LFT k , ´et (cid:1) → Prof ( S )associating to a higher stack X on the ´etale site of affine k -schemes of finite type a profinitespace b Π ´ et ∞ ( X ) called its profinite homotopy type , and an even more generalized Riemannexistence theorem is proven: Theorem 7.2. [9, Theorem 4.13]
Let X be higher stack on affine schemes of finite type over C , then there is a canonical equivalence of profinite spaces b Π ´ et ∞ ( X ) ≃ b Π ∞ ( X top ) N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES between its profinite ´etale homotopy type and the profinite homotopy type of its underlyingtopological stack X top in the sense of Theorem 3.1. Now let X be a log scheme locally of finite type over C . Its infinite root stack ∞ √ X is apro-object in Sh ∞ (cid:0) Aff LFT C , ´et (cid:1) . Notice that the functor b Π ´ et ∞ canonically extends to a functor b Π ´ et ∞ : Pro (cid:0) Sh ∞ (cid:0) Aff LFT k , ´et (cid:1)(cid:1) → Prof ( S ) . In light of the above theorem, we conclude that there is a canonical equivalence of profinitespaces b Π ´ et ∞ (cid:16) ∞ √ X (cid:17) ≃ b Π ∞ (cid:16) ∞ √ X top (cid:17) between the profinite ´etale homotopy type of the infinite root stack ∞ √ X and the profi-nite homotopy type of the underlying topological stack of the infinite root stack ∞ √ X top . Combining this with Theorem 6.4 yields the following theorem.
Theorem 7.3.
Let X be a log scheme locally of finite type over C . Then the following threeprofinite spaces are canonically equivalent: i) The profinite completion d X log of its Kato-Nakayama space. ii) The profinite homotopy type b Π ∞ (cid:16) ∞ √ X top (cid:17) of the underlying topological stack of itsinfinite root stack ∞ √ X. iii) The profinite ´etale homotopy type b Π ´ et ∞ (cid:16) ∞ √ X (cid:17) of its infinite root stack ∞ √ X. In light of the above theorem, we make the following definition:
Definition 7.4.
Let X be a log scheme over a ground ring k. Then the profinite homotopytype of X is the profinite ´etale homotopy type of its infinite root stack ∞ √ X. Appendix
A.In this appendix we gather some definitions and results about log schemes, analytification,the Kato-Nakayama space, root stacks and topological stacks.A.1.
Log schemes.
Log (short for “logarithmic”) schemes were first defined and studiedsystematically in [24]. A modern introduction (with a view towards moduli theory) can befound in [2].
Remark A.1.
We will give definitions and facts in the algebraic category, but we will applythem to the complex-analytic context as well. The only difference is that instead of the ´etaletopology we will be using the analytic topology.
Definition A.2. A log scheme is a scheme X with a sheaf of monoids M on the small´etale site X ´ et and a homomorphism α : M → O X of sheaves of monoids, where O X is seenas a monoid with respect to multiplication of regular functions, such that α induces anisomorphism α | α − ( O × X ) : α − ( O × X ) → O × X . Note that the last condition gives us a canonical embedding O × X ֒ → M as a subsheaf ofgroups.We denote a log scheme by ( X, M, α ) or sometimes simply by X . Example A.3. • Any scheme X is a log scheme with M = O × X and α the inclusion. This is the triviallog structure on X . • Any effective Cartier divisor D ⊆ X induces a log structure, by taking M to be thesubsheaf of O X given by functions that are invertible outside of D . • If P is a monoid, the spectrum of the monoid algebra X P .. = Spec k [ P ] has a naturallog structure. The sheaf M is obtained by considering the natural map P → k [ P ] =Γ( O X P ) and taking the “associated log structure” (see below for a few more details).Log structures can be pulled back and pushed forward along morphisms of schemes. Inparticular • any open subscheme of a log scheme can be equipped with the restriction of the logstructure, • if we have a morphism of schemes f : X → Spec k [ P ] we get an induced log structureon X . This happens in the following way: f gives a morphism of monoids P → O X ( X ), that induces e α : P → O X where P is the constant sheaf. It is typically nottrue that e α induces an isomorphism between e α − O × X and O × X , but there is a procedureto fix the behaviour of the units, and this produces a log structure α : M → O X . See[24, Example 1.5] for details. Remark A.4.
We remark that in the situation of the last bullet, the quotient M/ O × X isobtained from P by locally “killing the sections of P that become invertible in O X ”, so inparticular all the stalks of M/ O × X are quotients of the monoid P .We consider only coherent log structures, which are those that, ´etale locally, come bypullback from the spectrum of the monoid algebra of a monoid. Definition A.5.
A log scheme X is quasi-coherent if there is an ´etale cover U i of X ,monoids P i and morphisms of log schemes f i : U i → Spec k [ P i ] that are strict, i.e. the logstructure on U i is pulled back from Spec k [ P i ] via f i . The monoid P i and the map f i are a chart for the log structure over U i .A log scheme X is coherent (resp. fine , resp. fine and saturated ) if the monoids P i above can be taken finitely generated (resp. finitely generated and integral, resp. finitelygenerated, integral and saturated).A morphism such as f i in the definition above that identifies the pullback of the logstructure on the target with the one of the source will be called strict .We are interested only in fine and saturated log schemes. Proposition A.6 ([38, Proposition 2.1]) . Let X be a fine saturated log scheme and x ageometric point. Then there exists an ´etale neighborhood U of x over which there is a chartfor the log structure with monoid P = ( M/ O × X ) x . N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES This says in particular that if X is fine and saturated, we can locally find charts with P fine, saturated and sharp.The quotient sheaf M = M/ O × X is called the characteristic sheaf of the log structure.Taking the quotient (in an appropriate sense) by O × X of the map α , we get an alternativedefinition of a (quasi-integral) log scheme, introduced in [7].Let us denote by Div X the fibered category over X ´ et whose objects over U → X are pairs( L, s ) where L is an invertible sheaf of O U -modules on U and s is a global section. Thisis a symmetric monoidal fibered category, where the monoidal operation is given by tensorproduct. Definition A.7. A log scheme is a scheme X together with a sheaf of monoids A and asymmetric monoidal functor L : A → Div X with trivial kernel.The phrasing “trivial kernel” in the definition means that if a section a is such that L ( a )is isomorphic to ( O X ,
1) in Div X , then a = 0.Given a (quasi-integral) log scheme ( X, M, α ), by taking the “stacky quotient” of α : M → O X by O × X we get the functor L : A = M = [ M/ O × X ] → [ O X / O × X ] = Div X . Quasi-integralityensures that the quotient [ M/ O × X ] is actually a sheaf. Of course integral log structures arequasi-integral. See [7, Theorem 3.6] for details.One can give a notion of charts in this context as well. For many purposes these twonotions of chart can be used indifferently. We mostly use charts as in the first definitionabove. These are called “Kato charts” in [7]. Remark A.8.
A fist approximation of how one should “visualize” a log scheme is by thinkingabout the stalks of the sheaf M . This sheaf is locally constant on a stratification of X (seeProposition A.27) and the stalks are fine saturated sharp monoids. Of course this disregardsthe particular extension M of M by O × X and the map α (or equivalently the functor L ), soit is indeed just a crude approximation.A.2. Analytification.
We are mainly concerned with log schemes locally of finite type over C , and with their analytifications.Recall that if X is a scheme locally of finite typer over C , the associated analytic space X an is defined as a set as the C points X ( C ) = X (Spec C ) of X . This has an “analytic”topology coming from the local embeddings into C n . Moreover this construction extendsalso to algebraic spaces locally of finite type over C (see [3, 49]).If X is a log scheme locally of finite type over C , the analytication X an inherits a logstructure, because of the relationship between the ´etale topos of X and the analytic toposof X an . An ´etale morphism X → Y induces a local homeomorphism X an → Y an , thatconsequently has local sections in the analytic topology. This gives a functor from the ´etalesite of X to the analytic site of X an , and induces a morphism of topoi. The log structureon X an is obtained via this functor. Thus, in what follows every time something holds ´etalelocally for the log scheme X , it will hold analytically locally for the log analytic space X an .We will use this without further mention, and we will use the same letter to denote thesheaf of monoids M on X and the induced one on X an . This should cause no real confusion. Definition A.9.
For a monoid P we denote by C ( P ) the analytification of the spectrum ofthe monoid algebra Spec C [ P ]. As sets we have C ( P ) = Hom( P, C ), the set of homomorphisms of monoids, where C isgiven the multiplicative structure.A basis of opens of C ( P ) (where P is fine, saturated and sharp) can be described as follows:call p , . . . , p k the indecomposable elements of P (see [37, Proposition 2.1.2]), and choose opendisks D i in the complex plane C . Then the set of homomorphisms φ ∈ Hom( P, C ) such that φ ( p i ) ∈ D i is open in C ( P ). Letting the disks D i vary we get a basis for the open subsets of C ( P ). Lemma A.10 ([49, p. 12]) . Analytification commutes with finite limits.
We will need the following result on the topological properties of analytifications of schemeslocally of finite type over C . As references, we point out [28]. Proposition A.11.
Let X be an affine scheme of finite type over C and Y ⊆ X be a closedsubscheme. Then there exist compatible triangulations of X an and Y an , realizing Y an as asubcomplex. We can apply this iteratively to a stratification, to get compatible triangulations of theambient affine scheme and of all the (closed) strata.A.3.
Kato-Nakayama space.
From now on all log schemes will be fine and saturatedunless we specify otherwise. Just for this subsection, X will denote an analytic space ratherthan a scheme.The Kato-Nakayama space X log of a log analytic space X (for example of the form Y an for some log scheme Y locally of finite type over C ) is a topological space introduced in[26]. The idea is to define a topological space that “embodies” the log structure of X in atopological way (i.e. without using the sheaf of monoids, but only “points”).What comes out is a topological space X log (that also comes with a natural sheaf of rings,but we do not use this in the present work) with a continuous map τ : X log → X thatis proper and surjective. Moreover if U ⊆ X is the trivial locus of the log structure (thelargest open subset over which O × X ֒ → M is an isomorphism), the open embedding i : U → X factors through τ , so that X log can be considered as a “ relative compactification” of theopen immersion i .Let us denote by p † the log analytic space given by the point pt = (Spec C ) an with monoid M = R ≥ × S , and map α : M → C described by ( r, a ) r · a . Note that this log structureis not integral.As a set we have X log = Hom( p † , X ), the set of morphisms of log analytic spaces from thelog point p † to X . By unraveling this one can also write X log = (cid:26) ( x, c ) | c : M gp x → S is a group hom. such that c ( f ) = f | f | for all f ∈ O × X,x (cid:27) . In particular one can see that C ( P ) log = Hom( p † , C ( P )) = Hom( P, R ≥ × S ), and theprojection τ : C ( P ) log → C ( P ) is given by post-composition with R ≥ × S → C .Note that from the above description C ( P ) log has a natural topology, that by means oflocal charts for the log structure gives a topology on X log in general [26, Section 1.2]. N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES From the description one sees easily that for x ∈ X an , the fiber τ − ( x ) is homeomorphicto ( S ) r where r is the rank of the stalk M x , defined to be the rank of the free abelian group M gp x .The construction of the Kato-Nakayama space is clearly functorial, and is also compatiblewith strict base change. Proposition A.12 ([26, Lemma 1.3]) . Let f : X → Y be a strict morphism of fine saturatedlog analytic spaces. Then the diagram of topological spaces X log / / (cid:15) (cid:15) Y log (cid:15) (cid:15) X / / Y is cartesian. The description of X log as a set can actually be enhanced to a description of its functorof points (see Section 3.4).We now prove that following proposition. Proposition A.13.
For any log scheme X, the Kato-Nakayama space X log is locally Haus-dorff, locally contractible and locally compact. We will start by assuming that X is affine and has a global chart X → Spec C [ P ] for afine saturated sharp monoid P , and will prove that X log is locally compact, Hausdorff andlocally contractible. This implies the conclusion for arbitrary X .Note that since f : X → Spec C [ P ] is strict, there is a Cartesian diagram of topologicalspaces X log / / (cid:15) (cid:15) C ( P ) logτ (cid:15) (cid:15) X an f an / / C ( P ) . Our proof will be as follows: we note that X an and C ( P ) are semialgebraic, and the map X an → C ( P ) is a semialgebraic function (this part of the diagram is even algebraic). Wewill check that C ( P ) log is semialgebraic, and that the projection to C ( P ) is a semialgebraicfunction.After we do that, it will follow that X log is semialgebraic as well (being the inverse imageof the diagonal C ( P ) ⊆ C ( P ) × C ( P ), a semialgebraic set, through a semialgebraic map( f an , τ ) : X an × C ( P ) log → C ( P ) × C ( P ), see [6, Proposition 2.2.7]), hence triangulable(by the results of [28]), and any triangulable locally semialgebraic set is locally compact,Hausdorff and locally contractible [18]. Lemma A.14.
The space C ( P ) log is semialgebraic, and the projection C ( P ) log → C ( P ) isa semialgebraic map.Proof. We will check this by writing out these spaces explicitly. Let p i be a finite set ofgenerators for P (for example the indecomposable elements), and assume to have a finite number of relations that present the monoid P , of the form P j r ij p j = P j s ij p j . Say there’s k generators and h relations.Then we have a map C ( P ) = Hom( P, C ) → C k given by φ ( φ ( p i )). This is anembedding, and the closed image is the Zariski closed subset with equations Q j ( z j ) r ij = Q j ( z j ) s ij obtained from the h relations of the chosen presentation of P , and ( z j ) are thecoordinates of C k .In the exact same way we have a map C ( P ) log = Hom( P, R ≥ × S ) → ( R ≥ × S ) k givenby ψ ( ψ ( p i )). To describe the image, let us note that we have R ≥ × S ⊆ R in a naturalway, as a semialgebraic subset. If we denote by ( ζ j ) the “coordinates” of ( R ≥ × S ) k , thenthe (isomorphic) image of C ( P ) log is again described by the equations Q j ( ζ j ) r ij = Q j ( ζ j ) s ij ,so it is semialgebraic (the equations translate into algebraic equations on ( R ) k ).Of course the diagram C ( P ) log / / (cid:15) (cid:15) ( R ≥ × S ) k (cid:15) (cid:15) C ( P ) / / C k commutes.From this, it suffices to check that the map ( R ≥ × S ) k → C k is semialgebraic, and thisis easy: in coordinates (where we see ( R ≥ × S ) k ⊆ ( R ) k and C k ∼ = ( R ) k ) it is given by( a i , b i , c i ) ( a i · b i , a i · c i ). (cid:3) A.4.
Root stacks.
Root stacks of log schemes were introduced in [7]. The infinite rootstack, an inverse limit of the ones with finitely generated weight system, is the subject of[48]. We briefly recall the functorial definition and the groupoid presentations coming fromlocal charts.Let us fix a natural number n and a log scheme X with log structure L : A → Div X .We can consider a sheaf n A of “fractions” of sections of A : the sections of n A are formalfractions an where a is a section of A . There is a natural inclusion i n : A → n A .Note that n A is isomorphic to A via a an . Through this isomorphism, the inclusion i n corresponds to multiplication by n : A → A . The fact that this map is injective follows fromtorsion-freeness of stalks of A , which are fine saturated sharp monoids. Definition A.15.
The n -th root stack n √ X of the log scheme X is the stack over Sch , thecategory of schemes (with the ´etale topology), whose functor of points sends a scheme T tothe groupoid whose objects are pairs ( φ, N, a ) where φ : T → X is a morphism of schemes, N : n φ ∗ A → Div X is a symmetric monoidal functor with trivial kernel and a is a naturalisomorphism between φ ∗ L and the composite N ◦ i n . φ ∗ A / / (cid:15) (cid:15) a (cid:0) (cid:8) ✠✠✠✠✠✠✠✠ Div X n φ ∗ A : : tttttttttt Morphisms are defined in the obvious way.
N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES In other words the n -th root stack parametrizes extensions of the symmetric monoidalfunctor L : A → Div X to the sheaf n A . The pair ( N, a ) in the definition above could becalled an “ n -th root” of the log structure L : A → Div X .Every time n | m there is a morphism m √ X → n √ X , and by letting n and m vary, thesemaps give an inverse system of stacks over Sch . Definition A.16.
The infinite root stack ∞ √ X of the log scheme X is the pro-algebraicstack ( n √ X ) n ∈ N . Remark A.17.
In [48] the infinite root stack is defined as the actual limit of the inversesystem in the 2-category of fibered categories, but in the present paper it will always be thepro-object. We remark that the two contain the same information, since by the results of[48, Section 5] the limit of the system of n -th root stacks recovers the log scheme completely,and hence recovers the pro-object as well.The n -th root stack n √ X is a tame Artin stack with coarse moduli space X . Moreoverthere are presentations of n √ X for each n that assemble into a pro-object in groupoids inschemes, and can be regarded as a presentation of the pro-object ∞ √ X . This follows fromthe following local descriptions as quotient stacks [48, Corollary 3.12].Let us fix a monoid P , and let us denote by C n the cokernel of the injective map P gp → n P gp . Furthermore denote by µ n ( P ) the Cartier dual of C n . This acts on the monoid algebraSpec k [ n P ] ( k here is some base field, but this works the same way over Z ).If X is a log scheme with a global chart X → Spec k [ P ], then there is a cartesian diagram n √ X / / (cid:15) (cid:15) [Spec k [ n P ] /µ n ( P )] (cid:15) (cid:15) X / / Spec k [ P ]presenting n √ X as a quotient stack [ X n /µ n ( P )], where X n = X × Spec k [ P ] Spec k [ n P ].As we mentioned, these quotient stack presentations are all compatible, in the sense thatthey give a pro-object in groupoids in schemes ( X n × µ n ( P ) ⇒ X n ) n ∈ N , that can be seen asa groupoid presentation of ∞ √ X .If X does not have a global chart we cover it with ´etale opens U i where there is a chartwith monoid P i and assemble together the corresponding groupoid presentations. Proposition A.18 ([7, Proposition 4.19]) . The n -th root stack n √ X is a tame Artin stack,and is Deligne–Mumford when we are over a field of characteristic . A.5.
Topological stacks.
The main reference for this section is [34].The two preceding subsections were about the objects that we would like to compare,namely the Kato-Nakayama space and the infinite root stack of a log scheme locally of finitetype over C . Note that the former is of topological nature, and the latter is algebraic. Inorder to find a map between them, we carry over the root stacks to the topological side. One can talk about stacks over any Grothendieck site. Algebraic stacks (a.k.a. Artinstacks) are stacks on the category of schemes over a base with the ´etale topology that admita representable smooth epimorphism from a scheme and whose diagonal is representableby algebraic spaces (and often one imposes some conditions on this diagonal morphism, likebeing quasi-compact or locally of finite type). Equivalently, one can describe algebraic stacksas stacks of (´etale) torsors for certain groupoid objects in algebraic spaces, whose structuremaps are smooth.If instead of schemes over a base with the ´etale topology we start from topological spaceswith the ´etale topology (where covers are local homeomorphisms), and we require a repre-sentable epimorphism from a topological space, we obtain the theory of topological stacks .Such a stack will always have diagonal representable by a topological space. As on the alge-braic side, a topological stack can be defined through a groupoid presentation: a topologicalstack is a stack of principal G -bundles for G a topological groupoid, and much of the basicyoga that one learns when working with algebraic stacks carries over in close analogy in thiscontext.In particular if G is a topological group acting on a space X , the functor of points ofthe quotient stack [ X/G ] is described as principal G -bundles (the topological analogue of G -torsors) with an equivariant map to X . In the same fashion, if R ⇒ U is a topologicalgroupoid, one can characterize the associated stack [ U/R ] as the stack of principal bundlesfor this groupoid.There is a procedure to produce a topological stack starting from an algebraic one, thatextends the analytication functor. We apply this in particular to the n -th root stacks of alog scheme.Denote by A lgSt LF T C the 2-category of algebraic stack locally of finite type over C and by TopSt the 2-category of topological stacks.
Proposition A.19 ([34, Section 20]) . There is a functor of -categories ( · ) top : A lgSt LF T C → TopSt that associates a topological stack to an algebraic stack locally of finite type over C . In Section 3, we extend Noohi’s results to produce a left exact colimit preserving functorfrom ∞ -sheaves (a.k.a. stacks of ∞ -groupoids) on the algebraic ´etale site, to hypersheaveson a suitable topological site. See Theorem 3.1 and Corollary 3.11.This functor has several nice properties. We point out the ones that we use:1. If X is a scheme (or algebraic space) locally of finite type over C , then X top ≃ X an isthe analytification2. The functor ( · ) top preserves all finite limits (i.e. is left exact).3. The preceding properties give us a procedure for calculating X top for an algebraicstack X . If R ⇒ U is a groupoid presentation of X where R and U are locallyof finite type and the maps are smooth, then by the first property we can apply Sometimes, rather than working with the ´etale topology, one defines algebraic stacks with the fppf topology. However, the resulting 2-category of stacks is the same, cfr. [46, Tag 076U]. In [34], Noohi demands further conditions for such a stack to be called a topological stack, however insubsequent papers (e.g. [35]), he relaxes these conditions to the ones just described.
N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES the analytification functor to the diagram, and by the second one, this will resultin another groupoid, namely the groupoid in topological spaces R an ⇒ U an . Thetopological stack X top is then the associated stack [ U an /R an ].In particular if X = [ U/G ] for an action of an algebraic group locally of finite type G ona scheme locally of finite type X , we have X top = [ U an /G an ]. Definition A.20.
Let X be a log scheme locally of finite type over C . The topological n -throot stack of X is the topological stack n √ X top . As for the algebraic ones, the topologicalroot stacks form an inverse system. The pro-topological stack ∞ √ X top .. = ( n √ X top ) n ∈ N is the topological infinite root stack of X .A.6. The rank stratification.
In this section we will prove that the characteristic sheaf M is locally constant on a stratification over the log scheme X . This is used in the mainbody of this article to prove that the Kato-Nakayama space and the infinite root stack are“stratified fibrations” over X , and that the map that we construct between them induces anequivalence of profinite completions.The results of this part are probably known to experts, and we are including them becauseof the lack of a suitable reference. Definition A.21.
By a stratification of a topological space T we mean a collection ofclosed subsets S = { S i ⊆ T } i ∈ I where I is partially ordered, and the following are satisfied: • if i ≤ j , then S i ⊆ S j , and • the stratification is locally finite: every point t ∈ T has an open neighborhood U such that only finitely many of the intersections U ∩ S i are non-empty.The locally closed subsets S j \ S i will be called the strata of the stratification.If in the above definition T is the underlying topological space of a scheme X and each S i is Zariski closed, we will say that S is an algebraic stratification of the scheme X . Note thatan algebraic stratification on X will induce a stratification on the analytification X an . Definition A.22.
Let T be a topological space equipped with a stratification S , and let f : T ′ → T be a morphism, where T ′ is a topological space or stack. We will say that f is a stratified fibration with respect to S if the restrictions of f to the strata of S are fibrations(in our case, this will always mean “locally the projection from a product”).Now let X be a log scheme locally of finite type over a field k . We will describe an algebraicstratification of X over which the sheaf M is locally constant.The basic idea is that we are stratifying by the rank of the stalks M gp x of the sheaf ofabelian groups M gp . Lemma A.23 ([38, 3.5]) . The sheaf M gp is a constructible sheaf of Z -modules [1, IX 2.3] .This means that (Zariski locally) there is a decomposition of X into locally closed subsetsover which M gp is a locally constant sheaf. Lemma A.24 ([37, Theorem 2.3.2]) . If ξ is a generalization of η in X , meaning that η ∈ { ξ } ,then there is a natural morphism of the stalks M η → M ξ , and this is surjective (morespecifically, it is a quotient by a face ). This last lemma follows from Proposition A.6 and from the explicit description of thestalks of the monoid M of the log structure obtained from a chart, see Remark A.4.In particular the rank “only jumps up in closed subsets”, i.e. for every n ∈ N the subset R n of points of X where the rank of the group M gp x is ≥ n is closed: it is constructible byLemma A.23, and stable under specialization by Lemma A.24, so it is closed. Note also that R n +1 ⊆ R n . Definition A.25.
The rank stratification of a log scheme X is the algebraic stratification R = { R n } n ∈ N , where R n = { x ∈ X | rank Z M gp x ≥ n } . We will denote the strata by S n .. = R n \ R n +1 .For example, R = X and the complement X \ R is the open subset of X where the logstructure is trivial (which might be empty). In general S n is the locally closed subset of X over which the rank of M gp x is equal to n .We claim that both sheaves M and M gp are locally constant on the strata S n .To check this, let’s describe the canonical log structure M P → Div X P on X P = Spec k [ P ]in more detail: the log structure is induced by the morphism of monoids P → k [ P ], whichgives a morphism of sheaves of monoids P → O X P (here P denotes the constant sheaf), fromwhich we get the sheaf M P by killing the preimage of the units in O X P . More precisely,denote by { p i } i ∈ I the finitely many indecomposable elements of the fine saturated monoid P ; these are generators of P . For a geometric point x → X P call S ⊆ I the subset of indicessuch that the image of t p i ∈ k [ P ] is invertible in the residue field k ( x ). Then the the stalk( M P ) x is the quotient P / h p i | i ∈ S i .In particular we note the following: Lemma A.26.
The only point x of X P where the stalk ( M P ) x has rank n = rank Z P gp isthe “vertex” v P , the point given by the maximal ideal h t p i | i ∈ I i generated by the variablescorresponding to the indecomposable elements of P . The point v P is also sometimes referred to as the “torus-fixed point”. Proof.
Since P gp ∼ = Z n for some n , as soon as at least one of the indecomposable elements p i is killed, the rank will drop at least by 1. The only point in which no indecomposable iskilled is exactly the maximal ideal generated by all the t p i . (cid:3) Proposition A.27.
For every n and every point x of S n = R n \ R n − there is an ´etaleneighborhood U → S n of x such that the sheaves M | S n and M gp | S n are constant sheaves.Proof. If we equip R n with the reduced subscheme structure, it is a (fine saturated) logscheme with the log structure pulled back from X , and the same is true for the open subset S n ⊆ R n . Consequently there is an ´etale neighborhood U → S n of x and a chart U → Spec k [ P ] for the induced log structure on U , where P = M x (Proposition A.6). If M P isthe sheaf of monoids for the canonical log structure on Spec k [ P ], there is exactly one pointwhere the stalk has rank n = rank P (=rank Z P gp ), corresponding to the vertex v P (LemmaA.26). N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES This implies (since over U the rank of the stalks of M is always n ) that the morphism U → Spec k [ P ] sends everything to v P , and in turn that the sheaf M | U , being a pullbackfrom Spec k [ P ], is constant. This implies that M gp | U is constant as well, and concludes theproof. (cid:3) Note that if k = C , the algebraic stratification of X we just constructed induces a strati-fication of the analytification X an , and the sheaves M and M gp of the log analytic space arelocally constant over the strata. References [1]
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N-SPACES, INFINITE ROOT STACKS, AND THE PROFINITE HOMOTOPY TYPE OF LOG SCHEMES Department of Mathematical Sciences, George Mason University, 4400 University Drive,MS: 3F2, Exploratory Hall, Fairfax, Virginia 22030, USA
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