Asymptotics and Beilinson–Drinfeld Grassmannians in Differen

arXiv:1803.02982v2 [math.DG] 10 Nov 2019 nt eso points. of sets finite n r obida neetn ha fmdlson modules of sheaf interesting an build to try and aifigtefcoiaincniin hsgoercob geometric Grassmannian This Beilinson–Drinfeld condition. the factorization the satisfying enda h ouisaeo principal of space moduli the as defined hspoet eet h atthat fact the reflects property This hr sagoercdsrpino etxoeao algebra operator vertex of description curve geometric complex a is There Introduction n h motn ls feape fte oe rmaparti a from comes them of examples of class important the and eg 1]§0,ie emti object geometric a i.e. §20), [13] (e.g. punctures. [email protected] Windsor Ave, Sunset 401 Canada Windsor, of University Borisov Dennis hsi fcus h hoyo hrlagba rfactorizat or algebras chiral of theory the course of is This multiplicative hr saln udeo h elno–rnedGrassmannia Beilinson–Drinfeld the on bundle line a is there atrzto rpry eso httkn lblsection global taking on algebra that factorization show a We obtain property. factorization ty ebs ieetal ooooy hnhmtp,lo homotopy, thin cohomology, descent differentiable gerbes, etry, Keywords: codes: MSC elno–rnedGasanasi differential in Grassmannians Beilinson–Drinfeld epoethat prove We eodprubto :aypoisand asymptotics 2: perturbation Beyond X osdrtemdl space moduli the consider , elno–rnedGasanas o-rhmda differ non-Archimedean Grassmannians, Beilinson–Drinfeld k 25,2E7 30,5T0 80,5K5 81R10 58K55, 58A03, 57T10, 53C08, 22E67, 22E57, − 1 ∀ greo i group Lie a on -gerbe k ≥ 2 ie mohcompact smooth a given oebr1,2019 12, November Ran N geometry Gr . Y X G Abstract G −→ bnlson -bundles a h tutr famni,gvnb no of union by given monoid, a of structure the has ( X 1 ) Ran G Ran where , odtc d xodO26GG [email protected] OX2 Oxford Rd, Woodstock UK Oxford, of University Kremnizer Kobi oehrwt nitgal connection, integrable an with together X Ran X fuodrdfiiest fpit in points of sets finite unordered of X X G k hthstefcoiainproperty. factorization the has that , dmninlmanifold -dimensional ihcoe rvaiain outside trivializations chosen with ssm ru forcoc.I is It choice. our of group some is eti e nte ouispace: moduli another yet is ject pgop,Brown–Gersten groups, op [3 1-0:satwt a with start §19-20): ([13] s o lersdvlpdi [4], in developed algebras ion fti iebnl we bundle line this of s n cular Gr G ( N atrzto space factorization ) nilgeom- ential aigthe having N n a and X , It is easy to describe GrG(X) in terms of stacks (e.g. [14] §0.5.3), and it is rather clear that there is a canonical Gr (X) Ran , realizing Gr (X) as a factorization space (e.g. G → X G [13] §20.3). Then one usually considers a line bundle on GrG(X) and, pushing it forward to a sheaf on RanX , obtains a factorization algebra, or equivalently a chiral algebra. In this paper we apply these techniques in differential geometry. Specifically we show how to build principal -bundles on Gr (X) (considered as a stack on the usual site of C∞- A G manifolds) starting with any differentiable Deligne cohomology class of a Lie group G with values in an abelian Lie group . The dimension of X has to be 1 less than the dimension A of the cohomology class. Other than that there is no restriction on dimensions. This is very different from algebraic geometry, where X has to be a curve. The reason we can do this is that there is no Hartogs’ extension theorem in differential geometry, and the moduli space GrG(X) can be non-trivial for X of any dimension. The price that we pay for working with C∞-functions is that the polar behavior, which in the theory of vertex algebras is described by Laurent series, is considerably more complicated in differential geometry.

Asymptotic manifolds To deal with poles of C∞-functions we are forced to develop a geometric description of asymptotic behavior. By geometric description we mean using algebraic geometry of C∞- rings (e.g. [25]). When looking at C∞(R) from the point of view of an algebraic geometer, one immediately notices that the corresponding geometric object is much more than just R. For example the ideal of compactly supported functions has to be contained in some maximal ideals, which cannot correspond to points of R. These “asymptotic points” correspond to maximal filters of closed subsets of R (e.g. [24]). The set of all such filters parameter- izes the ways to approach the two infinities in R. The arithmetics of this non-Archimedean geometry of C∞(R) is rather complicated (e.g. [1] and references therein). Instead of doing algebraic geometry with all these complicated points, we choose to group them together into what we call asymptotic manifolds. Instead of maximal filters of closed subsets we take intersections of directed systems of open subsets. In terms of R-points such intersections might be empty, but algebraically they correspond to non-trivial C∞-rings obtained as colimits of C∞-rings of functions on the open subsets. A directed system can be finite, in which case the corresponding asymptotic manifold is just a usual manifold. Thus we obtain an enlargement of the category of manifolds to a bigger full subcategory of C∞Rop, where C∞R is the category of finitely generated C∞-rings. We organize asymptotic manifolds into a site, using the Zariski topology, since we need C∞-rings representing germs of punctures. Our definitions allow us to develop the usual machinery of open, closed submanifolds, intersections, unions, complements, dimension theory, density structure etc. In fact we manage to develop enough of the usual geometric techniques to be able to define and integrate connections around germs of punctures, and as a result to produce line bundles on Beilinson–Drinfeld Grassmannians by transgression, starting from differentiable Deligne cohomology classes on the group (e.g. [7], [15]). Here germs of punctures play the same role as spheres do in the usual transgression constructions (e.g. [8]). Asymptotic manifolds can be very different from the usual manifolds, and not only be-

2 cause they might have no R-points at all. One of the important differences is in the notion of compactness. If one tries to capture the asymptotic behavior around a point, or any compact puncture, it is enough to consider sequences of open neighbourhoods of the puncture. If the puncture is not compact, e.g. a line in a plane, the germ cannot be computed by a sequence of open neighbourhoods, but one needs an uncountable directed system. This is a consequence of the very well known fact (e.g. [18]) that any countable set or orders of growth can be dominated by one order. Because of this asymptotic manifolds are not necessarily first-countable, a property we find very useful in proving Brown–Gersten descent theorems for thin homotopy groupoids.

Thin homotopy and connections The strategy of building a line bundle on the affine Grassmannian, starting with a gerbe on the group, is of course by using transgression. This means that we start not just with a gerbe, but also with a connection on it. As we would like to integrate connections around germs of punctures, it is not enough, in general, to work with the usual infinitesimal theory based on nilpotent elements. We use the much more powerful infinitesimal theory in the geometry of C∞-rings developed in [5]. It is based on -nilpotents. The difference between nilpotents and -nilpotents,∞ is that we obtain 0 by evaluating a ∞ monomial of sufficiently high degree on the former, while for the latter, in order to obtain 0, we might need a C∞-function that decays faster than any monomial. In particular - ∞ infinitesimals are much more than first order infinitesimals, and this complicates the theory of connections. Whichever infinitesimals one chooses to use, defining a flat infinitesimal connection on some geometric object (e.g. a morphism from X into some classifying space Y ) is equivalent to postulating insensitivity to dividing out infinitesimals. For example a flat connection on a morphism X Y consists of a factorization through the de Rham space of X (e.g. [16]). → If one wants to allow connections that are not necessarily flat, one should not use the de Rham space but, for example, the free groupoid generated by X over the de Rham space. However, then even connections along curves within X will not be flat. Something similar happens when one wants to define higher order connections: one needs to postulate one dimensional flatness separately (e.g. [12]). In terms of polynomial infinitesimals this construction of a free groupoid and subsequent flattening of curves was performed in [22]. In the case of k 1-gerbes there is no reason to stop with flatness along curves, it makes sense to require flatness− all the way to dimension k. This means that we need to postulate that the morphism into the classifying space of the gerbe factors not through the de Rham space of X, but de Rham space of the k-thin homotopy groupoid of X. The k-thin homotopy groupoid of X consists of maps ∆n X that locally (on ∆n) factor through something of dimension k. The notion of thin→ homotopy is well known (e.g. [3], [10], [9], [26]). In our setting we would≤ like to compute the thin homotopy groupoids explicitly, i.e. to find their fibrant representatives in the category of pre-sheaves of simplicial sets on the Zariski site of asymptotic manifolds. This is the reason we need to develop the theory of Brown–Gersten descent, which in turn requires the theory of dimension for asymptotic manifolds.

3 Here are the contents of the paper: Asymptotic manifolds are defined in Section 1.1.1 and in Section 1.1.2 it is proved that they have well defined dimensions. Just as for the usual manifolds, there is the notion of regular values for functions on asymptotic manifolds, and Sard’s theorem tells us that almost all values are regular. This allows us, in Section 1.1.3, to define asymptotic submanifolds as solutions to equations and weak inequalities. In Section 1.1.4 we single out compact and locally compact asymptotic manifolds. The full subcategory of C∞Rop consisting of locally compact asymptotic manifolds, together with the Zariski topology, is the site we will be using. In order to prove descent over this site we need to measure dimensions of complements of open asymptotic submanifolds. Such complements are not asymptotic manifolds themselves, but they have enough structure to be given well defined dimensions. We call such complements asymptotic spaces and describe them in Section 1.1.5. In Sections 1.2.1 and 1.2.2 we prove that dimensions of asymptotic spaces, together with asymptotic submanifolds, give our site a bounded cd structure, allowing us to use the Brown– Gersten descent. This implies that to prove that a sheaf of Kan complexes is a homotopy sheaf, it is enough to show that it is soft, i.e. inclusions of closed submanifolds translate into fibrations. In Sections 2.1.1 and 2.1.2 we develop the machinery of smooth simplices and k-thin maps. The goal here is to obtain, for each k Z and an asymptotic manifold X, a ∈ ≥0 homotopy sheaf of smooth families of k-thin simplices in X. All these constructions are rather standard, but are quite tedious. Then in Sections 2.2.1-2.2.3 we use Brown–Gersten descent again to obtain infinitesimal k-thin groupoids, and moreover to give an explicit description of them. Connections and integrable connections are defined then as factorizations of morphisms. We also show that, if X is nice enough, e.g. compact and of dimension k, integrable infinitesimal k- connections factor through germs of diagonals. This allows us≤ later to integrate connections in a combinatorial way. Section 3 contains the main results of this paper. In the first part we give the precise formulation of the problem of constructing line bundles on Beilinson–Drinfeld Grassmanni- ans, and we solve this problem in the second part using asymptotic manifolds. The third part contains a simple observation that in the differential geometric setting the fibers of Beilinson–Drinfeld Grassmannians are affine, if we allow all convenient algebras in addition to C∞-rings. This immediately gives us a way to take global sections of factorizable line bundles and obtain factorization algebras. The Appendix contains necessary technical facts concerning C∞-schemes. Some notation: As we deal with pre-sheaves a lot, we need different notation for different Hom-functors: Hom stands for the usual set of morphisms, Hom denotes the simplicial set of morphisms in simplicial categories, Hom is the internal Hom-functor in a category of pre-sheaves of sets, while Hom is the internal Hom-functor in a category of pre-sheaves of simplicial sets.

4 Contents

1 Asymptotic manifolds and Brown–Gersten descent 5 1.1 Asymptotic manifolds and spaces ...... 6 1.1.1 Deﬁnition of asymptotic manifolds with corners ...... 6 1.1.2 Dimension of asymptotic manifolds with corners ...... 9 1.1.3 Asymptoticsubmanifolds...... 10 1.1.4 Compactness and local compactness ...... 12 1.1.5 Asymptoticspaces ...... 14 1.2 Density structure and Brown–Gersten descent ...... 16 1.2.1 Densityandcdstructure ...... 17 1.2.2 Brown–Gerstendescent...... 18

2 Groupoids and connections 20 2.1 Groupoids...... 20 2.1.1 Smoothsimplices ...... 20 2.1.2 Fundamentalandthingroupoids ...... 23 2.2 Inﬁnitesimal connections ...... 27 2.2.1 Inﬁnitesimalgroupoids ...... 27 2.2.2 Relationtopairgroupoids ...... 28 2.2.3 Connections and integrability ...... 30

3 Beilinson–Drinfeld Grassmannians and factorization algebras 31 3.1 The problem of constructing factorizable line bundles ...... 31 3.2 Thesolution...... 32 3.3 Takingglobalsections ...... 35

A Subschemes of C∞-schemes 36 A.1 Urysohn’slemma ...... 36 A.2 Complementsandgerms ...... 38

1 Asymptotic manifolds and Brown–Gersten descent

We denote by C∞Sch the opposite category of the category C∞R of finitely generated C∞- rings (e.g. [25] §I). Objects of C∞Sch will be called C∞-schemes.1 For A C∞R, C∞Sch we write Spec(A), C∞( ) to mean the corresponding objects in C∞Sch and∈ C∞R.X The ∈ empty scheme will be denotedX by ∅ := Spec(0). ∞ We equip C Sch with the Zariski topology (e.g. [25] §VI), and write C∞Sch, C∞Sch to mean the categories of pre-sheaves of sets and respectively simplicial setsZ on thisZ site. We will work with several sites equipped with Zariski topology, thus we keep the site as part of the notation. 1We require our C∞-schemes to be Hausdorff, implying that all of them are affine.

5 We do not restrict our attention only to sheaves or to homotopy sheaves,2 but we always consider categories of pre-sheaves together with the notion of local equivalence. In particular ∞ comes with two model structures: local projective and local injective (e.g. [11]). ZC Sch Homotopy descent can be complicated, but everything is simplified, if one can use some descent theorems, e.g. Brown–Gersten descent. This result requires existence of an appropriate theory of dimension producing a bounded density structure (e.g. [27]). Because of the presence of fractals, the notion of dimension for C∞-schemes is more complicated than the one in algebraic geometry. Instead of dealing with fractal dimensions we choose to work with full sub-categories of C∞Sch, consisting of C∞-schemes that satisfy some regularity conditions, ensuring integrality of the dimension. We would like to stress that it is not enough for us to work only with manifolds or manifolds equipped with infinitesimal structure, since we would like to use germs of subschemes and germs of punctures, the latter not even having any R-points. This leads us to C∞- schemes that we call asymptotic manifolds. We describe them in the first two parts of this section, and then prove Brown–Gersten descent for them.

1.1 Asymptotic manifolds and spaces

For us a (classical) n-dimensional manifold (n Z≥0) is a non-empty, Hausdorﬀ, second countable topological space with a chosen equivalence∈ class of C∞-atlases consisting of Rn- charts. The category of manifolds M is a full subcategory of C∞Sch (e.g. [25] Thm. I.2.8). We will arrive at asymptotic manifolds with corners by enlarging M to a bigger full subcategory of C∞Sch. First we add corners and inﬁnitesimal structure.

1.1.1 Definition of asymptotic manifolds with corners Since we would like to do algebraic geometry with C∞-schemes, as with any algebraic geometry over R, we are naturally led to consider solutions to inequalities, both strict and weak. This means that, as a first step, we need to enlarge the category of manifolds to include manifolds with corners. Different from several definitions based on local models (e.g. [21]), we use inequalities themselves as the basis for our definition. However, we need to impose some regularity conditions in order to have a well defined dimension.

∞ Deﬁnition 1. Let be a manifold, and let f1,...,fm C ( ). We will say that 0 is a regular value for Mf m , if j 0 is regular value{ for f and} ⊆ alsoM for restrictions f j j=1 j j T Mfi=0 { } ∀ |i∈S for each S 1,..., j,...,m . A non-empty⊆{ C∞b-scheme} is an n-dimensional manifold with corners, if there are an X ∞ n-dimensional manifold and a set f1,...,fm C ( ) having 0 as a regular value, s.t. writing for in C∞MSch/ we have{ 3 } ⊆ M ∩ × M

= \ fj ≤0 = \ fj <0. (1) X ∼ M M 1≤j≤m 1≤j≤m

2By homotopy sheaves we mean pre-sheaves of simplicial sets that satisfy the conditions of hyper-descent (which we call homotopy descent). 3Deﬁnitions of closures and solutions to inequalities are given in Def. 26 and 27.

6 ∞ The empty C -scheme ∅ is the 1-dimensional manifold with corners. The data , f1,...,f2 will be called a realization of .− The full subcategory of C∞Sch consisting of manifolds{M with } corners will be denoted by MXc. Notice that a choice of realization is not part of the structure of a manifold with corners. However, the ability to choose one will be used often. The requirement in (1) that is the closure of the set of solutions to strict inequalities implies that for n 0 every n-dimensionalX ≥ manifold with corners contains a dense subscheme, that is an n-dimensional manifold. As ∅ is the only 1-dimensional manifold with corners, it is clear then that manifolds with corners have well− deﬁned dimensions. Deﬁnition 2. Let Mc, and let f C∞( ). An r R is a regular value for f, if there X ∈ ∈ X ∈ ∞ is a realization , f1,...,fm of and an extension f C ( ) of f, s.t. 0 is a regular value for f r,{M f ,...,f . } X e ∈ M { − 1 m} Remark 1. Given a regular value r for f on a manifold with corners , it is clear that, c X c if f≤r = f

∞ Deﬁnition 3. Let C Sch, and let OX be the category of non-empty open subschemes of and inclusions. AX regular ∈ system of open subschemes of is given by a functor op O X X D → X where = is a directed category, s.t. for any i j in the corresponding inclusion D 6 ∅ → D j ֒ i factors through j i. We deﬁne T i := lim i, where the limit is taken in U → U U ⊆ U i∈D U i U C∞Sch.

If the directed category is ﬁnite, it has a maximal element. Then T i is just a non- D i∈DU empty open subscheme of . Also in the inﬁnite case it is true that T i ≇ ∅, as the X i∈DU following simple lemma shows.

∞ Lemma 1. Let C Sch, let i i∈D be an inﬁnite regular system of open subschemes of X ∈ ∞ {U } ∞ . Then T i ≇ ∅ and C ( ) C ( T i) is surjective. X i∈DU X → i∈DU Proof. Consider another directed category ′, that is obtained from by splitting each i into i′ i. We deﬁne a functor from ′opDto the category of all non-emptyD subschemes∈D of → D 7 ′ and inclusions as follows: i i and i i. This shows that T i = T i′ . For X 7→ U 7→ U i∈DU i′∈D′\DU ∞ ∞ ∞ ∞ each i let Ai := Ker(C ( ) C ( i)), then C ( T i′ ) ∼= C ( )/ S Ai. As i ∈ D X → U i′∈D′\DU X i∈D ∀ i = ∅, clearly 1 / S Ai, i.e. it is a proper ideal and T i ≇ ∅. U 6 ∈ i∈D i∈DU

We would like to have infinitesimal tools available, hence we need to work with non- reduced C∞-schemes. We would like to use the much richer infinitesimal structure from [5]. ∞ ∞ ∞ ∞ Given C Sch the corresponding reduced C -scheme is ( )red := Spec(C ( )/ √0), X∞ ∈ X X where √0 C∞( ) consists of -nilpotent elements ([5] Def. 2). ≤ X ∞ Definition 4. An C∞Sch is a reduced n-dimensional asymptotic manifold with corners X ∈ c (n 0), if there is an n-dimensional M and a regular system i i∈D of open ≥ M ∈ ∞ {U } subschemes of (Def. 3), s.t. ∼= T i. An C Sch is an n-dimensional asymptotic M X i∈DU X ∈ manifold with corners (n 0), if ( )red is a reduced n-dimensional asymptotic manifold with corners.4 ≥ X ∞ The 1-dimensional asymptotic manifold is the empty C -scheme ∅. The data , i i∈D will be called− a presentation of . If can be chosen to be a (usual) manifold,{M we{U will} } X M say that is an asymptotic manifold. We denote by AM C∞Sch the full subcategory consistingX of asymptotic manifolds with corners. ⊂

Remark 2. Notice that according to Lemma 1 an inﬁnite regular system of open subschemes produces a surjective localization map. Since open subschemes of manifolds with corners are themselves manifolds with corners, it follows that every AM has a presentation , , s.t. C∞( ) C∞(( ) ) is surjective. X ∈ {M {Ui}i∈D} M → X red Before considering examples we need the following lemma.

Lemma 2. Let C∞Sch be an n-dimensional asymptotic manifold with corners (n 0), X ∈ ≥ and let be an open subscheme, ≇ ∅. Then is an n-dimensional asymptotic manifoldU with ⊆ X corners. U U

∞ ∞ ∞ −1 Proof. Let f C ( ), s.t. C ( ) ∼= C ( ) f , and let , i i∈D be a presentation ∞∈ X ∞ U X { } {M {U } } ∞ of , s.t. C ( ) C (( )red) is surjective. We choose a pre-image f C ( ) of the X M∞ → X ′ ∞ −1 e ∈ M ′ image of f in C (( )red), and let := Spec(C ( ) f ). Clearly ( )red ∼= T ( i). X M M { e } U i∈D M ∩U The system ′ is regular, since if k s.t. ′ = ∅, then also = ∅. {M ∩ Ui}i∈D ∃ M ∩ Uk U ∼ Example 1. (a) All manifolds with corners are asymptotic manifolds with corners.

(b) Let be a manifold with corners, let V be a compact subset V = . We embed M m ⊆M 6 ∅ R and k N let k be the open subscheme consisting of points of ֒ M → 1 ∀ ∈ U ⊆ M distance < k from V . Then T k is an asymptotic manifold with corners – the germ k∈NU of at V . M 4In Section 1.1.2 we show that dimensions of asymptotic manifolds are well deﬁned.

8 (c) If V is not compact, we can write V = S Vs, where each Vs is compact. For each s∈N α: N N let α be the open subscheme consisting of p s.t. s N q Vs → 1 U ⊆M ∈M ∀ ∈ ∀ ∈ p q < α(s) . Then V := T α is an asymptotic manifold with corners – the germ k − k M α U of at V . M ∞ ∞ (d) Let f C ( ), s.t. p f =0 = V , and let f C ( V ) be the image of ∈ M { ∈M| } ◦ ∈ M ∞ −1 f. The germ of at the puncture V is V := Spec(C ( V ) f ). This is an asymptotic manifoldM with corners. M\ M { }

(e) Not all asymptotic manifolds with corners are germs. Consider S := fk k≥1 ∞ 2 2 1 1 ∞ 2{ }−1 ⊆ C (R ), s.t. fk = 0 exactly on R (( k , k ) R). Then Spec(C (R ) S ) is an asymptotic manifold, but it is not\ the− germ of×R2 at the y-axis (e.g. [25],{ p. 49).}

Examples 1.(c), 1.(d), 1.(e) exhibit a general fact: asymptotic manifolds with corners built as germs around compact sets can be constructed using regular sequences, while non- compact sets require regular systems (in particular uncountable). We take a closer look at this in Section 1.1.4.

1.1.2 Dimension of asymptotic manifolds with corners Even though dimension was part of the deﬁnition of asymptotic manifolds with corners, we have not shown yet that the same C∞-scheme cannot be an asymptotic manifold with corners of dimensions m, n simultaneously, with m = n. We do this in this section. In fact we show something more: a locally closed subscheme6 5 of an n-dimensional asymptotic manifold with corners cannot be an asymptotic manifold with corners of higher dimension.

.Proposition 1. Let ′ C∞Sch, and let Φ: ֒ ′ be a locally closed subscheme, s.t X ∈ X → X X is an m-dimensional and ′ is an n-dimensional asymptotic manifold with corners. Then m n. X ≤ Proof. As asymptotic manifolds with corners are deﬁned by putting conditions on their reduced parts, we can assume that both and ′ are reduced. According to Lemma 1 if n 0, anXn-dimensionalX asymptotic manifold with corners ≥ cannot be ∅, thus we can assume m, n 0. There is an open subscheme ′, s.t. Φ: is a closed embedding. As is an≥ asymptotic manifold with cornersU ⊆ of X dimensionXn →( U U ′ ′ Lemma 2) we can assume that Φ is a closed embedding. Let , , , ′ {M {Ui}i∈D} {M {Uj}j∈D } be some presentations of and ′ respectively. X X If there is at least one R-point p in , the claim trivially follows from comparing cotangent spaces. Therefore we can assume thatX does not have R-points. Also in this case we proceed X by comparing points, but this time “asymptotic points” rather than R-points.

Let n Z≥0 be the smallest with the property that m > n and a closed embedding ∈′ ′ ′ ∃ ′ Φ: as above. I.e. = T i, = T j, and , are m- and n-dimensional X → X X i∈DU X j∈D′U M M manifolds with corners. We choose a well ordering of the set of objects in , and construct D 5For locally closed subschemes see Def. 26.

9 f C∞( ) inductively as follows: let i be the first s.t. is disjoint from the union { k} ⊆ M ∈D Ui of supports of fk’s already constructed. Choose a function whose support is diffeomorphic to Rm and lies within , and add it to the end of the sequence f . Using transfinite Ui { k} induction we obtain in the end f := P fk s.t. k

i = , ∀ ∈D Mf6=0 ∩ Ui 6 ∅ m 6 ∞ ∞ and f6=0 is a disjoint union of copies of R . Let f C ( ) be the image of f in C ( ), let fM′ C∞( ′) be any pre-image of f, and let f ′ ∈C∞( X′) be any pre-image of f ′. AsX f ∈ X ∈ M′ has non-zero values on every i, clearly f6=0 ≇ ∅, hence f ′6=0 ≇ ∅, and they are m- and n-dimensional asymptotic manifoldsU respectivelyX (LemmaX 2). Explicitly

′ ′ ′ f6=0 = \( f6=0 i), f ′6=0 = \ ( f ′6=0 j). X ∼ M ∩ U X ∼ M ∩ U i∈D j∈D′

m By construction f6=0 is a disjoint union of a countable set of copies of R . Informally f is a bunch of “asymptoticM points” in Rm. X ∞ By assumption m > n 0, so let x C ( f6=0) be the function that restricts to x1 on m ≥ ∞ ∈ M ′ ∞ ′ each copy of R . Let x C ( f6=0) be the image of x, x C ( f ′6=0) any pre-mage of ′ ∞ ′ ∈ X ′ ∈ X x, and x C ( f ′6=0) any pre-image of x . For any r R the closed subscheme of f6=0 ∈ M ′ ′ ∈ X deﬁned by x r is not ∅, therefore x : f ′6=0 R is surjective. ′ − M → Since f ′6=0 is an n-dimensional manifold with corners, by Sard’s theorem we can choose M ′ r R, s.t. f ′6=0,x′=r is an n 1-dimensional manifold with corners. Since every value of x on∈ isM regular, clearly − is an m 1-dimensional manifold. Then we have Mf6=0 Mf6=0,x=r − ∞ \( f6=0,x=r i) = Spec(C ( f6=0)/(x r)), (2) M ∩ U ∼ X − i∈D

′ ′ ∞ ′ ′ \ ( f ′6=0,x′=r j) = Spec(C ( f ′6=0)/(x r)). (3) M ∩ U ∼ X − j∈D′ As (2) is an m 1-dimensional asymptotic manifold, sitting as a closed subscheme in (3), which is an n 1−-dimensional asymptotic manifold with corners, this contradicts minimality − of n.

Remark 3. Since identity is a closed embedding, Prop. 1 shows that any asymptotic manifold with corners has a well deﬁned dimension dim . X X 1.1.3 Asymptotic submanifolds As with usual manifolds we would like to be able to deﬁne asymptotic submanifolds as solutions to equations, subject to some regularity conditions. Since we are working over R, we would like to also have solutions to weak inequalities. We start with regularity conditions.

6 The sum fk is countable (because is second-countable), but it can be parameterized by an ordinal Pk M larger than N.

10 Deﬁnition 5. Let AM. For an f C∞( ) an r R is a regular value, if there is a X ∈ ∈ X ∈ ∞ presentation , i i∈D of , s.t. r is a regular value for a pre-image in C ( ) of the image of f in{MC∞(({U)} ) (Def.} X 2). The set of regular values for f will be denoted byMReg(f). X red The classical theorem of Sard immediately implies the following. Proposition 2. For any AM and f C∞( ) the set R Reg(f) has Lebesgue measure 0. X ∈ ∀ ∈ X \ The following statement is only slightly more complicated. ∞ Proposition 3. Let AM and let r Reg(f) for f C ( ). Then f

∞ / ∞ C ( ) / C ( f

=∼ / ( )red / T i. X i∈DU

In particular ′ is an asymptotic manifold with corners of dimension dim . X X Proof. Since localization commutes with reduction, we can assume that is reduced. Let ∞ ′ X fk k∈N C ( ) be s.t. k k = fk6=0 and fk factors fk+1. For each k denote Ak := { } ∞ ⊆ X∞ ′ ∀ U X ′ ′ ∞ Ker(C ( ) C ( k)). Regularity of k k∈N implies that k there are fk C ( ), X → ′ U {U } ′ ∀ ∞ ∈ X αk Ak+1 s.t. fkfk =1+ αk. We choose pre-images fk, f k, αk k∈N C ( ) s.t. k fk ∈ ∞ {∞e e e } ⊆ M ∀ e factors fk+1. For an i let Ai := Ker(C ( ) C ( i)), k N let ik , βik Aik s.t. e ∈D M → U ∀ ∈ ∈D ∈ ′ ′ f f =1+ α + β , k >k i ′ > i , α =0. k k k ik k k k Mfe 6=0∩Uik+1 e e e ⇒ e | k+1 We deﬁne ′ := (k, j) k N, j s.t. j i and ′′ := . This is a regular k k,j fk6=0 j system of openD subschemes{ | ∈ of ∈D. The image≥ of} the obviousU projectionM e ∩ U ′ is coﬁnal in ′′ M ′′ D →D′ . =∽ hence T k,j ֒ factors through , and therefore T k,j , D (k,j)∈D′U →M X (k,j)∈D′U X

1.1.4 Compactness and local compactness Now we come to the distinction between objects in AM defined as intersections of regular systems and as intersections of regular sequences (Ex. 1). In the world of asymptotic objects this distinction reflects the notion of compactness for the usual manifolds. First we recall ∞ ∞ n some terminology (e.g. [25] §I.4): a C -ring A is point determined, if A ∼= C (R )/A, and f C∞(Rn) that vanishes at the common zeroes of A belongs to A. ∀ ∈ Definition 7. Let AM. We will say that is compact, if there is a presentation X ∈ X′ ′ i i∈D of and a dense embedding ֒ , s.t. is point determined, and there , {M {U } } X ′ 9 M → X X ′ is a compact K with ( )red ∼= K. The data , ,K will be called a compact presentation of ⊆. X X M {M X } X Example 2. (a) Every manifold with corners, that is compact in the usual sense, is also compact according to our definition.

(b) Germs at compact subsets are clearly compact, as well as germs at such punctures.

(c) The germ of the open upper half-plane at the x-axis is not compact. It is natural to expect the following Lemma. Lemma 4. Let AM be compact, and let ′ be an asymptotic submanifold (Def. 6). Then ′ is compact.X ∈ X ⊆ X X 9 Recall (Def. 31) that K means the germ of at K. M M 12 ′′ ∞ ′ Proof. Let ( , ,K) be a compact presentation of . Let f C ( ) be s.t. = fa)red (respectivelyX X ( ) ). Let K K consist of p, s.t. thereU ⊆M is an openU ∩VX ′′ containingX Xf6=a red 1 ⊆ ⊆ X p with V . Clearly K K1 is closed, and hence compact. It is easy to see that ( ′) is the∩M⊆U germ of at K\ K . X red M \ U \ 1 As examples 1.(e) and 2.(c) show there are plenty of asymptotic manifolds with corners, that are not compact, but can be broken into compact pieces. We would like to formalize this.

m ∞ Deﬁnition 8. An AM is locally compact, if we can ﬁnd m N and fi i=1 C ( ), X ∈ m m m∈ { } ⊂ X s.t. for any set of pairs ai, bi i=1 R of regular values for fi i=1 the asymptotic submanifold is compact.{ } ⊂ { } T ai

The most important example of an asymptotic manifold for us is ( S)S, which is the germ of a manifold at a puncture, obtained by removing a ﬁnite unionM\ of submanifolds M S . This is clearly a locally compact asymptotic manifold. ⊆M ∞ Proposition 4. Let AMl.c. C Sch be the full subcategory consisting of locally compact asymptotic manifolds with corners.⊂ Then (a) AM any open subscheme of is also in AM , ∀X ∈ l.c. X l.c. (b) AM any asymptotic submanifold of is also in AM , ∀X ∈ l.c. X l.c. (c) , AM , computed in C∞Sch, belongs to AM . ∀X1 X2 ∈ l.c. X1 × X2 l.c. ∞ Proof. (a) We choose f C ( ) s.t. = f<0. Then for any regular values a

′ ′ ′′ ′′ (c) We can assume that 1, 2 are reduced. Let , i i∈D′ and , j j∈D′′ be X X {M {U }′ } ′′ {M′ ′′{U } } some presentations of , respectively. Consider , ′ ′′ , X1 X2 {M ×M {Ui × Uj }(i,j)∈D ×D } where ′ ′′ is computed in C∞Sch. This clearly defines an object in AM. Suppose we areM given×M another reduced AM with a presentation , , and let X ∈ {M {Uk}k∈D} Φ1 : 1, Φ2 : 2 be any morphisms. As manifolds are finitely presented, both X → X X → X ′ ′′ ′ ′′ Φ1 and Φ2 lift to , , and then the corresponding M→M′ ′′ M→M M→M ×M maps to T ( i j ). This shows that product of objects in AM, computed in X (i,j)∈D′×D′′ U ×U ∞ C Sch, belongs to AM. Starting with objects of AMl.c., means that we can decompose ′, ′′ into pieces, that are germs at compacts. Since direct products of compact M M point determined C∞-schemes are again compact, this finishes the proof.

Plenty of objects in AM are not locally compact. Example 3. Let be the germ of R2 0 at 0. The y-axis deﬁnes an asymptotic submanifold ′ . Let ′′ X be the germ of \at ′. It is easy to see that ′′ is not locally compact. X ⊂ X X ⊂ X X X X 13 The previous example shows an important fact: even locally compact asymptotic manifolds with corners are not ﬁrst countable, meaning that germs at closed subschemes do not have to have a fundamental system of open neighbourhoods. The following lemma shows something more: quite often intersection of any regular sequence of open neighbourhoods contains another open neighbourhood.

′ Lemma 5. For a compact AMl.c. without R-points let be an asymptotic X ∈ ′ X ⊆ X ′ submanifold. Then for any regular sequence i i∈N of open subschemes of , s.t. ′ {U′ } ′ X X ⊆ T i , there is an open subscheme , s.t. T i . i∈NU U ⊆ X X ⊆ U ⊆ i∈NU

Proof. We can assume that is reduced. Let , i i∈N be a presentation of . Let ∞ ′ X {M {U } } X f C ( ) be s.t. = f

The category AMl.c., together with the Zariski topology, will be our site, where we will be defining connections and performing transgression. We would like to use Brown–Gersten descent, which requires a density structure based on the notion of dimension. This means that we need to be able to measure the dimension of a complement of an open subscheme within an asymptotic manifold with corners. Such complements do not have to belong to AMl.c., for example the union of intersecting lines is not a manifold with corners. In this section we enlarge AM to include asymptotic spaces, which will be such complements. Not all complements will be allowed though. The defining property is existence of strat- ification by objects of AM. Our intuition comes from cell complexes in topology. We will use asymptotic spaces only to define the density structure, i.e. we need them only to count dimensions. In particular, all of the asymptotic spaces will be reduced.

Deﬁnition 9. A reduced C∞Sch is an asymptotic space, if there are closed embeddings X ∈ ,∅ = ֓ . . . ֓ ֓ = X X0 ← X1 ← ← Xt ∼ 10 s.t. i 1 i ∼= i−1 i, where i i−1 is open and belongs to AM. The data ∀ , ≥ Xwill beX called− Ua C∞-cell decompositionU ⊆ X of . {{Xi} {Ui}} X The choice of a C∞-cell decomposition is not part of the structure, only existence thereof. We do not require dimensions of i’s to satisfy any inequalities. For example a disjoint union of a ﬁnite set of manifolds is an asymptoticU space, and we can choose a C∞-cell decomposition

10The operation of subtraction is described in Def. 28.

14 for any ordering of their dimensions. However, our ﬁrst goal is to deﬁne dimensions of asymptotic spaces as the maximum of the dimensions of asymptotic manifolds involved. For this we have the following lemma.

′ ′ ∞ Lemma 6. Let be an asymptotic space, let i , i , j , j be any two C -cell X {{X′ } {U }} {{X } {U }} decompositions. Then max dim i = max dim j. i U j U Proof. For any j consider ′ , where is the fiber product over . This is an open Uj ∩ U1 ∩ X subscheme of ′, hence ′ = ∅ or ′ is an asymptotic manifold with corners of Uj Uj ∩ U1 ∼ Uj ∩ U1 dimension dim ′ (Lemma 2). In the latter case let ′ be the maximal open extension of ′ Uj Uj Uj ′ ′ ′ e ′ in (Prop. 16), then j 1 = j 1 j−1, and j 1 1 is locally closed. Hence X ′ U ∩ U Ue ∩ U ∩ X U ∩ U ⊆ U dim j dim 1 (Prop. 1). U ≤ U ′ ′ In the former case j factors through 1 (Lemma 20). Then j 2 is an open ′ U → X X U ∩ U ′ subscheme of j (here means fiber product over 1). As before we have either dim j ′ U ∩ X U ≤ dim 2, or j 1 factors through 2. After a finite number of steps we stop and conclude U U′ → X X that dim j max dim i. U ≤ i U Definition 10. Let be an asymptotic space, and let , be a C∞-cell decompo- X {{Xi} {Ui}} sition of . The dimension of is dim := max dim i. X X X i U We would like to glue asymptotic spaces and obtain asymptotic spaces again. The following two lemmas are essential for this.

∞ Lemma 7. Let be an asymptotic space, and let i, i be a C -cell decomposition. Let be an openX subscheme, then , {X isU } a C∞-cell decomposition of , in U ⊆ X {U ∩Xi U ∩ Ui} U particular is an asymptotic space and dim dim . U U ≤ X Proof. Clearly is an asymptotic manifold with corners (of dimension dim ). Since U1 ∩U ≤ U1 1 = 1 (Prop. 16) we can continue within 1. After a ﬁnite number of steps weU−U∩U stop. U ∩ X X Lemma 8. A reduced non-empty C∞Sch is an asymptotic space, if and only if there are open subschemes X ∈ ∅ = ... = , (4) Ut ⊆ ⊆ U1 ⊆ U0 X s.t. i 1 ( ) is an asymptotic manifold with corners.11 ∀ ≥ X − Ui ∩ Ui−1 t ∞ Proof. Given (4) i we deﬁne i := t−i, and choose fi i=0 C ( ) s.t. i = t . Since ∀ , ∞ (fX) ∞X(f − U) (Lemma 18), and{ we} have⊆ closedX embeddings{U Xfi6=0}i=0 Ui ⊆ Ui−1 p i ≤ p i−1 Let f C∞( ) be the image of f , then . = ֒ ֒ . . . ֒ = ∅ Xt → → X1 → X0 X t−i ∈ Xi−1 t−i i = i−1 ( i−1) , and ( i−1) = i−1 t−i =( t−i+1) t−i is an asymptotic X X − X ft−i X ft−i X ∩ U X − U ∩ U manifold. Conversely, let , t be a C∞-cell decomposition. Let t be the maximal {Xi Ui}i=1 {Ui}i=1 extensions of t to open subschemes of (Prop. 16). We havee = , in {Ui}i=1 X Xi X − Ui particular i . Then ( ) = (Rem. 8) and taking ∅ t ewith ∀ Ui ⊆ Ui+1 X − Ui−1 ∩ Ui ∼ Ui { }∪{Ui}i=1 the opposite ordere e we get (4). e e e

11Recall that means fiber product over . ∩ X 15 As with asymptotic manifolds with corners, also on asymptotic spaces we would like to consider regular values of functions. Definition 11. Let be an asymptotic space and let f C∞( ). We say that r R is a X ∞ ∈ X ∈ regular value for f, if there is a C -cell decomposition i , i of , s.t. r is a regular value for restriction of f to each . The set of regular{{X values} will{U }} be denotedX by Reg(f). Ui As finite unions of sets of Lebesgue measure 0 also have Lebesgue measure 0, it is clear that almost all r R are regular for a given f. However, solving equations and inequalities, given by regular values∈ of functions, does not always give us asymptotic submanifolds (Prop. 3, Rem. 4). Similarly for asymptotic spaces. Hence we need the following proposition. Proposition 5. Let be an asymptotic space and f C∞( ). Let S R consist of r s.t. = ∅ or is an asymptoticX space of dimension dim∈ andX = ∅⊆or is an asymptotic Xf≤r X Xf=r space of dimension dim 1. Then R S has Lebesgue measure 0. X − \ Proof. Let , be a C∞-cell decomposition of , let f be the restrictions of f {{Xi} {Ui}} X { i} to i . Let r T Reg(fi) and consider {U } ∈ i .∅ ֓ ֓ . . . ֓ ֓ Xf=r ← X1 ∩ Xf=r ← ← Xt−1 ∩ Xf=r ← Clearly =( ) ( ), i.e. this would be a C∞-cell decomposition, Xf=r ∩Xi Xf=r ∩ Xi−1 − Xf=r ∩ Ui if each f=r i was an asymptotic manifold with corners. It can be that f=r i is not an asymptoticX ∩ U manifold with corners: let , be a presentation ofX , and∩ U let {M {Uj}j∈D} Ui f C∞( ) be a pre-image of f, s.t. r is a regular value for f on the interior of and e ∈ M M on each piece of the boundary. Then the submanifold f=r can have intersections with the boundary of that are isolated. M e M Since there are at most countably many such corners, excluding such values from T Reg(fi) i we get an S s.t. R S has Lebesgue measure 0. We claim that r S also is an asymp- \ ∀ ∈ Xf≤r totic space. We define a new sequence of closed subschemes of : X i ′ := , ′′ := ( ) ′ . ∀ Xi Xi ∩ Xf≤r Xi Xi ∩ Xf=r ∪ Xi+1 Then ′′ = ′ ( ), ′ = ′′ ( ). Therefore Xi Xi − Ui+1 ∩ Xf

∞ As a subcategory of C Sch, AMl.c. inherits the Zariski topology. According to Prop. 4 open subschemes of an AM are themselves in AM , hence ﬁnite open covers constitute a X ∈ l.c. l.c. basis for this topology on AM . Let AM , be the categories of pre-sheaves of sets l.c. Z l.c. ZAMl.c. and of simplicial sets on AMl.c. respectively.

Recall (e.g. [11]) that the global projective model structure on AMl.c. is given by ob- jectwise weak equivalences and ﬁbrations. The local projective modelZ structure is given by

16 global cofibrations and local weak equivalences, i.e. morphisms that induce isomorphisms on sheaves of homotopy groups. We would like to be able to characterize fibrant objects in the local projective model structure, which we call homotopy sheaves. Also we would like to explicitly compute homotopy pullbacks in . For this we use Brown–Gersten descent. To prove Brown–Gersten ZAMl.c. descent in we need to introduce cd and density structures. ZAMl.c. 1.2.1 Density and cd structure Definition 12. A pullback square

/ X1 ∩ X2 X2

/ X1 X in AMl.c. will be called 1. open, if each , is an open subscheme and = , X1 → X X2 → X X X1 ∪ X2

2. closed, if 1 is an open subscheme and 2 is an asymptotic submanifold X → X ′ X → X (Def. 6), s.t. there is an open subscheme 2 factoring through 2, and = ′. X → X X X X1 ∪ X2 Clearly each of the two kinds of squares is stable with respect to isomorphisms of square diagrams. Hence we have two cd structures ([27] Def. 2.1): the open and the closed.

Proposition 6. Both of the cd structures on AMl.c. are complete and regular, and each generates the Zariski topology. Proof. The open cd structure generates Zariski topology by deﬁnition of this topology. Since closed squares can be reﬁned by open squares, it is clear that the topology generated by closed squares is not stronger than the Zariski one. To prove the opposite inequality consider a Zariski covering: = . X U1 ∪ U2 According to Prop. 15 f C∞( ) and r < r R, s.t. = , = . ∃ ∈ X 2 1 ∈ U1 Xfr2 According to Prop. 2 we can choose r1

Now we deﬁne the density structure on AMl.c. ([27] Def. 2.20).

Definition 13. For an AMl.c. an open subscheme is m-dense, if is an asymptotic space of dimensionX ∈ max( 1, dim( ) m).U ⊆ X X − U ≤ − X − 17 For Brown–Gersten descent the most important property of a density structure is whether it is reducing for a given cd structure. Even before that we need to show that Def. 13 does give us a density structure that is locally of finite dimension. This is the purpose of parts 1-3 the following lemma. Lemma 9. Let AM be of dimension n 1. Then X ∈ l.c. ≥ − ;is 0-dense; if ֒ is m +1-dense, it is also m-dense ֒ ∅ .1 → X U → X ;is n +1-dense, iff it is an isomorphism ֒ .2 U → X ;if ֒ and ′ ֒ are m-dense, the composite ′ ֒ is m-dense .3 U → X U → U U → X 4. let be an m-dense open subscheme, and let be any open subscheme, U1 ⊆ X U2 ⊆ X ;then ֒ is m-dense U1 ∩ U2 → U2 . ֒ if , ֒ are m-dense open subschemes, so is .5 U1 U2 → X U1 ∩ U2 → X Proof. Parts 1. and 2. are obvious. For part 3. let be a sequence of open subschemes {Ui} of as in Lemma 8, and let i be the maximal extensions to open subschemes of X − U {Ue } ′ ′ X (Prop. 16), in particular i i. Let j be a sequence of open subschemes of as ′ ∀ U ⊆ Ue ′{U } U − U in Lemma 8. Then j , i ( ) is a sequence of open subschemes as in Lemma 8, realizing ′ as{{U an} asymptotic{Ue ∩ X − space. U }} X − U The asymptotic manifolds with corners in the C∞-cell decomposition we have constructed are the same as in decompositions of and ′. Therefore, since dim( ) dim( ) (Lemma 7) we finish the proof of 3. X − U U − U X ≥ U

To prove 4. we note that 2 ( 1 2)= 2 ( 1) (Prop. 16), and open subschemes of asymptotic spaces are asymptoticU − U ∩ spaces U ofU smaller∩ X − or U equal dimension (Lemma 7). Part 5. follows from 3. and 4.

1.2.2 Brown–Gersten descent In this section we show that the density structure from Def. 13 is reducing ([27] Def. 2.22) for both of our cd structures. As in the beginning of the proof of Prop. 2.10 in [28] we start with the following lemma.

Lemma 10. Let AMl.c. and let 1, 2 in AMl.c. be asymptotic submanifolds (Def. or open subschemesX ∈ containing anX openX cover→ X , of . Let ֒ , ֒ be (6 {U1 U2} X W1 → X1 W2 → X2 m-dense open subschemes. There is an m-dense open subscheme and an open cover ′ , ′ , s.t. ′ , ′ . U ⊆ X U1 U2 → U U1 ⊆ U1 ∩ W1 U2 ⊆ U2 ∩ W2 Proof. According to Prop. 15 we can ﬁnd f C∞( ) and r < r R s.t. = , ∈ X 2 1 ∈ U1 Xfr2 . By assumption 1 1, 2 2 are asymptotic spaces, hence we can ﬁnd U X′ ′ X − W X − W r2

:= ( (( 1 1) ( 1) ′ )) ( (( 1 2) ( 2) ′ )). U X − X − W ∩ X f≤r1 ∩ X − X − W ∩ X f≥r2 Being an intersection of two m-dense open subschemes of , also ֒ is m-dense. We ′ ′ X U → X deﬁne := ′ , := ′ . U1 Ufr2 18 The previous lemma, together with the following one, shows that the density structure is reducing for the open cd structure.

Lemma 11. Consider a cartesian diagram of open subschemes in AMl.c.:

/ U1 ∩ U2 U2

/ . U1 X

Let 1 2 be an m-dense open subscheme. There is an m +1-dense open subscheme ′ U ⊆and U ∩ an U open cover ′ , ′ ′ s.t. ′ , ′ and ′ ′ . U ⊆ X U1 U2 → U U1 ⊆ U1 U2 ⊆ U2 U1 ∩ U2 ⊆ U ∞ Proof. According to Prop. 15 we can ﬁnd f C ( ) and r2 < r1 R, s.t. 1 = f1r2 . Let := ( 1 2) . According to Prop. 5 there is r (r2,r1), s.t. U ′′ X ′ X U ∩ U − U ′ ∈′ := f=r is an asymptotic space of dimension < dim( ). Since 1 2 is X X ∩ X ′′ X X ⊆ U ∩ U a principal closed subscheme and f=r 1 2, is also principal, and clearly ′ := ′′ is an m +1-denseX ⊆ open U ∩ subscheme. U X ⊆ X We deﬁne ′ := ′ , ′ := U X − X ⊆ X U1 U f

Theorem 1. The open and the closed cd structures on AMl.c. are bounded. Proof. We need to show that every distinguished square

/ U ∩ W W

/ , U X where is open and is either open or an asymptotic submanifold, is reducing with respect ֒ to theU density structureW ([27] Def. 2.21). According to Lemma 10 we can assume W → X to be open and only need to show that, for any m-dense open subscheme ′ , there is an m +1-dense open ′ and a distinguished square on ′ that is containedU ⊆U∩W in , X ⊆ X X U W and ′. Lemma 11 gives us this ′ and a distinguished open square on it. Making one part slightlyU smaller, we obtain a closedX distinguished square.

Theorem 1 allows us to use Brown–Gersten descent (e.g. [27]): one declares a pre-sheaf of simplicial sets to be flasque, if it turns every distinguished square into a homotopy limit square. Since the usual pullback of simplicial sets is a homotopy pullback, if every corner is fibrant and at least one of the original arrows is a fibration, it is natural in our situation to switch from flasque to soft pre-sheaves. Definition 14. An AMl.c. is a soft sheaf, if it is a sheaf of Kan complexes and AM and for any asymptoticF ∈ Z submanifold (Def. 6) ′ ֒ the map ( ) ( ′∀X) is ∈ a l.c. X → X F X → F X fibration.

Theorem 2. Any soft is a homotopy sheaf. F ∈ Z AMl.c.

19 Consider a diagram of soft sheaves in ZAMl.c. (5) F2

/ F1 F s.t. for any 1 2 in AMl.c. the map 2( 2) ( ) is a ﬁbration. Then the usual limit of (5) is alsoX a→ homotopy X limit in F. X →F X ZAMl.c. Proof. According to [6] Lemma 4.1, to prove that a pre-sheaf is a homotopy sheaf it is enough to show that it takes values in Kan complexes and turns distinguished squares into homotopy pullbacks. According to Def. 14 soft sheaves do take values in Kan complexes, and moreover softness allows us to identify usual pullbacks with homotopy pullbacks. Finally, being sheaves they turn every distinguished square into a pullback square. To compute a homotopy pullback of (5) it is enough to substitute with a F2 → F ﬁbrant replacement and take the usual limit. According to [27] Lemma 3.5 a local weak equivalence between soft sheaves is necessarily a global weak equivalence, hence, since limits of pre-sheaves are computed object-wise, the usual limit of (5) is also a homotopy limit.

2 Groupoids and connections

As we have discussed in the Introduction, when working with higher order differential operators there is no automatic flatness of connections even along curves. On the other hand, some- times we would like to have say 2-dimensional flatness, but not necessarily the 3-dimensional one. This means we need to distinguish parts of manifolds according to their dimension. In terms of fundamental groupoids this amounts to looking at thin maps. We use C∞- realizations of the standard topological simplices to measure thinness of a morphism. To organize these simplices into an -groupoid we need to smoothen them at the edges and ∞ corners. This occupies the first part of this section. Then we define fundamental groupoid up to a given level of thinness and finally construct the infinitesimal versions of everything by taking infinitesimal neighbourhoods of the diagonals.

2.1 Groupoids 2.1.1 Smooth simplices

∞ ′ g ∞ Recall (Def. 31) that given C Sch and a subscheme the ideal mX ,X ′ C ( ) consists of functions that haveX ∈0-germs at . X ⊆ X ≤ X X Definition 15. For n Z let ∆ AM be defined as follows: ∈ ≥0 n ∈ l.c. ∞ n+1 g ∆ := Spec(C (R )/A), A = (1 x )+ m n+1 , n X i Rn+1,R − ≥0 0≤i≤n i.e. A is generated by functions that vanish on the hyper-plane P xi = 1, together with 0≤i≤n functions that have 0-germs at p Rn+1 x 0 n . { ∈ |{ i ≥ }i=0} 20 ∆ ∆ n n Notice that 0 ∼= Spec(R), while for n> 0 n is the germ of R at ∆n (R . Remark 5. Definition 15 gives us a functor n ∆ from the category of finite non- 7→ n empty ordinals and weakly order preserving maps to AMl.c.. Composing with the Yoneda ∞ embedding AMl.c. AMl.c. we obtain the C -realization functor ρ: SSet AMl.c. as a left Kan extension of n→ Z∆ along the canonical embedding n ∆ SSet→. Z 7→ n 7→ n ∈ 2 Notice that ρ does not preserve direct products, e.g. ρ(∆1) ρ(∆1) R , while ρ(∆1 ∆1) is only piece-wise smooth. To compensate for this we need to× smoothen⊂ our simplices× at the edges. The standard way to do it is by using collarings. First some notation: AM ∀X ∈ l.c. . ∆ and µ: k ֒ n we write ∆ to mean the corresponding image of ∆ in ∀ → X × µ X × k X × n Definition 16. Let , n Z and AM . A collaring on γ : ∆ F ∈ ZAMl.c. ∈ ≥0 X ∈ l.c. X × n →F is given by the following data: for any k′

1. invariance of γ: we have commutative diagrams

∆ / ∆ γ / ( µ)(X×∆ ′ ) / n ;/ (7) X × µ◦µ X × ✇✇; F ✇✇ πµ,µ′ ✇✇ ✇✇ γ ✇✇ ∆ ′ / ∆ , X × µ◦µ X × n

where πµ,µ′ is given by the obvious projection on the r.h.s. of (6); associativity: for any k′′

κ ′ ′′ µ,µ ◦µ k−k′′ ∆ / ∆ ′ ′′ ( µ)(X×∆ ′ ′′ ) / µ◦µ ◦µ R (8) µ◦µ ◦µ ❢3 0 X × ❢❢❢X❢❢ ×3 × ❢❢❢❢❢ κµ,µ′ ❢❢❢❢❢ ❢❢❢❢κ ′ ′′ ×Id ❢❢❢ µ◦µ ,µ k−k′ ( ∆ ′ ) ∆ R . X × µ◦µ (X× µ◦µ′◦µ′′ ) × 0

Remark 6. Here is the reasoning behind Def. 16: µ: k n we can choose an open neighbourhood ∆ of ∆ , and a projection∀ π→: ∆ representing Uµ ⊆ X × n X × µ µ Uµ → X × µ πId,µ. Let := S µ, then associativity implies that, making smaller if necessary, we can U µ U U ﬁnd an ι: Sn−1 ֒ and π : ι( Sn−1), s.t. π ι contracts an open neighbourhood of X × → U n−1 U → X × ◦ π( Skn−2(∆n)) in ι( S ) and is the identity away from this neighbourhood. Moreover, if X ×is represented by X′ × C∞Sch, invariance of γ with respect to contracting germs extends F X ∈ to invariance with respect to contracting open neighbourhoods, and γ = γ n−1 π. |U |ι(X×S ) ◦

21 Deﬁnition 17. Let , n Z . For any AM a smooth -family of n- F ∈ ZAMl.c. ∈ ≥0 X ∈ l.c. X simplices in is any γ : ∆n that admits a collaring (Def. 16). The pre-sheaf of smooth n-simplicesF in Xwill × be denoted→ F by Hom(∆ , ). F n F Notice that a choice of collaring is not part of the deﬁnition, only existence thereof. Thus Hom(∆ , ) is a sub-pre-sheaf of Hom (∆ , ). If is a sheaf, also Hom (∆ , ) is a n F Z n F F Z n F sheaf, and then Hom(∆n, ) is automatically a mono-pre-sheaf. To show that Hom(∆n, ) is in fact a sheaf, we needF to prove that collarings can be glued. F

Proposition 7. Let , AM , n Z and γ : ∆ . Given F ∈ ZAMl.c. X ∈ l.c. ∈ ≥0 X × n → F an open covering = , s.t. γ , γ have collarings, there is a collaring on γ, X U1 ∪ U2 |U1 |U2 whose restrictions to 1, 2 equal restrictions of the collarings on γ U2 and γ U1 respectively. X − U X − U | |

Proof. We argue inductively on the simplicial dimension, starting with dimension 0. The beginning of the induction and each step are based on the following: for k < n let Φ: Rn n n k n−k → R be an automorphism s.t. Φ Rk×0 = Id, where we choose R = R R ; then there is n n | k× Θ: [0, 1] R R , s.t. t [0, 1] Θt is an isomorphism ﬁxing R 0, Θ1 = Φ and Θ0 is an automorphism× −→ over R∀k.∈ This Θ is constructed as follows: denoting× a point in Rn by (p, q) Rk Rn−k, and correspondingly Φ=(Φ , Φ ) we deﬁne ∈ × k n−k 1 Θ(t, p, q) := (Φ (p, tq), Φ (p, tq)). (9) k t n−k

k Since Φn−k vanishes on R 0, (9) is defined also for t =0, where it equals (p, (∂qΦn−k)(p, 0)). It is easy to see that (9) is× smooth and satisfies all the conditions we wanted. Moreover, if ∞ dΦ Rk×0 is orientation preserving, it is clear that Θ 0 is C -homotopic to IdRn . | | ∞ Coming back to γ : ∆n we choose f C ( , [0, 1]), s.t. f<1 1 and f>0 . Starting with k =0Xwe × use→F (9) and f to construct∈ aX collaring aroundX ⊆ USk (∆X ), s.t.⊆ U2 X × k n restrictions of this collaring to f≤ 1 , f≥ 2 equal the collarings from 1 and 2 respectively. X 3 X 3 U U Going from Skn(∆n) to Skk+1(∆n) is also by using (9), since Skk+1(∆n) away from X × X × n k+1 Skk(∆n) is a disjoint union of germs of R at R . The resulting collaring on γ satisfies the invariance condition, since (9) is constructed by moving along the fibers of one collaring, then the other collaring, and finally the first one again. Each of these three steps produces an automorphism of γ, and hence so does their composition.

Proposition 7 immediately implies that n Z≥0 Hom(∆n, ) is a sheaf, if itself is a sheaf. We would like to argue that smooth∀ ∈ simplices abounFd. The followingF simple proposition shows that any map ∆ can be reparameterized into a smooth simplex. n →F

Proposition 8. For any n N there is a smooth Φ: ∆n ∆n, s.t. it is the identity morphism outside an open neighbourhood∈ of the boundary, and→ it deﬁnes a surjective map between the underlying topological spaces.

n n Proof. We have deﬁned ∆n as the germ of R at a linear embedding ∆n R . For any k < n and µ: k ֒ n we can choose coordinates in an open neighbourhood of⊆ µ(∆ ) s.t. all → k

22 n 1-faces of ∆n that intersect in µ(∆k) lie in coordinate hyperplanes. Moreover, we can make− these choices in a compatible way, i.e. k′

2.1.2 Fundamental and thin groupoids Deﬁnition 18. The fundamental -groupoid of a sheaf is ∞ F ∈ ZAMl.c. Π( ) := Hom(∆ , ) . F { n F }n∈Z≥0 Since we smoothen by pre-composing with contractions, the fundamental -groupoid is ∞ functorial in . To justify calling Π( ) an -groupoid, we need to show that smooth horns extend to smoothF simplices. F ∞

Notation 1. For any 0 k < n, applying the C∞-realization functor, we obtain the k-th ∞ ≤ Λ ∆ C -horn in AMl.c. , that we denote by n,k n. Z n ⊂ Realizing ∆n in R s.t. the center of the k-face is the origin, the k-th face itself lies n−1 within R = xk = 0 , and the k-th vertex is the point xk = 1, x6=k = 0 we have the { } n n−1 ﬂattening Ξn,k : ∆n ∆n−1 given by the projection R R . In particular we have Ξ : Λ ∆ . → → n,k n,k → n−1 Our deﬁnition of collarings uses germs of boundary, but to extend smooth horns we will need to approximate them by simplices. This requires extending collarings from germs to open neighbourhoods. In some cases this extension is always possible.

′ Lemma 12. Let AMl.c. s.t. for any ﬁnite set S and any s s∈S there is C∞Sch, a morphismF ∈ of Z pre-sheaves ′ and s S a factorization{X → F} ′ X ∈. X → F ∀ ∈ Xs → X → F Then for any n>k Z≥0, AMl.c., if the restriction of γ to each ∆n−1 in Λn,k is smooth, there is a∈ factorizationX ∈ X × X ×

γ Λ / n,k❖ t9 X × ❖ ′ tt F ❖❖❖ γ tt ❖❖❖ tt Id×Ξ ❖❖ tt n,k ❖' tt ∆ , X × n−1 s.t. γ′ is smooth.

Λ ∞ Proof. By deﬁnition n,k is a colimit of C -simplices computed within AMl.c. . Since γ ′ ′ ∞ Z (factors through Λn,k ֒ , with C Sch, γ factors through colim ( ∆m × X × → X →F X ∈ ∆m֒→Λn,k X ∞ with the colimit computed within C Sch. The assumption that each γ X×∆n−1 has a collaring immediately implies then that γ factors through Id Ξ (gluing over| germs). × n,k Proposition 9. Let be as in Lemma 12. Then Π( ) is a sheaf of Kan complexes. F ∈ ZAMl.c. F 23 Proof. From Lemma 12 we immediately conclude that every Λ , whose restriction X × n,k →F on each ∆n−1 is smooth, can be extended to a smooth ∆n (just compose ∆ X × with the ﬂattening morphism). X × → F X × n−1 →F

We would like to prove that for some AMl.c. of interest to us, the pre-sheaf of simplicial sets Π( ) is a soft sheaf (Def. 14).F Of ∈ course Z the main tool in proving such results F is by using a partition of unity. The following lemma formalizes this procedure.

′ ′ Lemma 13. Let AMl.c., , open subschemes, s.t. . Suppose we have smooth γ : Λ X ∈ , γ′ : U′ U∆⊆ X , s.t. U ⊆ U X × n,k →F U × n →F ′ γ ′ Λ = γ ′ Λ . |U × n,k |U × n,k ′′ ′′ ′′ ′ There is a smooth γ : ∆ s.t. γ Λ = γ, γ ∆ = γ ∆ . X × n →F |X× n,k |U× n |U× n Proof. Making ′ smaller, if necessary, we can assume that we can choose collarings on γ and γ′ that areU equal where they are both deﬁned. Using Lemma 12 we see that γ factors through the ﬂattening morphism

(Ξ : Λ ֒ ∆ ∆ . (10 n,m X × n,m → X × n → X × n−1 ′ ′ ′ As in Rem. 6 we can find an open neighbourhood Λn,m of Λn,m in ∆n and ′ ′ ′ ′ U × ′ U × U × ι: ∆n−1 ֒ Λn,m, s.t. ι( ∆n−1) ( Λn,m)= ∂ Λn,m, and the collaring U ′ × → U × U × ∩ U × U × on γ ′ Λ defines a projection |U × n,m π : ′ Λ ι( ′ ∆ ), U × n,m −→ U × n−1 ′ ′ ′ ′ ∆ ′ ′ Λ s.t. γ U ×Λn,m = γ ι(U × n−1) π. Since Ξn,m U and π U × n,m are defined using the same collaring| on γ′, we have| a factorization◦ | |

′ π / ′ Λn,m / ι( ∆n−1) U × ❥❥4U × ❥❥❥❥ Ξn,m ❥❥❥ ❥❥❥❥ ❥❥ ′ ∆ . U × n−1 Therefore, applying Ξ to all of ′ ∆ we have an embedding n,m U × n . ∆ ′ ֒ ( ∆ ′ )α: ′ ∆ ι U × n → U × n−1 → U × n ′ ′ Using convexity of ∆n we can find a homotopy h: [0, 1] ∆n ∆n between α (at ′′ × U × ′′ →′ U × ′ ′′ 0) and IdU ×∆n (at 1). Let be an open subscheme, s.t. = and = ∅. We can choose f C∞( U, [0,⊆1]) X, s.t. ′′ , .U Now∪U weX defineU a composite∩ U ∈ X U ⊆ Xf=0 U ⊆ Xf=1 morphism ν : ′ ∆ [0, 1] ′ ∆ ′ ∆ , U × n −→ × U × n −→ U × n where the first arrow is f Id Id and the second arrow is h. It is clear that ν U ′′∩U ′ = α U ′′∩U ′ , × × ′ ′′ ′ ∆ | | while µ U = Id. By construction γ U ′′∩U ′ ν U ′′∩U ′ equals ( ) n obtained | | ◦ | ′ U ∩ U ′′× → F from γ using (10). Therefore we have an extension of γ ν to γ : ∆n s.t. ′′ ◦ X × → F γ Λ = γ. |X× n,m 24 ′ In order to use Lemma 13, we need to show that for certain AMl.c. a smooth - family of n-simplices can be extended to a smooth -family whereF ∈′ Z is an asymptoticX submanifold and is an open neighbourhood.U X ⊆ X U ⊇ X If is represented by a manifold , we can choose a good covering = S s, which F M M s∈SU is a locally finite open covering, s.t. each non-empty intersection is isomorphic to Rk, k = dim . Then we can try to extend families locally, starting with the smallest intersections. M Since we have plenty of objects in AMl.c., that are not germ-determined (mostly due to the infinitesimal structure), we need to demand the good coverings to be finite.

′ ′ ′ Lemma 14. Let AMl.c., and let be an asymptotic submanifold. Any Φ : , where is aX manifold ∈ admittingX a finite⊆ X good covering, can be extended to Φ: X →, M M U→M where ′ is an open subscheme of . U ⊇ X X ′ −1 Proof. Let s s∈S be a finite good covering on , and denote s := Φ ( s). Consider {U } ′ M U U the germ ′ = ′ . For any S S, if = ∅, also ′ = ∅, therefore we can X S Us T s T Us X s∈SX ⊆ s∈S′U s∈S′X ′ ′ extend Φ to X ′ . Now using the fact that is finitely presentable, we see that Φ extends to anX open→ neighbourhood M of ′ in . M X X ′′ ′′ Theorem 3. For AMl.c. Π( ) is a soft sheaf of Kan complexes in either one of the following cases: X ∈ X

(a) ′′ is a manifold admitting a ﬁnite good covering, X M (b) ′′ is the germ of at a puncture, where is a manifold admitting a ﬁnite good coveringX M M

Proof. (a) Let n N, 0 m n, and suppose we are given a smooth family ∈ ≤ ≤ µ: ( Λ ) ( ′ ∆ ) . X × n,m ∪ X × n −→ M

Lemma 14 tells us that µ can be extended to an open neighbourhood ∆n of ( Λ ) ( ′ ∆ ), and then, according to Lemma 13, we can ﬁndU⊆X× an extension X × n,m ∪ X × n to all of ∆ . X × n (b) Suppose we are given

. ֒ ′′ ( ∆ ′ ) ( µ: ( Λ X × n,m ∪ X × n −→ X →M Using the previous part we can extend µ to

µ′ : ∆ . X × n −→ M ′′ Let i i∈D be a regular system of open submanifolds of , s.t. ∼= T i. We {U } M X i∈DU would like show that there is an open neighbourhood

( Λ ) ( ′ ∆ ) ∆ , X × n,m ∪ X × n ⊆U⊆X× n

25 ′ ′′ s.t. i µ ( ) i. By assumption is locally compact, hence working with pre-images∀ ∈ D of compactU ⊆ U pieces, we can assumeX that all of ′′ is compact, i.e. is a ′−1 X D sequence. Then µ ( i) i∈D is a regular sequence of open neighbourhoods of ( ′ { U } X × Λn,m) ( ∆n). Also ∆n is locally compact, and working on each compact piece separately,∪ X × we can assumeX × that all of ∆ is compact. Finally absence of X × n R-points in means that also ∆n cannot have R-points, and we can use Lemma 5. Having foundX we apply LemmaX × 13. U

Now we would like to describe the ﬁltration of fundamental groupoids by the level of thinness that the simplices are allowed to have.

We start with some notation. For AMl.c. we have the constant simplicial pre-sheaf , i.e. equals in each simplicial dimension.F ∈ Z If is a sheaf, is a homotopy sheaf. We F F F F F∆ will call the trivial -groupoid on . The unique projections n pt n∈Z≥0 define the { → }.obvious F ֒ Π( ), whose∞ image consistsF of degenerate simplices F → F ∞ n n We have defined ∆n as the C -germ of R at the topological simplex ∆n R . Therefore ⊆ n for any R-point p ∆n, the germ (∆n)p of ∆n at p is isomorphic to the germ of R at a point. In particular for any∈ k n there are many possible C∞-morphisms κ: (∆ ) (∆ ) . ≤ n p → k p ∆ Definition 19. Let AMl.c., n k Z≥0, AMl.c. . A smooth -family n ∆ X ∈ ≥ ∈ F ∈ Z X X × →F is k-thin, if p n there is a finite open covering = S s, for each s Sp a morphism ∀ ∈ X s∈SpU ∈ κ : (∆ ) in AM / with dim k, and a factorization s Us × n p → Us × Xs l.c. Us Xs ≤ ∆ / ∆ / s ( n)p / n ;/ U × ❖❖ X × ✈✈; F ❖❖❖ ✈✈ ❖❖❖ ✈✈ κs ❖❖ ✈✈ ❖' ✈✈ . Us × Xs It is immediately clear that for any morphism ′ a k-thin -family of simplices X → X X induces a k-thin ′-family of simplices, and moreover k-thin simplices are stable with respect to simplicial operators.X

Definition 20. Let , k Z , the k-thin -groupoid of is the sub-pre-sheaf F ∈ Z ∈ ≥0 ∞ F Θk Π( ), consisting of k-thin families of simplices. F ⊆ F Since we allow arbitrary finite coverings in Def. 19, and our topology is Zariski, it is clear that Θk is a sheaf of simplicial sets, whenever Π( ) is a sheaf. In the proof of Lemma 12 F F we extended smooth families of horns to smooth families of simplices by pre-composing with the flattening morphism. Therefore we immediately have the following result.

k Proposition 10. Let be as in Lemma 12, k Z≥0, AMl.c. ΘF ( ) is a Kan complex. F ∈ Z ∀ ∈ ∀X ∈ X

Also in the proof of Thm. 3 we have used pre-composition. Partially with the ﬂattening map, and partially with projection on an n 1-simplex contained in an n-simplex. Hence we have the following corollary. −

26 Theorem 4. Let be as in Thm. 3, then Θk is a soft sheaf k Z . F F ∀ ∈ ≥0 Clearly ∆ is 0-thin, iﬀ it factors through the projection ∆ . X × n → F X × n → X Therefore Θ0 = . When we let k grow, we allow more and more maps ∆ , and F ∼ F X × n →F thus we have a sequence of inclusions

= Θ0 Θ1 ... Θk ... Π( ). F ∼ F ⊆ F ⊆ ⊆ F ⊆ ⊆ F k k Moreover, since Π( )n = ΘF n for all n k, we have Π( ) = colim ΘF with the colimit F ∼ ≤ F k∈Z≥0 computed in . Z AMl.c. 2.2 Inﬁnitesimal connections 2.2.1 Inﬁnitesimal groupoids

∞ ∞ Recall ([5], §3) that there are are two functors Rnil, R∞ : C Sch C Sch that send A C∞R to A/ nil√0, A/ ∞√0 respectively, where nil√0, ∞√0 are the nil-→ and -radicals ([5], §2).∈ ∞ Correspondingly to every ∞ there are associated two de Rham spaces: F ∈ ZC Sch := R , := R . FdR F ◦ nil FdR∞ F ◦ ∞ According to our deﬁnition AM , if and only if R ( ) is a locally compact reduced X ∈ l.c. ∞ X asymptotic manifolds with corners. This immediately implies that AM R ( ) X ∈ l.c. ⇒ ∞ X ∈ AMl.c. and Rnil( ) AMl.c.. Therefore we can deﬁne dR, dR∞ also for AMl.c. . Both de RhamX ∈ space constructions are functorialF in .F On the otherZ hand there are F canonical natural transformations

R Id , R Id . nil −→ AMl.c. ∞ −→ AMl.c.

Definition 21. For any AMl.c. , k Z≥0, we define the formal k-thin and the infinitesimal k-thin groupoids of F ∈to Z be the respective∈ homotopy pullbacks (computed in ): F ZAMl.c. k / k k / k XF / ΘF YF / ΘF (11)

/ Θk / Θk , F dR F dR F dR∞ F dR∞

k .where use use the canonical inclusion ֒ ΘF as degenerate simplices k F → Using Π( ) instead of ΘF we obtain the formal and inﬁnitesimal fundamental groupoids ∞ ∞ F XF , YF respectively.

k k We would like to describe sections of XF and YF explicitly. This means computing the k homotopy pullbacks. We have already seen that in some cases ΘF is a soft sheaf and in k particular fibrant, while is always fibrant, if is a sheaf. We need to see when ΘF dR and k F F ΘF dR∞ are fibrant. Lemma 15. , Lemma 15 and If is a soft sheaf, so are and . F ∈ ZAMl.c. FdR FdR∞ 27 Proof. Since reductions of Zariski open subschemes are again Zariski open subschemes, it is clear that ( )red is a sheaf. The fact that ( )red( ) is a Kan complex is obvious, since takes values in KanF complexes. As asymptoticF submanifoldsX reduce to asymptotic submanifolds,F the same reasoning shows that ( ) is soft. F red Theorem 5. Let AM be as in Thm. 3, then X ∈ l.c. Π( )( ′) Π( ) ( ′), Π( )( ′) Π( ) ( ′), X X −→ X dR∞ X X X −→ X dR X Θk ( ′) Θk ( ′), Θk ( ′) Θk ( ′) X X −→ X dR∞ X X X −→ X dR X are fibrations ′ AM , k Z . ∀X ∈ l.c. ∀ ∈ ≥0 ′ ′ ′ Proof. We need to show that, given γ : Λn,m , γ : ( )red ∆n , that agree ′ X ×′′ ′ → X X × → X on ( )red Λn,m, there is an extension to γ : ∆n . X × ′ ′ X × → X ∞ ′ ( ) Since the inclusion ( )red ֒ corresponds to a surjective morphism C ∞ ′ X → X ′ ′ X → C (( )red), we can use the finite good covering on to extend γ γ to ∆n . ThisX shows the first case. M ∪ X × → X ֒ If is the germ of at a puncture, we first extend the composite of γ γ′ with ′ X∆ M ∪ X →M to n , and then notice that the image has to be contained in T i, where i i∈D X × →M i∈DU {U } is a regular system computing the germ of the puncture, because this is true in the reduced case.

k ∞ k ∞ Using Thm. 5, Lemma 15 and Thm. 4 we can compute XX , XX , YX , YX explicitly for each k Z≥0, simply by computing the usual pullbacks in (11). For example given ′ ∈ k ′ ′ ∆ AMl.c. the simplicial set YX ( ) consists of k-thin maps n , s.t. the . compositeX ∈ ( ′) ∆ ֒ ′ ∆ X factors through the projectionX × on (→′) X X red × n → X × n → X X red 2.2.2 Relation to pair groupoids Now we would like to compare different groupoids and infinitesimal groupoids associated to the same AM . The first statement is obvious. X ∈ l.c. Lemma 16. Let AM be of dimension k Z . Then k′ k we have an equality12 X ∈ l.c. ∈ ≥0 ∀ ≥ ′ Θk = Π( ). X X In addition to the fundamental groupoids of various thinness we have the pair groupoid ×n P AMl.c. associated to each AMl.c. . In simplicial dimension n P is just . The correspondingF ∈ Z infinitesimal groupoids,F ∈ Z are obtained as homotopy pullbacksF F

/ / ∆F / P ∆F / P b F e F

/ P / P . F dR F dR F dR∞ F dR∞ 12This is not just an isomorphism, but an actual equality of sheaves.

28 In the case is represented by an with a ﬁnite good covering, the same argument as in the case ofF thin groupoids showsX that the usual pullbacks are also homotopy pullbacks. Then ∆F and ∆F are the same groupoids as in [5] Prop. 6. I.e. ∆F consists of formal b e b neighbourhoods of the diagonal and ∆F in dimension n Z≥0 is the colimit (computed in ′ ×n ′ e ∈×n AMl.c. ) of , s.t. R∞( ) is the diagonal ∆ . Moreover, as it was shown in Z X ⊆ X X ⊆ Xn [5], the reﬂection of ∆ in AM / is the groupoid × , i.e. it consists of the germs X l.c. X {X∆ }n∈Z≥0 around diagonals. e Evaluating at the vertices of simplices we have morphisms of groupoids (natural in ): F ν : Π( ) P . F F −→ F The following lemma is obvious.

k k Lemma 17. For any k Z ν k : Π(R ) PR is a trivial fibration. ∈ ≥0 R → ∞ ∞ ∞ As XF , YF , ∆F , ∆F are defined as homotopy pullbacks, we have the induced νF : XF ∞ b e → ∆F , νF : YF ∆F . For an arbitrary manifold the groupoid Π( ) can be complicated, b → e ∞ ∞ M M but the infinitesimal versions XM, YM depend only on local data, hence we can use Lemma 17 and obtain the following.

Proposition 11. Let AM be a manifold admitting a ﬁnite good covering, then X ∈ l.c. ∞ ∞ νF : XX ∆X , νF : YX ∆X (12) −→ b −→ e are weak equivalences.

Proof. The requirements that we have put on allow us to compute all four groupoids X involved as simple pullbacks. In particular the morphisms in (12) are over and FdR FdR∞ respectively. Therefore it is enough to look at the morphisms between the ﬁbers over dR and . Taking pre-images of the contractible charts on we obtain a ﬁnite open coveringF of FdR∞ X ′, on each piece of which the two maps are weak equivalences, hence the global morphism ofX simplicial sheaves is a weak equivalence.

If we admit punctures in Rk, the statement in Lemma 17 ceases to be true, of course. However, by restricting to the germs around the diagonals in PRk we can ﬁnd a section of νRk . Proposition 12. Let AM be any asymptotic manifold. We can choose η : ∆ X ∈ l.c. X → Π( ), s.t. ν η = Id∆ . X X ◦ X Proof. Suppose ﬁrst that is a manifold. We choose a metric on and obtain isomorphisms between germs of 0’s in tangentX spaces and germs of the manifoldX itself. Then any morphism ′ (∆X )n can be seen as n +1 morphisms into then germ of 0 in the corresponding Xtangent→ space. Using the linear structure on the tangent space we extend it to ′ ∆ . X × n If is the germ of a manifold at a puncture, we have ∆X ֒ ∆M as well as Π( ) X֒ Π( ). Since our extensionsM to simplices never leave germs of→ points, it is clear thatX any→ η asM above for maps ∆ Π( ). M X → X

29 ′ ′ Looking at what happens if we go from to ( )red in the proof of Prop. 12, we see that the inﬁnitesimal part of P lifts to theX inﬁnitesimalX part of Π( ). Of course, if we X ∞ ∞ X want to conclude something about XX of YX , we should be careful that our constructions do compute the required homotopy pullback. Hence the assumptions in the following.

Theorem 6. Let be as in Thm. 3. Then we can choose sections of X ∞ ∞ νX : XX ∆X , νX : YX ∆X . −→ b −→ e In each case any two possible choices are homotopically equivalent.

Proof. The only part we have not explained yet is why any two choices of sections in each case should be homotopically equivalent. This would immediately follow from νX being a weak equivalence in each case. Since infinitesimal simplices (of both kinds) have to be contained in Rk-charts of , and we have only finitely many of these charts, it is clear that X each νX consists of trivial fibrations.

2.2.3 Connections and integrability

′ Deﬁnition 22. Let Φ: be a morphism in AMl.c. and let k Z≥0 or k = . A k-ﬂat connection on Φ isF a →F factorization of Φ as followsZ ∈ ∞

Φ / ′ ♣8 F ♣♣♣ F ♣♣♣ ♣♣♣ ♣♣♣ k ΘF .

An inﬁnitesimal k-ﬂat connection on Φ is a factorization of Φ as follows

Φ / ′ ♣8 F ♣♣♣ F ♣♣♣ ♣♣♣ ♣♣♣ k YF .

k k Instead of YF we could have used XF , which contains strictly less information, in general. In that case we would use the adjective formal instead of infinitesimal. Since we are mostly interested in the infinitesimal connections, we will suppress the formal ones from now on. For some ′ it is natural to expect a k-connection for any morphism into ′, for example ′ k F F if = B for an abelian group in AMl.c. . We will not address this question here, but simplyF assumeA that we can start withZ a given connection. k However, we would like to single out integrable infinitesimal connections. Recall that YF k k is defined as the pullback of ΘF over dR∞ ΘF dR∞ . In particular there is the canonical embedding F → Yk Θk . (13) F −→ F k ′ Definition 23. An infinitesimal k-flat connection YF is integrable, if it factors through (13). →F

30 In the Section 2.2.2 we have seen that, if is representable by an asymptotic manifold admitting a finite good covering, an integrableF infinitesimal k-connection on ′ will factor through the groupoid ∆ consisting of germs around the diagonals in F →F×n . F {F }n∈Z≥0 Moreover, from [5] we know that such factorization is unique, if it exists. In the next section we assume that the infinitesimal connections we work with are always integrable, allowing us to use germs of diagonals. This is a somewhat lazy approach. Why use infinitesimal structure at all, if we have integrable connections from the start? However, we would like to argue that infinitesimal k-flat connections are interesting in their own right, with or without the integrability assumption. In this paper we make this assumption, for the lack of space, if not anything else. In general there are two questions that come to mind immediately: what is the cohomology theory governing obstructions to integrability? The corresponding deformation theory should be quite unusual, as it should describe obstructions for all orders of vanishing, not just the polynomial ones. The second question is whether we can integrate non integrable infinitesimal connections in a wider class of functions than just the C∞-functions. Even more interesting would be to find solutions in classes that still have the C∞-ring structure.

3 Beilinson–Drinfeld Grassmannians and factorization algebras

3.1 The problem of constructing factorizable line bundles

Recall that M AMl.c. AM denote the sites of the usual manifolds, the locally compact asymptotic manifolds⊂ with⊂ corners, and all asymptotic manifolds with corners, equipped with

Zariski topologies. We denote by M, AMl.c. , AM, M, AMl.c. , AM the corresponding categories of pre-sheaves with valuesZ is SetZ and SSetZ respectively.Z Z Z Let be an abelian group-object in . We assume that and all of its classifying A ZAM A spaces Bk , k N are obtained by a left Kan extension from an abelian group object over M, equippedA ∈ with the usual (not Zariski) topology. In particular AM any Bk factors through an object in M. This assumption is automatically∀X satisfied, ∈ if X → A A is representable by an abelian finite dimensional Lie group. We fix a k 2 and a k-dimensional M admitting finite good covering. Given a ≥ N ∈ group object AMl.c. and a multiplicative k-gerbe with integrable infinitesimal k-flat connection G ∈ Z α∇ : Xk Θk Bk+1 , BG −→ G −→ A we would like to construct an -torsor on the Beilinson–Drinfeld Grassmannian corresponding to and , that has theA factorization property. N G First we need to explain what we mean by Beilinson–Drinfeld Grassmannians in our setting. We follow [14] §0.5.3. Let FSet be the category of finite non-empty sets and surjections. For every S FSet we write S for the S-fold direct product. Every π : S S in FSet ∈ N 1 → 2 defines S2 S1 by mapping the copy of corresponding to s S diagonally into N → N N ∈ 2

31 π−1(s) op . In this way we obtain an FSet -diagram ∆ in M, and using Yoneda embedding weN consider it as a diagram in . N ZM Deﬁnition 24. The Ran-space Ran corresponding to is the colimit of com- N ∈ ZM N N∆ puted in . ZM

Given S M, a morphism Φ: S RanN is an equivalence class of sets of maps Φi : S n ∈ → { → i=1 with arbitrary n N, where the equivalence is given by renumbering. We write ΓΦi N}for the graph of Φ in S ∈ , and Γ := Γ S . Clearly Γ is independent of the i Φ S Φi Φ ×N 1≤i≤n ⊆ ×N numbering. Now we take the germs

◦ Γ˙ := (S ) , Γ := (S Γ ) . Φ ×N ΓΦ Φ ×N \ Φ ΓΦ

These are objects in AMl.c., and we deﬁne pre-sheaves on M:

◦ ◦ ˙ ˙ Gr( , )(S) := G Hom(ΓΦ, ), Gr( , )(S) := G Hom(ΓΦ, ). N G G N G G Φ∈RanN (S) Φ∈RanN (S)

◦ The obvious projections Gr(˙ , ) RanN , Gr( , ) RanN have groups as ﬁbers, and the former are subgroups of theN G latter.→ N G →

Deﬁnition 25. The Beilinson–Drinfeld Grassmannian corresponding to , is GrG( ) := ◦ N G N Gr( , )/Gr(˙ , ). N G N G

Recall (e.g. [13] §20.3) that GrG( ) is the moduli space of principal -bundles on , trivialized away from ﬁnite sets of points.N Indeed, without dividing by Gr(˙ G , ) we obtainN N G bundles on glued out of trivial bundles on the germs and on the complements. Dividing leaves a chosenN trivialization only on the complements.

3.2 The solution ◦ The strategy for constructing an -torsor on GrG( ) is by constructing a bundle on Gr( , ), A N ◦ N G that restricts to a trivial bundle on Gr(˙ , ). We would like not just any bundle on Gr( , ), N G ◦ N G but a central extension (relative to Ran ), hence we need Gr( , B ) deﬁned as follows N N G ∈ ZM ◦ ◦ 13 Gr( , B )(S) := G Hom(ΓΦ, B ). N G G Φ∈RanN (S)

◦ Fixing a Φ: S Ran we have a simplicial set B (Γ ). We would like to promote this → N G Φ simplicial set to a pre-sheaf of simplicial sets on AMl.c./S:

◦ ◦ S GrΦ( , B )( ) := HomS( ΓΦ, B S), ∀X → N G X X ×S G × 13Recall that Hom denotes the simplicial set of maps.

32 ◦ ◦ where we use the composition Γ ֒ S S to realize Γ in AM /S, and the subscript Φ → ×N → Φ l.c. in HomS means computing the mapping space in AMl.c./S. Now we have the evaluation morphism Z ◦ ◦ ev : ΓΦ GrΦ( , B ) B S. ×S N G −→ G × Just as with B we lift α∇ to α∇ : Xk S Bk+1 S, and composing with the evaluation G S BG × → A× morphism we have

◦ ∇ k k α k+1 (X◦ ) GrΦ( , B ) XBG S B S. (14) ΓΦ/S ×S N G −→ × −→ A ×

◦ ∇ k Since α is integrable, the composite morphism in (14) factors through (Θ◦ ) GrΦ( , B ), ΓΦ/S ×S N G and therefore (Section 2.2.3) through

◦ k+1 ∆◦ GrΦ( , B ) B S, ΓΦ ×S N G −→ A ×

◦ where ∆◦ is the groupoid consisting of germs around diagonals in direct powers of ΓΦ. We ΓΦ would like to rewrite this map slightly diﬀerently:

◦ Hom ∆ k+1 14 GrΦ( , B ) S( ◦ , B S). (15) N G −→ ΓΦ A × Now we are ready to perform transgression. Proposition 13. There is a morphism

Hom ∆ k+1 2 S( ◦ , B S) B S, (16) ΓΦ A × −→ A × that is uniquely defined up to a unique homotopy equivalence. Proof. To perform transgression we need to integrate a connection around a cycle. To do ◦ this we would like to substitute ΓΦ with an asymptotic manifold, whose homotopy type over S is that of a disjoint union of k 1-spheres. By this we mean an asymptotic manifold, that can be broken into contractible pieces− over S (i.e. germs of manifolds contractible to S), s.t. the diagram of the pieces is that of a disjoint union of k 1-spheres. − We choose a metric on , and obtain a geodesic coordinate neighbourhood around each point. For any p S Nwe can choose a small enough open p, s.t. over each ∈ U ∋ U connected component of ΓΦ is contained in the geodesic neighbourhood around the image of p. Therefore, restricting to we can assume that S is connected and there is a chosen U coordinate system around each component of ΓΦ. We choose a numbering of the copies of S in each connected component of ΓΦ. Then, within each connected component, we connect by curves consecutive punctures in the fibers of S S.15 We denote the resulting subset of S by Γ . Clearly fibers of Γ S are N × → N × Φ Φ → 14Recall that Hom denotes the simplicial pre-sheaf of mapping spaces. 15For example to choose the shortest path from the i-th puncture to the previous i 1 punctures. − 33 disjoint unions of contractible topological spaces. Let S be the germ of S at ( S) ΓΦ. As is k-dimensional, it is clear that S/S has the homotopy type ofN ×a disjointN × union\ of k N1-spheres. − ◦ The inclusion S ֒ Γ gives us → Φ Hom ∆ k+1 Hom ∆ k+1 S( ◦ , B S) S( S, B S). ΓΦ A × −→ A × Now we use the assumption on , that it is a left Kan extension of a sheaf on M. This A implies that we can regard S as k-dimensional manifold, and instead of ∆S we have a simplicial manifold • = n n∈Z≥0 consisting of open neighbourhoods of the diagonals in S×n . U {U } { }n∈Z≥0 For any AM an n-simplex in Hom ( , Bk+1 S) over is a morphism X ∈ l.c. S U• A × X k+1 ∆n • (B ) S, X ×S U → A ×

i.e. it is a morphism S (Bk+1 )∆n S together with a chosen trivialization on for eachX contractible × → AS. Breaking× S into contractible pieces over S and choosingX×W→X one point in each piece, weW obtain ⊆

Hom ∆ k+1 Hom k−1 k+1 2 S( S, B S) S( G S , B S) = Y B S, A × −→ A × ∼ A × 1≤i≤m 1≤i≤m

where m is the number of k 1-spheres in the disjoint union. Now, using the group structure on B2 , we obtain altogether− A Hom ∆ k+1 2 S( ◦ , B S) B S. ΓΦ A × −→ A × We need to show that this morphism is independent of the choices that we have made, up to a homotopy equivalence. We chose four things: the numbering of copies of S in ΓΦ, curves that connected punctures, decomposition of S into contractible pieces over S, points in each of the contractible piece. Changing the latter two choices clearly results in a unique connecting homotopy equivalence, since the trivializations are uniquely defined on each contractible piece of S. Having two different choices of the curves, connecting the punctures, and the corresponding two different S, S′, we can choose surfaces connecting the curves, and then take the complement S′′ of the resulting union of contractible pieces. Clearly S′′ S S′, and ⊆ ∩ the corresponding three asymptotic manifolds have the same homotopy type over S. In this way S′′ provides the required homotopy equivalence. This homotopy equivalence is dependent on the choice of the connecting surfaces, but choosing a filling of dimension 3, we obtain a 2-homotopy, etc.16 The process stops when we reach the dimension of . N Finally we need to show that our construction is independent of p S and the neigh- ∈ bourhood around it that we have chosen. In other words we need to show that the U 16 The ability to always choose a filling exists because every connected component of ΓΦ lies within an Rk S-chart on S. × N × 34 morphism into B2 over a set of distinct punctures (restricting to a subset of ) is the sum of morphisms overA individual punctures (computing the morphism over the subsetU itself). The difference between the two is in using of connecting curves. I.e. integrating the connection separately around each puncture and then adding up, or connecting them all and integrating around the resulting contractible space. But adding up the results of integration around two punctures is the same as choosing a trivialization over the disjoint union of two contractible pieces, each at a different puncture. Since is abelian, any such choice of trivialization produces the same result (the conjugation isA trivial). Working with connected components over the entire just provides one possible choice. U ◦ k+1 ֒ If we restrict (16) to the image of Hom (∆ ˙ , B S), given by the inclusion Γ S ΓΦ A × Φ → Γ˙ Φ, we clearly obtain trivial gerbes, because the entire Γ˙ Φ/S is a disjoint union of contractible pieces. The factorization property is obvious from the construction.

3.3 Taking global sections Constructing factorization algebras from factorizable line bundles on Beilinson–Drinfeld Grassmannians is fairly straightforward in differential geometry. The reason for this is that the fibers of Gr ( ) Ran become affine, if we allow all convenient algebras, and not G N → N just C∞-rings. ◦ Let Φ: S Ran be represented by an n-tuple Φ : S , and let Γ S → N { i →N}1≤i≤n Φ ⊆ N × be the punctured germ around the union ΓΦ S of the graphs of Φi 1≤i≤n. For any ◦ ⊆ N × ◦ { } ◦ ′ ′ ′ S M the value of GrΦ( , ) on S S is HomAMl.c. (ΓΦ S , ). By definition ΓΦ is a limit of∈ a system U , whereNU G= V ×Γ and V is an open× neighbourhoodG of Γ in S. { α} α α \ Φ α Φ N × Assuming that is a finite dimensional Lie group we immediately see that G ◦ ′ ′ HomAM (ΓΦ S , ) = lim HomM(Uα S , ). (17) l.c. × G ∼ α × G

Suppose that each HomM(Uα, ) is representable, i.e. it is the spectrum of some ring, then the right hand side of (17) is alsoG representable, if our category of rings has enough limits. ∞ This is of course not true for the category of finitely generated C -rings. Also HomM(Uα, ) are not representable in this category in general. We need to consider much larger categories.G Let Born be the category of complete bornological spaces over R, and let (Born) be the category of commutative, associative unital algebras in Born, considered togetherR with ∞ the usual (projective) tensor product. As for any manifold the ring C (Uα) has a natural bornology, given by the canonical Fréchet topology on C∞(U ), i.e. we have C∞(U ) α α ∈ (Born). In particular we obtain the category of quasi-coherent sheaves on Uα as the R ∞ category of modules in Born over the algebra C (Uα). The natural bornologies on rings of smooth functions on finite dimensional manifolds are very special. They are the topological bornologies, these spaces are often called convenient. One of the properties of the subcategory Conv Born of convenient spaces, is that there are two tensor products present, and they agree, if⊂ one of the factors is nuclear. It so happens that the rings of smooth functions on finite dimensional manifolds are nuclear. So in our case ∞ ∞ ′ we can substitute C (Uα) C (S ) with the injective tensor product. This tensor product ⊗b 35 preserves all limits, and therefore, using the special adjunction theorem, we conclude the following

Proposition 14. The functor

◦ ∞ ∞ op Hom(C ( ), C (ΓΦ) ): M Set G ⊗−b −→ ◦ is representable by a commutative ring coHom(C∞( ), C∞(Γ )) in the category of conve- G Φ nient spaces over R. A similar statement holds for Γ˙ Φ.

Notice that we have three different levels: our site M (as well as AMl.c.) consists of finitely generated C∞-rings, but we also consider convenient rings, which are not in this site, on the other hand the modules over all of our rings can be be any bornological spaces, not necessarily convenient. Going back to line bundles over the Beilinson–Drinfeld Grassmannians we can view ◦ Gr ( , B ) as the spectrum of a (simplicial) convenient ring P . This ring is obtained Φ N G • using Prop. 14 and then tensoring with C∞(S), hence it is a C∞(S)-algebra. In a similar fashion let G• be the simplicial convenient ring whose spectrum is Gr˙ Φ( , B ). A morphism 2 N G Spec(P•) B defines a module M• over P•. As the two fibers over Φ are both groups, → A ◦ the rings P , G have comultiplications, and since M is defined over the nerve of Gr( , ), 0 0 • N G M0 is a bimodule over P0. Consider the following diagram:

M0 ⇒ M0 P0 M0 G0, (18) ⊗b → ⊗b where the ﬁrst two arrows are the comultiplication and the trivial comultiplication, and the ◦ last arrow is given by the restriction Gr(˙ , ) Gr( , ). Let K be the equalizer of the N G → N G two composite maps in (18). This is the space of global sections of the line bundle over the ∞ ﬁber of GrG( ) RanN over Φ. It carries a natural C (S)-module structure, so altogether we obtain theN following→

Theorem 7. Any factorizable line bundle on the Beilinson–Drinfeld Grassmannian GrG( ), constructed in section 3.2, induces a factorization algebra on . N N

A Subschemes of C∞-schemes

In this appendix we collect some definitions and statements, used throughout the paper. All these constructions are rather standard, either in differential or in algebraic geometry. The only original input that we put here is by using the -radicals from [5]. They allow us to translate many standard algebraic-geometric results into∞ differential geometry.

A.1 Urysohn’s lemma Definition 26. 1. Let C∞Sch, f C∞( ), an open subscheme of C∞Sch defined by f is :=X Spec(C ∈ ∞( ) f∈−1 ). X X ∈ Xf6=0 X { } 36 2. A closed subscheme of is Spec(C∞( )/A) for some ideal A C∞( ). Given two X X ∞ ≤ X closed subschemes 1, 2 , defined by A1, A2 C ( ), their union is 1 2 := Spec(C∞( )/(A +XA X)). ⊆ X ≤ X X ∪ X X 1 2 3. A morphism Φ: ′ is a locally closed subscheme, if there is an open , s.t. X → X U ⊆ X . ֒ Φ factors through a closed embedding X → U 4. The closed image of a morphism Φ: ′ is Φ( ′) := Spec(C∞( )/I ), where I X → X X X Φ Φ is the kernel of C∞( ) C∞( ′). X → X An open subscheme is equipped with the canonical embedding into , that we will denote by . We will write , if factors through X . Similarly for Xf ⊆ X Xf1 ⊆ Xf2 Xf1 ⊆ X Xf2 ⊆ X closed subschemes. The closed image of an open subscheme will be denoted by . U ⊆ X U Remark 7. Let , ′ be open subschemes, the composite ′ ֒ is an open U ⊆ X∞ U ⊆ U ′ U → X subscheme, i.e. f C ( ), s.t. ∼= f (e.g. [24] Thm. 1.4(i)). Given , ∃ ∈ andX an openU subschemeX the set , is a Zariski cover Xf1 Xf2 ⊆ X U ⊆ X {Xf1 Xf2 } of , iff = f f := 2 2 (e.g. [25] Lemma VI.1.2). U U ∼ X 1 ∪ X 2 Xf1 +f2 Lemma 18. Let C∞Sch, and let f , f C∞( ). Then X ∈ 1 2 ∈ X ∞ (f ) ∞ (f ).17 Xf1 ⊆ Xf2 ⇔ p 1 ≤ p 2 ∞ ∞ ∞ ∞ −1 Proof. Consider φ: C ( ) C ( )/(f2), and let f1 := φ(f1). The C -ring (C ( )/(f2)) f1 is a⇒ solution to two universalX → problems:X inverting f C∞( ) and killing (f X) { } 1 ∈ X 2 ≤ C∞( ). Therefore X (C∞( )/(f )) f −1 = (C∞( ) f −1 )/(f ), (19) X 2 { 1 } ∼ X { 1 } 2 ∞ −1 where f2 C ( ) f1 is the image of f2. By assumption f2 becomes invertible in ∞ ∈−1 X { } ∞ ∞ ∞ C ( ) f1 , therefore the C -rings in (19) are trivial, i.e. f1 √0 C ( )/(f2), X { } ∞ ∞ ∈ ≤ ∞X implying that f1 p(f2) ([5] Prop. 3). Then (f1) p(f2) and hence p(f1) ∞ ∈ ≤ ≤ p(f2) ([5] Lemmas 2,4).

∞ ∞ −1 Since f1 p(f2), the ideal of C ( ) f1 generated by the image of f2 contains 1 ⇐ (see (19)),∈ i.e. X { } Xf1 ⊆ Xf2

∞ ∞ ∞ Definition 27. Let C Sch, f C ( ). For r R let αr C (R) be any function s.t. p R α =0 =(X ∈ ,r]. Then∈ we defineX ∈ ∈ { ∈ | r } −∞ := Spec(C∞( )/ ∞ (α (f))), := Spec(C∞( ) α (f)−1 ), Xf≤r X p r Xf>r X { r } similarly we define , , and, taking intersections, , etc. Xf≥r Xfr αr. 17 The -radical ∞√ was defined in [5] Def. 1. ∞ 37 Proof. Choosing generators we find a surjective C∞(RS) C∞( ) for some set S. Let ∞ S ′ ∞ → X f C (R ) be any pre-image of f. Let αr, αr C (R) be any two choices as in Def. e ∈ ′ ∈ ′ 27. Since αr, αr have the same 0-sets in R, also αr(f), αr(f) have the same 0-sets. Then C∞(RS) α (f)−1 = C∞(RS) α′ (f)−1 . Using [24] Prop.e 1.2e we immediately conclude that { r } { r } = ′ e . This implies thate also is well defined (Lemma 18). Xαr(f) Xαr(f) Xf≤r ∞ Proposition 15. Let C Sch, ≇ ∅, and let 1, 2 be open subschemes, s.t. = . Then thereX are∈ f C∞X( ), r r2 n Proof. According to [25] Lemma VI.1.2 we can find n Z≥0, an open set U R , a surjective ∞ ∞ ∞ ∈ ⊆ morphism C (U) C ( ), and f1, f2 C (U), s.t. 1 (f1, f2) and 1 = f1 , 2 = f2 , where f , f C∞→( ) areX the images of∈f , f . Let U ,∈U U be theU openX subschemesU X 1 2 ∈ X 1 2 1 2 ⊆ defined by f1, f2. Then U1 U2 = U, (U U1) (U U2)= . Using the smooth version of ∪ \ ∩ \ −1∅ −1 Urysohn’s lemma we can find f : U [0, 1], s.t. U U1 = f ( 1 ), U U2 = f ( 0 ). Put r := 0, r := 1 and take the image→ of f in C∞( ).\ { } \ { } 2 1 X A.2 Complements and germs Lemma 18 implies that the following definition makes sense. Definition 28. Let C∞Sch, f C∞( ). The complement of is Spec(C∞( )/ ∞ (f)). X ∈ ∈ X X − Xf Xf X p The boundary of in is ∂ := . Xf X Xf Xf − Xf ∞ −1 Example 4. Let ∼= Spec(C ( ) f ) be an open subscheme of a manifold , s.t. 0 is a regular valueU of f C∞( M), then{ ∂} is the submanifold p f =0 .M Indeed, regularity of 0 implies that∈ (f)M C∞( ) Uis R-Jacobson radical{ ([5]∈M| Def. 1). In} particular ∞ ≤ M p(f)=(f) ([5] Prop. 1). ,(Definition 29. Let ′ ֒ be a closed subscheme (i.e. φ: C∞( ) C∞( ′) is surjective X → X X → X it is a principal closed subscheme, if f C∞( ), s.t. ∃ ∈ X ∞ ∞ pKer(φ)= p(f). (20) We do not require ′ to be reduced (the radical is on both sides of (20)). X Proposition 16. Let C∞Sch. X ∈ 1. Let ′ be a principal closed closed subscheme. For any open ′ there is an openX ⊆ X , s.t. ′ = and ′ with this property weU have⊆ X ′ . If Ue ⊆ X Ue ∩ X ′U ∀ U ⊆ X U ⊆ Ue ∼= ∅, we will write := ∅. U X − X e 2. Let , be open subschemes, then ( ) = ( ), where U1 U2 ⊆ X U1 − U1 ∩ U2 U1 ∩ X − U2 ∩ means the fiber product over . X Remark 8. Proposition 16 immediately implies that for any open subscheme we U ⊆ X have ( )= . On the other hand, given an open subscheme ′ , we have X − X − U U U ⊆X−U ( ) ′ = ′. X − U − U X − Ue 38 Lemma 20. Let Φ: ′ be a morphism in C∞Sch, and let be an open subscheme, s.t. Φ−1( )= ∅. ThenX → the X composite morphism ( ′) ′ U ⊆ Xfactors through . U X red → X → X X − U Definition 30. A morphism Φ: ′ in C∞Sch is a principal locally closed subscheme of X → X , if there is an open subscheme , s.t. Φ factors through ′ , which is a principal Xclosed subscheme. U ⊆ X X → U

Deﬁnition 31. Let Φ: ′ be a morphism in C∞Sch, the germ of at the image of Φ ∞ Xg → X g X is ′ := Spec(C ( )/m ), where m is the ideal of 0-germs at the image of Φ, i.e. XX X Φ Φ mg := a C∞( ) a′ C∞( ) s.t. aa′ =0 and a′−1 C∞( ′) . Φ { ∈ X | ∃ ∈ X ∃ ∈ X } In some cases, e.g. that of a locally closed subscheme, we write ′ to mean the germ of ′ g Xf at the image of the inclusion ֒ , and we write mX ,X ′ for the corresponding ideal of 0X-germs. X → X

Proposition 17. Let ′ ֒ be a principal locally closed subscheme, and let ( ′) be the X → X U X set of open subschemes of containing ′. Then X X ′ = lim . Xf ∼ U∈U(X ′)U Proof. Let ′′ be a cone on the diagram of open subschemes of , containing ′. X → {U} X X Since this diagram is filtered, there is a well defined morphism Φ: ′′ . Since this morphism factors through each , for every f C∞( ), that becomesX invertible→ X in C∞( ′), U ∈ X g X ′ φ(f) is invertible. This immediately implies that φ maps m ′ to 0, i.e. Φ factors through . X X We need to show that ′ factors through each containing ′. This is equivalentf to showing that each f XfC∞→( X), that becomes invertibleU in C∞( ′X), is also invertible in ∈ X X C∞( ′). Since ′ is principal locally closed, taking intersections with an open neighbourhood X X of f′, we can assume that ′ is a principal closed subscheme, i.e. we have f ′ C∞( ), X ′ X ⊆′ X ′′ ∈ ∞ X s.t. = ′ . Let , then ′ = . Using Prop. 15 we can find f C ( ), X − X Xf Xf ⊇ X Xf ∪Xf X ∈ X ′′ ′ ′′ s.t. f = f r2 for some r1,r2 R, and then f becomes invertible on X X ′ X X ∞ ∞ ∈ g r +r r +r f ′′≤ 1 2 . The kernel of C ( ) C ( f ′′≤ 1 2 ) is contained in mX ′ . X 2 ⊇ X X → X 2 Lemma 21. Let C∞Sch, and let ′, ′′ two principal closed subschemes. Then ′ ′′ is aX principal ∈ closed subscheme.X X ⊆ X X ∪ X ⊆ X Proof. Let φ′ : C∞( ) C∞( ′), φ′′ : C∞( ) C∞( ′′) be the corresponding morphisms of C∞-rings. ThenX by definition→ X X → X

′ ′′ := Spec(C∞( )/(Ker(φ′) Ker(φ′′))). X ∪ X X ∩ ∞ ′ ′′ ∞ ′ ∞ ′′ According to [5] Lemma 5 pKer(φ ) Ker(φ )= pKer(φ ) pKer(φ ). By assumption ∞ ∞ ′ ∩∞ ∞ ′′ ∩∞ f1, f2 C ( ), s.t. pKer(φ ) = p(f1), pKer(φ ) = p(f2). Clearly inverting an ∃ ∈ X ∞ ∞ element of (f1) (f2) implies inverting f1f2, i.e. (f1) (f2) p(f1f2). Hence p(f1) ∞ ∞ ∩ ∩ ⊆ ∩ p(f2)= p(f1f2).

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