MATH 465-2: Factorization Algebras

Transcription

1 ATH 465-2: Factorization Algebras Taught by John Francis, Notes by Brian Williams Winter 2014 In the following all merits of the material and ideas are those of the instructor. oreover, any typos or mathematical errors are due to the note-taker. 1 Lecture 1: Introduction, 8 Jan 2014 Roughly, this class will be about studying one fundamental object in two different guises. Let X be an algebraic curve over a field k, and let G be some algebraic group over X. We assume for this lecture that G splits trivially as G 0 X, but this is not necessary for what we will do in the course. The fundamental object we will study is the collection of G-bundles on X. We will consider two approaches to understanding this example: from a topological standpoint and from an algebraic geometric one. First, let s work topologically. For simplicity let k = C. Then the space of G-bundles on X is equivalent to the mapping space: Bun G (X ) ap(x (C),BG 0 (C)). The first thing one could hope to compute is the cohomology of this space. A pedestrian approach would be to calculate this using the Serre spectral sequence. Indeed, consider the fibration given by evaluation at a C-point of X : ap (X (C),BG 0 (C)) ap(x (C),BG 0 (C)) BG 0 (C). For simplicity we assume that the genus of X is at least one so that X (C) = K(π 1 X (C),1). In this case the space of pointed maps has the form ap (X (C),BG 0 (C)) ap Grps (π 1 X (C), G 0 (C)). Note that even to compute the connected components of the above space one needs to understand the group homomorphisms from a surface group to a Lie group. This is a highly nontrivial problem. The E 2 page of the Serre spectral sequence for the above fibration looks like E p,q 2 = H p (BG 0 (C)), H q ap Grps (π 1 X (C), G 0 (C))) where the group π 1 BG 0 (C) = π 0 G 0 (C) acts on the coefficient system by conjugation. The moral of the story is that this is really hard, and a priori gives a pretty nasty answer. But we know the answer is simple, so in this retrospect we conclude such a method cannot be the most efficient. Instead we take the following approach. We wish to study the above mapping space locally in X. Intuitively, this should manifest in something like sheaf cohomology. Let U(X ()) be the category of opens in X () with inclusions as morphisms. Consider the composite functor U(X (C)) op ap(,bg 0 (C)) Spaces. Ch C ( ) 1

2 There are two natural questions: is the above functor a (homotopy) sheaf; and if so, does C compute sheaf cohomology? The answer to the first question is no, in general. This can be seen in the first nontrivial case, namely when X () consists of two points. In this case the cochains split as C ap(,bg 0 (C)) C BG 0 (C) C BG 0 (C) But the sheaf condition would require this to split as a direct sum not a tensor product. Now, fix an n-manifold. Consider the category Disk 1 ( ) of embeddings R n with morphisms embeddings over. That is a morphism is a triangle of embeddings R n R n. There is natural inclusion Disk 1 ( ) U(), so that we can restrict any presheaf F : U() op Ch to this subcategory. Denote the resulting functor by res F for the time being. We have the following Lemma 1.1. Suppose F is a sheaf, then C (,F) lim res F. Disk 1 () Less suprisingly, we can recover sheaf cohomology as a limit over a larger category Disk(). Objects are embeddings I Rn and morphisms are embedding triangles analogous to the above. Then we have a sequence of inclusions Disk 1 ( ) Disk( ) U() so that we can restrict along the right inclusion as well. As promised C (,F) lim Disk 1 () res F lim Disk( ) when F is a sheaf. The idea is to use this fact to define a notion of presheaf cohomology. Namely, when F is merely a presheaf on we define the factorization cohomology of a presheaf F to be: res F F := lim res F. Disk() Note that when F is a sheaf this corresponds with sheaf cohomology. There is a dual result that states Lemma 1.2. Suppose F is a cosheaf on. Then C (,F) colim res F colim res F. Disk 1 () Disk() This motivates the defintion of factorization homology of a precosheaf F: F := colim res F. Disk() The idea now is rather than consider the sheaf cohomology with coefficients in the sheafification of this functor U C ap(u,bg 0 (C)), we shall consider the factorization cohomology X () C (BG 0 (C)) = lim C (ap(u,bg 0 ())). U Disk() There is a corresponding dual picture in which we consider compactly supported maps. Namely, consider the composite U(X (C)) op Ch ap(,bg 0 (C)) Spaces. C ( ) 2

3 The above is not necessarily a cosheaf so we should consider the quantity C ap c (R 2,BG 0 (C)) = colim C ap c (U,BG 0 (C)). U Disk() X (C) The reason this is all useful is the following version of the Atiyah-Bott calculation of the cohomology of Bun G (X ). Theorem 1.3. (i) Suppose G 0 (C) is simply connected, then C ap(x (C),BG 0 (C)) X (C) C BG 0 (C). (ii) Suppose G 0 (C) is connected. Then C ap c (X (C),BG 0 (C)) X (C) C ap c (R 2,BG 0 (C)). The point is that the right hand quantities in the above result are fairly tractable. Here are the goals for the class: Topology First we will define and prove everything mentioned above. Then we will reformulate factorization homology as a type of sheaf cohomology on something called the Ran space. Algebraic geometry The next main pillar of the course will recast everything in the topology picture in terms of l-adic cohomology. We will prove the Weil conjecture that gives a formula for the number of points of Bun G (X ) in terms of local data on X. 2 Lecture 2: Disk n -algebras and factorization homology, 10 Jan 2014 We will start with the topological side of the course. Before we get into the meaty stuff, let s set some goals for the next couple of lectures. (0) Define factorization homology. (1) Prove non-abelian Poicaré duality. Namely when is an n-manifold, show that there is a natural map C (Ω n BG 0 ) C ap c (,BG 0 ) and it is an equivalence (quasi-isomorphism in this case) provided BG 0 is n-connective. (2) There is an equivalence C (Ω n BG 0 ) In fact, we will see how this is an encarnation of Koszul duality. C BG 0. (3) Complete the calculation by doing the relatively easy task of computing the right-hand side of the above equation. Let fld n be the topological category of manifolds with morphisms embeddings given the compact-open topology. There is a full subcategory Disk n spanned by the objects which are homeomorphic to disjoint unions of R n. Recall, a Disk n -algebra valued in a symmetric monoidal topological category C (whose monoidal structure we denote by ) is a functor continuous symmetric monoidal functor Disk n C. Note that we use the natural symmetric monoidal structure of disjoint union for the category Disk n. We denote by Alg Diskn (C) the topological category of algebras. Here are examples of coefficient categories we will consider 3

4 Example 2.1. (1) C = Spaces, the category of spaces with cartesian product. Almost equivalently, C = ssets with product. In this case, the mapping spaces are given by Hom (X, Y ) where the internel hom of simplicial sets has i-simplices given by Hom i (X, Y ) := Hom(X i, Y ). (2) C = Ch, the topological category of chain complexes with disjoint union. The mapping spaces are provided by the Dold-Kan correspondence and are given by Hom (V,W ) where Hom i (V,W ) := Hom(V C i,w ). We will see this example will be pretty boring since monoids in this category can be identified with the usual category of chain complexes. (3) A priori similar, but much different is C = Ch, the category of chain complexes with monoidal structure given by tensor product. Here are some examples of Disk n algebras. Example 2.2. (1) The following Disk n -algebra in spaces will be very important. Let X be a pointed space, denote by Ω n X the functor ap c (, X ) : Disk n Spaces. It is easy to see that this defines a Disk n -algebra in spaces. The reason for the notation is that as spaces we have the string of equivalences Ω n X ap((d n, D n ),(X, )) ap c (R n, X ). (2) Let Disk 1 n Disk n be the full subcategory spanned by objects homeomorphic to exactly one copy of Rn. Define the functor Disk 1 n Ch by sending all objects to V and all morphisms to id V. Extend this functor to Disk n as follows: on objects k Rn V, and all morphisms get sent to the addition map + : V k V. (3) Consider the composite functor Disk n Ω nx Spaces C Ch where Ω n X was defined above. One easily checks that this defines a Disk n algebra object in Ch. (4) Commutative algebra objects in C may be identified with symmetric monoidal functors Fin C. That is Alg Comm (C) = Fun (Fun,C). Note that a cdga may then be identified with a symmetric monoidal functor A : Fin Ch. Now for any C and any commutative algebra object A in C we may consider the composite That is, restriction along π 0 gives us a functor Disk n π 0 Fin A C. Alg Comm (C) Alg Diskn (C). We are now ready to define factorization homology. Let A Alg Diskn (C). We may left-kan extend A along the inclusion i : Disk n fld n : Disk n i fld n. A A C 4

5 That is A = i! A. Evaluation of this functor on a manifold : A := (i! A)( ) C is called the factorization homology of with coefficients in A. It is a standard fact that left-kan extensions may be computed as a colimit: A = colim A. Disk n/ Of course, everything above is in the -categorical sense. For instance, the mapping space from U to V in the category Disk n / is not just embeddings U V over. In fact this set would either be empty or a single point depending on whether U V or not. Rather the mapping space consists of an embedding U V together with an isotopy from the induced map U V to the original embedding U. ore explicitly it the mapping space from U to V over is given by the following (homotopy) pull-back Emb / (U,V ) Emb(U,V ) Emb(U, ). 3 Lecture 3: The Ran space, 13 Jan 2014 Last time we introduced the category Disk n/ = Disk(). By definition the mapping space from R n to R n was identified with the homotopy pullback below: ap Disk() (R n,r n ) Emb / (R n,r n ) {R n } Emb(R n,r n ) Emb(R n, ). Note that if we look at pointed maps we get another homotopy pullback square Emb ((R n,0),(, )) Emb(R n, ). Now, once we choose a small enough neighborhood in we can identify the above homotopy pullback with Emb (R n,r n ). Furthermore, this can be identified with Emb(R n,r n ). Combining the above two homotopy pullback squares we get a big homotopy pullback square Emb / (R n,r n ) Emb(R n,r n ) {R n } Emb(R n, ) thus giving a homotopy equivalence Ω Emb / (R n,r n ). As a corollary we see that the evaluation map gives a homotopy equvialence Disk 1 ( ) (1) as -groupoids (or just spaces). Or, if you like, an equivalence Disk 1 () Sing( ) of Kan complexes. There is an important class of subcategories similar to Disk 1 of Disk n n. Define the subcategory Diski to have n objects n-manifolds that are homeomorphic to exactly i copies of R n, and morphisms those embeddings that induce bijections on π 0. Define Disk i () to be the subcategory of Disk( ) to have objects that are homeomorphic to exactly i copies of R n and morphisms equal to embeddings over as above that induce bijections on π 0. 5

6 Proposition 3.1. Let be connected. Then Disk i () Conf i () Σi Recall the (ordered) configuration space Conf i () is, by definition, the space of all possible ways of choosing i points in. There is an obvious Σ i action and the resulting space of coinvariants is called the unordered configuration space. This is what appears on the right-hand side of the above. We should think of the above result as generalizing the homotopy equivalence in (1). Proof. We prove the above proposition. Consider the following diagram ap Disk i () ( i Rn, i Rn ) Emb ( i Rn, i Rn ) Emb( i Rn, ) Conf i ( ) Σi Here Emb denotes embeddings that induce bijections on π 0. Where do these squares come from? We have seen the top square above, and know that it is a homotopy pull-back. The bottom square is constructed as follows. It is known that if U is an open submanifold then evaluation at zero determines a homotopy pull-back square Emb( i Rn, U ) Emb( i Rn, ) Conf i (U ) Conf i ( ). Restricting to embeddings that induce bijections on connected components we get the bottom square above. We are now ready to define one of the most important spaces we will consider in this course. Definition 3.1. If X is a space we define the Ran space Ran(X ) to be, as a set Conf i ( ) Σi i 1 and the topology given by the finest one such that the obvious map i Ran() is continuous. An equivalent defintion is Ran( ) = colim ( ) Fin where Fin denotes the category of finite sets with morphisms given by surjections. This space is really interesting, as Belinson and Drinfeld proved: Theorem 3.2. For any connected space X the space Ran(X ) is contractible. oreover, for any S X the subspace Ran(X ) S/ of Ran(X ) consisting of subsets containing S is contractible. Proof. The first claim follows from the second. Consider the map u : Ran(X ) S/ Ran(X ) S/ Ran(X ) S/ given by union. This defines a topological monoid structure on Ran(X ) S/. But clearly u = id so that π k Ran(X ) S/ = 0 for all k. It remains to see that Ran(X ) has the homotopy type of a CW complex, we won t do that here. The following is a technical result that we won t prove right now. 6

7 Theorem 3.3. The evaluation at zero map B(Disk()) Ran() is a homotopy equivalence. The proof of this involves something called the inter-path space. Definition 3.2. A topological category X is said to be tensored in spaces provided there is a functor such that there is an equivalence of functors X Spaces: for all spaces K and all objects X. : Spaces X X ap X (K X, ) ap(k,ap X (X, )) Example 3.1. (1) The category of spaces is tensored in spaces with the usual product. (2) The category Ch is tensored in spaces with the operation K A C (A) A. Let s consider a more subtle example. Consider the adjunction Sym Alg Comm (Ch ) Ch. forget Using this, we will compute the tensor product in commutative algebras of a space with a free algebra. ap AlgComm (K Sym(V ), ) ap AlgComm (Sym(V ), ) K ap Ch (V, ) K ap Ch (K V, ) ap Ch (C K V, ) ap AlgComm (Sym(C K V, ). Thus K Sym(V ) Sym(C K V ). 4 Lecture 4: Tensor product of commutative algebras, 16 Jan 2014 Recall the tensor product We can compute this as follows: : Spaces Alg Comm (C) Alg Comm (C). If V is a an object of C and Sym(V ) is the symmetric algebra on V then If K is finite then K A A K. In general, if K K then K A A K. K Sym(V ) Sym(C K V ). Lemma 4.1. Let X be a topological category tensored in spaces and suppose A X and K Spaces. Then K A colim A K where A in the right hand side of the equation denotes the constant functor Sing(K) X. 7

8 As a corollary we see that if A is a commutative algebra in C then for all manifolds we have A colim A Disk 1 ( ) where here, on the right hand side, A denotes the composition Disk 1 () Disk( ) π 0 Fin A Alg Comm (C). In general colimits in a commutative algebra are hard to compute. A naive thing to ask is whether we get the same answer if we look at the resulting colimit in the category C using the obvious forgetful map Alg Comm (C) C. In general one should not expect this to be the same, but we will see conditions so that it is. Suppose we have any diagram B : J Alg O (C). Denote the resulting functor J Alg O (C C by B. Also, consider the composition J J B B Alg O (C) Alg O (C) C C C which we denote by B B. There is an obvious natural transformation B B B given by multiplication. We get a partial square of solid arrows colim B B colim J colim B B colim J J J J B B colim B J and we want to be able to fill in the right vertical dotted arrow. We can do this provided the arrows in the bottom corner are an equivalence. The left vertical arrow is an equivalence provided: ( ) The monoidal structure on C distributes over colimits. The bottom horizontal arrow is an equivalence provided: ( ) J is sifted, that is for all F : J J D the natural map is an equivalence. One can take this as the definition of sifted. That is colim J res F colim F J J Definition 4.1. A category J is sifted provided the diagonal map J J J is final. One can ask, how interesting can sifted categories be? The following lemma gives a partial answer. Lemma 4.2. If J is sifted then BJ. In practice, one checks that a functor is final using Quillen s Theorem A. It states that a functor F : A B is final iff A b/ is contractible for all objects b B. Here, the undercategory is defined to be the fiber product A b/ := A B B b/. (i, j )/ Example 4.1. (1) Let Z denote the poset of integers with the obvious ordering. This category is sifted as Z has an initial object for all i, j, hence it is contractible. (2) The ordinal category op is sifted. The proof of this involved the concept of baryacentric subdivision and is omitted. 8

9 The main technical result we will need is the following. Proposition 4.3. The category Disk() is sifted Proof. The evaluation at zero map BDisk() Ran() extends to a homotopy equivalence of undercategories BDisk() (U,V )/ Ran() (π 0 U,π 0 V ) and the right hand space is still contractible. We can now state the main result of this lecture. Theorem 4.4. For A Alg Comm (C) where C satifsies ( ) above, and for a manifold we have an equivalence A A. Proof. We have already seen that A colim(disk 1 ( ) A Alg Comm (C) where the functor inside the colimit is the obvious constant functor. Consider the left-kan extension with respect to the inclusion of Disk 1 () into Disk() Disk 1 () i Disk( ) A Alg Comm (C) i! A and it is computed as (i! A)( I R n ) = A I. oreover we have a string of equivalences A colim(disk 1 () A Alg Comm (C)) colim(disk( ) i! A Alg Comm (C)) colim(disk( ) A C) A. The second line follows from commutativity of colimits, the third line follows since Disk( ) is sifted. The last line is the definition of factorization homology. As a corollary of this and our computation of the tensor product in cdga s we have for V a chain complex: Sym(V ) Sym(C V ). This will turn out to be our main computational tool. 5 Lecture 5: Pushing forward, 17 January 2014 Our goal for the next couple of lectures is to prove the following version of non-abelian Poincaré duality. Let be a manifold, and recall that for X a space there is a Disk n -algebra Ω n X defined by ap c (, X ). Theorem 5.1. Suppose that X is n-connective. Then there is an equivalence Ω n X ap c (,X ). oreover we have C (Ω n X ) C (ap c (, X )). 9

10 The case = R n should be clear, as Disk n (R n ) is contractible so that colim Ω n X Ω n X. Disk n (R n ) The idea of the proof is to break up into pieces and to follow a local to global argument. This decomposition of should be thought of as giving a orse function, and studying the preimages will determine the decomposition and gluing data. The main technical tool we will need for starting this process is called the push-forward. Let f : N be a fiber bundle and consider the category fld n/ of n-manifolds over. Then f determines a functor f 1 ( ) : Disk(N) fld n/ given by taking preimages. The importance of being a fiber bundle is so that we can lift appropriate isotopies so that this preimage functor determines a well defined continous functor between topological categories. For A Alg Diskn (C), the fiber bundle f also determines a functor f A : Disk(N) C defined by U A. The following theorem relates the factorization homology of and that of N. f 1 (U ) Theorem 5.2. Let f be as above. Then there is a natural equivalence N f A The key to the proof is to consider the following category, denoted T f. Roughly it s objects consist of three pieces of data: an embedding of disks U, and embedding of disks V N, and an embedding V f 1 U over N. Explicitly this category is given as the limit (in Cat ) of the following diagram A. Disk() Fun( 1,fld n/ ) Disk(N) ev 1 ev 1 f 1 ( ) fld n/ fld n/. Lemma 5.3. Let A be a Disk n -algebra. Then there is an equivalence Proof. We first take the following left-kan extension colim A ev 0 A. T f N f ev 0 T f Disk() A C ev 1 Disk(N) (ev 1 )! (A ev 0 ). By universality of left-kan extensions we have a natural equivalence colim A ev 0 colim (ev 1 )! (A ev 0 ). T f Disk(N) It remains to compute the right-hand side. Left-Kan extensions can be pointwise computed as (ev 1 )! (A ev 0 )(U N) colim A. Disk n/ f 1 (U ) But the right-hand side is just the definition of A, so we are done. f 1 (U ) 10

11 Lemma 5.4. The functor ev 0 : T f Disk() is final. In particular A colim T f A ev 0. Proof. We use the fact that ev 0 is a cartesian fibration. So to check finality it is enough to check that pointwise inverses are contractible. That is, it suffices to check ev 1 ({V }). But it is clear that 0 B(ev 1 0 {V }) Ran(N)π 0 V /. And, we have already mentioned contractibility of the right-hand side. Theorem 5.2 is an immediate corollary of the above two lemmas. We note that the above push-forward formula works for manifolds with boundary with the following condition: the map f : N between manifolds with boundary must be a fiber bundle when restricted to both int(n) and N separately. It will be useful to consider the following category Disk of n-disks with boundary. It s objects are spaces U homeomorphic to disjoint unions n of (1) Euclidean spaces R n and (2) upper half-spaces R n 1 [0,1). Note that the closed unit disk is not an object in this category. We define Disk () := n Disk just as above for the boundary-less case. For a n/ Disk -algebra, i.e. a n symmetric monoidal functor A : Disk C, we define n A := colim A. Disk n () From the above remark we have the following result. Corollary. Let f : [ 1,1] be a map which is a fiber bundle over ( 1,1) and { 1},{1} separately. Then there is an equivalence A A. It turns out the factorization homology of the unit interval is not so bad; we will compute it next time. [ 1,1] 6 Lecture 6: Non-abelian Poincaré duality, 22 Jan 2014 This time we will finish the proof of non-abelian Poincaré duality. Let f : [ 1,1] be a map such that it is a fiber bundle when restricted to ( 1,1). We think of f as providing a decomposition of with = 0 ( 1,1) where = f 1 ( 1,1], = f 1 [ 1,1), and 0 = f 1 (0). Let X be a space. We claim that the restriction map is a fibration. oreover, the fiber of this map is Note that... ap c (, X ) ap c ( 0, X ) ap((, 0 ),(X, )) ap c (, X ) ap c (, X ) ap c (, X ). Theorem 6.1. Let R : Disk([ 1, 1]) C be an 1-disk (with boundary) algebra. Then R R([ 1,1)) R( 1,1) R(( 1,1]). [ 1,1] f 11

12 Remark. In the case of spaces with R = f Ωn X, with f : [ 1,1] the orse function as above, the theorem says that ap c (,X ) ap c (, X ) apc ( 0 ( 1,1),X ) ap c (, X ) where the quantity on the right is the diagonal quotient (not the fiber product). Proof. An arbitrary U Disk([ 1,1]) has the form U = [ 1, 1 + ε ) δ R i (1 ε,1] δ together with an embedding into [ 1,1]. Here δ,δ {0,1}. Let D Disk([ 1,1]) be the subcategory consisting of objects that surject onto the endpoints of [ 1,1]. We first claim that D is final in Disk([ 1,1]). Let V be any object of Disk([ 1,1]) and form V = [ 1, 1 + ε ) δ V (1 ε,1] δ where δ = 0,1 depending on whether V has a component of the form [ 1,1 + ε ) and similarly for δ. Then V V naturally. oreover given any other embedding V U there exists a factorization V V U which is unique up to a contractible choice since all embeddings must fix the endpoints by the defintion of D. Thus we see that D V / has an initial object so that B(D V / ). Thus D Disk([ 1,1]) is final. Next we show that there is an equivalence D op. First let s construct the map. Given U an object of D consider its complement [ 1,1] \ U. The components of the complement has a natural ordering so we certainly get a map on objects. What the functor does to embeddings is obvious, and it is also clear that this is an equivalence. Putting this all together we have [ 1,1] R colim Disk([ 1,1]) R colim D R colim R op and the right-hand side is computed by the desired two-sided bar construction. Now, by induction on the critical points of f : [ 1,1] we obtain the desired form of non-abelian Poinaré duality: Ω n X ap c (,X ). Finish P.D... 7 Lecture 7: Koszul duality, 24 Jan 2014 Today we construct a Koszul duality functor in the setting of Disk n -algebras. Namely we construct a functor Alg aug Disk n (C) coalg aug Disk n (C). First we mention the Bar construction, which is the n = 1 encarnation of this duality. Let Alg aug (C) denote the category of Disk 1 -alebras. There is a natural functor Alg aug (C) coalg aug (C) as above which sends an augmented algebra A 1 to the realization of its Bar complex The comultiplication comes from 1 A 1 := Bar(1, A,1). 1 A 1 1 A A A 1 1 A 1 A (1 A 1) 1 (1 A 1) 12

13 but it is not obvious why all the desired coherency conditions hold. A hint for fixing this comes from the identification 1 A 1 A proved above where A denotes the functor Disk([ 1,1]) that sends ( 1,1) A, [ 1,1) 1, ( 1,1] 1. The idea is to use a natural coproduct structure on manifolds to induce the desired comuliplication. Identify the interval [ 1,1] with boundary { 1,1} with the circle S 1 with a chosen basepoint. There is an obvious map [ 1,1] (S 1, ) (S 1 S 1, ) but this leads us out of the world of manifolds. This map is not far from being an embedding however. It is an embedding away from the basepoint, for instance. Another way of saying this is that it is a one-point compactifition of an embedding. This leads us to consider a category of manifolds with more general maps, that is wrong way maps. Definition 7.1. A zero-pointed n-manifold is a pointed Hausdorff space denoted such that := \ is an n-manifold. A map between two zero-pointed n-manifolds f : N is one such that f f 1 (N) is an embedding. The resulting topological category will be denoted Zfld n. This category inherits the obvious symmetric monoidal structure of disjoint union. One should think of maps in this category as embeddings where anything (or everything) can be sucked in to the point at infinity. There are some important functors that we should name at this point. First, consider ( ) : fld n Zfld n induced by adding a disjoint basepoint. We let the essential image of Disk n with respect to this functor be denoted Disk n,. There is also a functor ( ) + : fld op Zfld n n induced by one-point compactification. A generalization of the first functor is ( )/ ( ) : fld n Zfld n induced by quotienting out the boundary. Note that essentially by defintion Alg aug (C) = Fun (Disk Disk n,,c). For n such an augmented algebra A and a zero-pointed manifold, we define A C as the left Kan-extension Disk n, i A C A = i! applied to. Explicitly Now, consider the following two embeddings Zfld n A = colim A. Disk n, / Disk n, i Zfld n Disk op where i is the natural inclusion and j is the Thom collapse map induced by ( I R n ) I (R n ) +. Left-Kan extending and restricting respectively we get a composite functor: j n, Alg aug Disk n (C) Fun (Disk n,,c) i! Fun(Zfld n,c) res j Fun(Disk op n,,c) coalgaug Disk n (C). With symbols we write this composite as A A. (R n ) + Next we need the defintion of factorization cohomology. 13

14 Definition 7.2. Let C coalg aug Disk n (C) and Zfld n. The factorization cohomology of with coefficients in C is the right Kan extension evaulated on. We denote this by C C. As usual we may compute cohomology as Disk op n, i op Zfld op n C C (i op )! = ( ) C Example 7.1. When C = Ch, =, then C = lim (Disk op n, ) / C. C = C c (, C ) i.e. compactly supported cochains on with coefficients in C. *Not sure if this is right. Another construction is done in the next lecture*. Now, we claim there is an antiequivalence of categories ( ) : Zfld op Zfld n n Let A be an augmented Disk n -algebra. We can take its Koszul dual as above and we consider the following diagram. Disk op n, Zfld op n (R n ) + A Zfld n C ( ) A As the negation of n adjoin a point is one-point compactification, we see that the outside diagram commutes. Since the upper triangle is a right-kan extension we get a universal arrow as depicted. On objects we get a map A A. (R n ) + 8 Lecture 8: Poincaré-Koszul Duality, 3 Feb At this point it should be known that everything in the previous lecture and this one is to appear in a forthcoming paper of JF s and David Ayala. In this lecture we combine the Koszul duality functor discussed in the previous lecture to formulate a generalized Poincaré duality result. Let X be a locally compact Hasudorff space. A pointed extension of X is a pointed locally compact Hausdorff space X such that as a set X = X { } and such that there is an open inclusion X X. Let Point X Spaces X / be the subcategory of pointed spaces under X that are pointed extensions of X. It is clear that this category is a poset. There is an isomorphism ( ) : Point op Point X X induced by X (X ) + \ { }. E.g. we one point compactify X then we remove the original basepoint. Example 8.1. Suppose X + = X { } as topological spaces. Then (X + ) = X +. oreover (X + ) = X { } 14

15 The last example hints at the following, which is a pretty standard to prove if one wishes to get their hands dirty with some point-set topology. (Just like the good days) Lemma 8.1. ( ) ( ) Example 8.2. Let be a manifold with partitioned boundary +. Then (/ + ) = / +. We have already introduced the category of zero-pointed manifolds. From the above discussion, it is not hard to see that negation extends to an isomorphism of categories ( ) : Zfld op n Zfld n. Let fld n,+ be the full subcategory of Zfld n consisting of n-manifolds with a disjoint basepoint added. Let fld + denote the essential image of the one-point compactification functor in Zfld n n. A morphism in this former category f : + N + induces a morphism f : N + + in the latter category, and this is identified with the induced map of compactifications, namely the Pontryagin-Thom collapse map. Corollary. There is an equivalence induced by the collapse map: op Diskn,+/ Disk + / n ( I R k ) ( + I (R k ) + ). As before, the factorization cohomology of with coefficients in an augmented Disk n -coalgebra C is the right-kan extension C = lim C. Disk op n,+/ Using the equivalence above the right-hand side may be rewritten as lim (Disk + n ) Consider the restriction map along the inclusion Disk + n Zfld n : C. Fun (Zfld n,c) Fun (Disk + n,c). This admits a left adjoint and the unit of this adjunction gives us a maps A A. (R n ) + Example 8.3. Let A Ch Alg Diskn (Ch). Then if is an n-manifold + A C (,A). oreover + A = lim Disk op n/ A = C (,A) so that + A = C c (, A). The shift is encoded by the Koszul duality map (R n ) + A A[n]. It turns out the map above is the usual Poincaré duality map C (,A) C c (,A[n]). 15

16 The proof of the following will appear soon. Theorem 8.2. ( J. Francis, D. Ayala) (Poincaré-Koszul duality) Let be a compact n-manifold with boundary and let = int(). Suppose A is an augmented n-disk algebra whose augmentation ideal is connected. Then the map constructed above + A A. + (R n ) + is an equivalence. Remark. The augmentation ideal is the fiber of the map A 1. There is also a version of this theorem with the connectivity hypothesis removed. The idea of the proof is pretty awesome so we ll give a sketch. First we find a cofiltering of both sides. For the left-hand side we consider the Goodwillie tower for the functor A. It turns out ( ) this tower converges, i.e. P A A. The cofiltering on the left-hand side is more straightforward. For a coalgebra C set τ k C := lim (Disk k n,+/ ) op C where the limit is taken over the category of unions of disks with k or fewer components. Since τ k is a right adjoint we get a natural identification Next, we need to construct maps lim τ k C = C. k P k A τ k (R n ) + A. Let s do it for n = 1. It is a fact that the linearization of the factorization homology is given by P 1 A C () L 1 A where L 1 A is the cotangent complex of A evaluated at the augmentation A 1. oreover we have k L 1 A = (R n ) + A, and the desired map comes from usual Poincaré duality. 9 Lecture 9: Atiyah-Bott formula, 5 Feb 2014 Today we will combine the results thusfar to reproduce a famous calculation of Atiyah and Bott of the cohomology of the space of principal bundles over a fixed topological space. So far everything has held with coefficients in an abstract symmetric monoidal -category C satisfying some mild technical conditions. Today we will specialize to chain complexes over a field of characteristic zero, with monoidal structure given by tensor product. We first need one final ingredient that hasn t been mentioned yet. Proposition 9.1. Let B be an n-disk algebra in Ch k such that B is connective. H i (B) is finite dimensional for all i. Then for all n-manifolds we have B B. 16

17 Remark. One should think of the case B = Sym V where V is a postively concentrated chain complex that is levelwise finite dimensional. In fact, this is the only case in which we apply the above result. Proof. The definition of B is the following composite Disk n B Ch k ( ) Ch op k. While we can always form this functor, it is not always the case that B actually defines a coalgebra. That is, we always have the following picture B (B B) B B but constructing a map B B B requires our hypotheses. We state without proof that when this conditions are satisfied that the right-hand map is an equivalence so that we get the desired map. It is then routine to check this defines a Disk n -coalgebra. From here it is clear that B = colim B lim B = B. Disk() Disk() op Note that everything except for the last equality holds with no assumptions on B. That the limit computes cohomology comes from the fact that B is a coalgebra. Remark. The following example indicates why the assumptions are necessary. Take n = 1 and consider the tensor algebra B = T V = i 0 V i. Assuming V is finite dimensional we have (T V ) = (V ) i i 0 and while we have a map T V T V = (V ) i i 0 2 (V ) i (V ) j = (T V T V ) i, j 0 it certainly is not an equivalence when V is nontrivial in degree zero. But, when V is connective the tensor algebra is quasi-isomorphic to the direct sum of tensor powers. Since tensor products distribute over direct sums (Kunneth!) we get the desired equivalence. The following is a culmination of our main computations thusfar. Proposition 9.2. Let X be a space such that C (X ) is connective and let be an n-manifold. Then C (ap c (,X )) C (X ). Proof. We have the following string of equivalences C X C X Taking duals, we get the result. C (ap c (D n, X )) (D n, D n ) C Ω n X C Ω n X since D n is contractible non abelian Poincare duality Poincare Koszul duality C (ap c (,X )) Poincare duality. 17

18 We are now ready to prove the Atiyah-Bott calculation Theorem 9.3. Let be a closed surface and G a simply connected semisimple Lie group over C. Then where V G is the vector space such that H (BG) = Sym V G. H (Bun G ()) Sym(H (,V G )) Proof. Take X = BG as above. Then Bun G ( ) = ap(, BG) so we have C (Bun G ()) C BG. But, the right-hand side is Sym V G. We have compute factorization homology with coefficients in a symmetric algebra: Sym V G Sym(C V G ). So we get C (Bun G ( )) Sym C (,V G ). As the functor taking the symmetric algebra is exact in characteristic zero, we get the desired result. Algebraic Geometry This was the easy part. Now, the real class begins. We enter the realm of algebraic geometry. Our guiding principal will be to count the number of principal G-bundles on a given curve X over F q. There are always infinitely many, so we need think about what we mean by count. The right way of doing so seems to be Bun G (X ) := P 1 Aut(P) where P runs over isomorphism classes of principal G-bundles on X. One should think of this as a type of volume of the group, and is the proper notion of counting since we should think of Bun G (X ) as a category rather than just a set. There is a way of counting points on an algebraic variety due to Weil. If Y is quasi-projective over F q with an embedding Y P n for some n then we can consider the completion F q Y = Spec(F q ) SpecFq Y and the induced embedding Y P Fq. There is a type of Frobenius map with such data that takes the form ϕ : Y Y. Then there is a Lefshetz formula Y (F q ) q dim Y = ( 1) i tr ϕ 1 H i (Y ;Q l ) i 0 where the cohomology appearing is called rational l-adic cohomology. 10 l-adic cohomology, 7 Feb 2014 Today we will introduct l-adic cohomology and state our main focus for the remainder of the course. Consider the category Sch k of schemes over k. Recall, the étale topology (in the sense of Grothendieck) on Sch k consists of covers { α U α X } that are étale and surjective. As usual, let C be a symmetric monoidal -category. Define Shv(Sch k,c) Fun(Sch op k ) 18

19 to be the full subcategory spanned by functors which are sheaves in the étale topology. That is, all F such that for any cover { α U α X } the natural map F(X ) Tot F U α is an equivalence. There is a familiar adjunction α shfy Shv(Sch k,c) Fun(Sch forget k,c) F U α X U β ) α,β given by forgetting and sheafifying. For R C we let R denote the constant presheaf Sch k C that sends every object to R. Define the étale cohomology with coefficients in R to be the sheaf C ( ; R) := shfy(r). Now, let l be a prime in k that is invertible. For a fixed scheme X, we can then consider the following limit C (X,Z l ) := lim C (X ;Z/l n ) which we call the l-adic cohomology of X. (The precise reason for the completion outside the cohomology is unknown to the note-taker. One should ask a number theorist why the above definition is the right one, or just take it on faith.) We will also need a rational version C (X ;Q l ) := C (X ;Z l ) Zl Q l. For the time being, let R be one of the following rings: Z/l,Z l,q l. Note that if X, Y are quasi-projective k-schemes then we have maps C (X ; R) R C (Y ; R) C (X Spec k Y ; R) C (X Spec k Y ; R) C (X Spec k Y ; R). There is the following Künneth theorem Theorem Suppose k is an algebraically closed field in the setting above. Then the above map is an equivalence Remark. One proves this theorem by reducing it to the case R = Z/l n where it further reduces to the statement of the usual Künneth theorem for cohomology. The homology of X is the R-dual, C (X ; R) := C (X ; R). Theorem Suppose X is an affine curve. Then H i (X ;Z/r ) is zero for i > 0 and H 1 (X,Z/r ) = {etale Z/r torsors over X } / Prestacks We will need to deal with more general object than schemes. Taking the functor of points approach we view a scheme as a set valued functor X : Ring k Set. Stacks enter the picture when we want to remember automorphisms, so instead of considering sets, we want to consider categories. We will can view a prestack as a functor X : Ring k Grpd. One should think of this as a higher scheme. ore generally one can imagine remembering not only automorphisms, but also homotopies of, and higher homotopies, etc.. That is we can consider functors X : Ring k Space or X : Ring k sset 19

20 These two notions are equal, and a choice of one at a particular point is for mere convenience. Given such a higherer prestack, we can forget down and get a (pre)scheme as the picture Ring k sset Grpd Set indicates. One can also imagine going up a similar tower in the codomain category. This is the realm of derived algebraic geometry. You could say people have thought a little bit about this, but it won t concern us for the time being. Sometimes it s hard to write down a functor Ring k sset, as it requires a good amount of data. There is an alternative description of functors of this type, through sometime called the Grothendieck construction. For convenience let s consider Grpd-valued functors, but the same idea works for simplicial set valued functors. There is a (rough) equivalence straight {op fibrations over D} Fun(D,Grpd) which we briefly describe now. Given such a functor X : D Grpd, form its Grothendieck category Groth Groth(X ) = D /X which has objects pairs (R, a) where R is an object of D and a is an object of X (R). A morphism (R, a) (R, a ) is a pair f : R R in D and α : X ( f )(a) a in X (R ). Now, an op-fibration is a functor π : E D such that for all α 0 : π(e) D in D there is a lift to a cocartesian map α : E E. That is, π(α) = α 0. Given any prestack X that is Grpd/sSet valued on Ring k define its cohomology as C (X ; R) := lim C (, R) Ring k/x where R is one of Z/l n,z l,q l as above. Similarly its homology is C (X ; R) := colim C Ring op (, R). k/x Definition Let Bun G (X ) be the category whose objects are pairs (A, P) where A is a k-algebra and P is a principal G-bundle over X A := X Spec k Spec A. A morphism is a pair f : A A and ϕ : P = P XA X A. The category Bun G (X ) naturally sits over Ring k, and one can check that it is op-fibered. Thus, it determines a prestack. The main goal for the rest of the course is to compute its rational l-adic cohomology: H (Bun G (X );Q l ). The idea will be to do this locally in X, just as in the topological case. 11 Lecture 10: Algebrometic Ran space, 11 Feb 2014 entioned at the end of the last lecture, our goal is to come up with an algebra geometric analog of factorization homology. ore specifically, we want to find an analog of the following result: colim ap c (U, Y ) ap c (, Y ) U Disk( ) in the case that = X is a curve and Y = BG. The most naive replacement for open sets are Zariski open sets. These are no good as they are usually dense in the ambient variety. Étale opens don t work either. The idea is to think of ap(u, BG) as the mapping space of the following pair: ap c (U, BG) ap c ((, U ),(BG, )) 20

21 which we can think of and G-bundles on with a trivialization on U. The moral of this is that we should consider G-bundles on our curve X together with a trivialization on X {x} for each point x X. Definition We define the category Ran u (X ) as follows. Its objects are pairs (A, S A) where A is a k-algebra and S is required to be finite and nonempty. A morphism (A, S) (A, S ) is a ring map ϕ : A A such that ϕ S : S S is (well defined and) a surjection. This is obviously a category over Ring k, and it is, in fact, op-fibered. So we think of Ring k as a prestack. We will also need a straightened version. Definition Define the category Ran(X ) as follows. Its objects are pairs (A, f : S X (A) where S is a finite nonempty set. orphisms are pairs ϕ : A A, ϕ 0 : S S such that ϕ 0 is a surjection and such that the obvious square commutes. Again Ran(X ) has the structure of a prestack on Ring k. Finally we need a version of this which takes into account bundles on X. Definition Define the category Ran u (X ) as follows. Its objects are quadruples (A, P, S,γ) where G A is a k-algebra. P X A is a principal G-bundle. S X (A) is a finite nonempty subset. γ is a trivialization P XA S = (X A S ) G. Some explaining is in order for this one. Each element s S X (A) determines a section Spec A X A = Spec k Spec A. We let S X A be the union of the images of all such sections determined by elements in S. A morphism (A, P, S,γ) (A, P, S,γ ) is a homomorphism ϕ : A A together with an isomorphism ϕ := ϕ P = 0 P XA = P such that S X (A) ϕ S X (A ) commute. We also require γ restricts to γ in the sense that the composition ϕ XA S (ϕ P) 0 X A S P XA S γ (X A S ) G ϕ 0 (X A S ) G coincides with γ. In the topological setting non-abelian Poincaré duality equated factorization homology of X with coefficients in Ω 2 BG to maps ap c (X, BG) = Bun G (X ), at least when X was a manifold. We cannot ask for an equivalence here, but we get something close. Theorem Let X be a algebraic curve over k and let G be a (smooth affine??) group scheme over X whose generic fiber is semisimple and simply connected. oreover suppose X is connected. Then natural map Ran u G (X ) Bun G (X ) is an l-adic homology equivalence. That is, it is an equivalence after applying the functor H ( ;Z l ). Now, we will prove a case of the above theorem when G is the trivial group. This amounts to showing that the map Ran u (X ) Spec k is a homology equivalence. That is, Ran u (X ) has the same l-adic homology type as a point. 21

22 We start by showing H 0 is the right thing. For this we will need the two different versions of the Ran prestack. There are functors Ran(X ) F Ran u (X ) G Ran(X ). given as follows. F simply sends (A,µ : S X (A)) to (A,µ(S)). The functor G sends (A, S) to (A,id : S S X (A)). Clearly G is a right inverse for F. oreover there is a natural transformation id G F and so G and F produce mutual inverses isomorphisms H (Ran(X ); R) = H (Ran u (X ); R). Now, let Fin surj denote the category of finite sets with surjections as morphisms. There is an obvious functor Ran(X ) Fin surj which turns out to be a fibration of categories whose fiber over S can be identified with X S. Thus Ran(X ) = colim X S. Correspondingly we have C (Ran(X ); R) colim S Fin surj C (X S ; R). and this passes to an equivalence on H 0 since there is no cohomology in negative degrees. Since X is connected H 0 (X S ; R) = R. Thus H 0 (Ran(X ); R) = R. It remains to see that the higher homologies vanish. For this, we consider the prestack Ran u (X ) again. By induction we assume that H i (Ran u (X ); R) 0 for 0 < i < n. From the prestack K unneth theorem we have an isomorphism H n (Ran u (X ) Spec k Ran u (X )) H n (Ran u (X )) H n (Ran u (X )). Consider the composition Ran u (X ) diag Ran u (X ) Ran u (X ) m Ran u (X ) where m((r, S),(R, S )) = (R, S S ). This composition is clearly the identity. FINISH 12 Lecture 11, Setting up the proof Today we set up notation and outline the main steps in proving Theorem The idea is that we will prove that the map Ran u G (X ) Bun G (X ) is a fibration (in some sense) whose fibers are homologically trivial. The fibers coincide with maps that one could think of as rational maps as they are defined outside of a finite set of points. The correct notion of rational maps is as follows. Definition For schemes X, Y define the prestack ap u rat (X, Y )+ as follows. As usual we define this by providing an op-fibered category over Ring k. The objects are triples (A, S,γ) where A is a commutative k-algebra. S X (A) is a finite subset (possibly empty). And γ : X A S Y is a map of schemes. A morphism (A, S,γ) (A, S,γ ) is a map ϕ 0 : A A such that ϕ S maps into S (not necessarily surjectively). orever, the following diagram X A S ϕ 0 Y X A S is required to commute. 22

23 Recall G is a group scheme over X. Let G 0 denote the fiber. In the cases we care about G is split over X, and so has the form G = X G 0. We will not need this specificity yet, but it may be convenient in the upcoming lectures. To restate the main idea somewhat more precisesly, we would like to think of Ran u (X ) Bun G G (X ) as some type of principal bundle. The appropriate structure group of this principal bundle will be (as expected) ap u (X, G rat 0 ). Here are the main steps of the proof. For the remainder of this lecture, the notation above an arrow means an l-adic homological equivalence. For a trivial bundle P : Spec A Bun G (X ) we want to show that Sect u (P) + Spec k. We will then show the above equivalence for generically trivial P. We say P is generically trivial if there exists an open set U X A such that P U is trivial and such that the composite is surjective. U X A Spec A To complete the proof we will prove a theorem of Drinfel-Simpson which states that upon passing to a cover every G-bundle can be taken to be generically trivial. We will deduce the first bullet by first proving that there is an equivalence ap u rat (X, G 0 ) Spec k. The idea in proving this will be motivated by a very nice fact about rational maps. Namely, they have surprising locality in the target (unlike maps of spaces, say). Slightly more precisely, if Y U U U U and ap u rat (X, Z)+ Spec k for Z = U, U, U U, then it follows that ap u rat (X, Y )+ Spec k. From this we will deduce that rational maps into open subsets of affine space is homologically trivial. In order to globalize we will refer to the classical Bruhat decomposition. Namely we will show that G 0 can be covered by translates of the big cell in the decomposition. 23