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Derived Algebraic Geometry VI: E[K]-Algebras

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Derived Algebraic Geometry VI: E[K]-Algebras Derived Algebraic Geometry VI: E[k]-Algebras October 30, 2009 Contents 1 Foundations 5 1.1 The E[k]-Operads . 6 1.2 The Additivity Theorem . 8 1.3 Iterated Loop Spaces . 20 1.4 Coherence of the Little Cubes Operads . 26 1.5 Tensor Products of E[k]-Algebras . 29 1.6 Nonunital Algebras . 38 2 Applications of Left Modules 47 2.1 Algebras and their Module Categories . 48 2.2 Properties of ModA(C)........................................ 55 2.3 Behavior of the Functor Θ . 63 2.4 The 1-Operad LMod ........................................ 69 2.5 Centralizers and Deligne's Conjecture . 72 2.6 The Adjoint Representation . 82 2.7 The Cotangent Complex of an E[k]-Algebra . 89 3 Factorizable Sheaves 92 3.1 Variations on the Little Cubes Operads . 93 3.2 Little Cubes in a Manifold . 96 3.3 The Ran Space . 101 3.4 Topological Chiral Homology . 107 3.5 Properties of Topological Chiral Homology . 111 3.6 Factorizable Cosheaves and Ran Integration . 114 3.7 Digression: Colimits of Fiber Products . 120 3.8 Nonabelian Poincare Duality . 126 A Background on Topology 133 A.1 The Seifert-van Kampen Theorem . 134 A.2 Locally Constant Sheaves . 138 A.3 Homotopy Invariance . 141 A.4 Singular Shape . 145 A.5 Constructible Sheaves . 147 A.6 1-Categories of Exit Paths . 152 A.7 Exit Paths in a Simplicial Complex . 157 A.8 A Seifert-van Kampen Theorem for Exit Paths . 159 A.9 Digression: Complementary Colocalizations . 162 A.10 Exit Paths and Constructible Sheaves . 165 1 A.11 Embeddings of Topological Manifolds . 172 A.12 Verdier Duality . 176 B Generalities on 1-Operads 182 B.1 Unitalization . 183 B.2 Disintegration of 1-Operads . 185 B.3 Transitivity of Operadic Left Kan Extensions . 194 B.4 A Coherence Criterion . 197 B.5 Coproducts of 1-Operads . 207 B.6 Slicing 1-Operads . 212 B.7 Wreath Products of 1-Operads . 217 2 Introduction Let X be a topological space equipped with a base point ∗. We let ΩX denote the loop space of X, which we will identify with the space of continuous map p :[−1; 1] ! X such that f(−1) = ∗ = f(1). Given a pair of loops p; q 2 ΩX, we can define a composite loop p ◦ q by concatenating p with q: that is, we define p ◦ q :[−1; 1] ! X by the formula ( q(2t + 1) if − 1 ≤ t ≤ 0 (p ◦ q)(t) = p(2t − 1) if 0 ≤ t ≤ 1: This composition operation is associative up to homotopy, and endows the set of path components π0ΩX with the structure of a group: namely, the fundamental group π1(X; ∗). However, composition of paths is not strictly associative: given a triple of paths p; q; r 2 ΩX, we have 8 −1 8 >r(4t + 3) if − 1 ≤ t ≤ 2 >r(2t + 1) if − 1 ≤ t ≤ 0 < −1 < 1 (p ◦ (q ◦ r))(t) = q(4t + 1) if 2 ≤ t ≤ 0 ((p ◦ q) ◦ r)(t) = q(4t − 1) if 0 ≤ t ≤ 2 :> :> 1 p(2t − 1) if 0 ≤ t ≤ 1: p(4t − 3) if 2 ≤ t ≤ 1: The paths p ◦ (q ◦ r) and (p ◦ q) ◦ r follow the same trajectories but are parametrized differently; they are homotopic but not identical. One way to compensate for the failure of strict associativity is to consider not one composition operation but several. For every finite set S, let Rect((−1; 1) × S; (−1; 1)) denote the collection of finite sequences of maps ffS :(−1; 1) ! (−1; 1)gs2S with the following properties: (a) For s 6= t, the maps fs and ft have disjoint images. (b) For each s 2 S, the map fs is given by a formula fs(t) = at + b where a > 0. If X is any pointed topological space, then there is an evident map θ : (ΩX)S × Rect((−1; 1) × S; (−1; 1)) ! ΩX; given by the formula ( 0 0 ps(t ) if t = fs(t ) θ(fpsgs2S; ffsgs2S)(t) = ∗ otherwise. Each of the spaces Rect((−1; 1) × S; (−1; 1)) is equipped with a natural topology (with respect to which the map θ is continuous), and the collection of spaces fRect((−1; 1) × S; (−1; 1))gI can be organized into a topological operad, which we will denote by C1. We can summarize the situation as follows: (∗) For every pointed topological space X, the loop space ΩX carries an action of the topological operad C1. Every point of Rect((−1; 1) × S; (−1; 1)) determines a linear ordering of the finite set S. Conversely, if we fix a linear ordering of S, then the corresponding subspace of Rect((−1; 1) × S; (−1; 1)) is contractible. In other words, there is a canonical homotopy equivalence of Rect((−1; 1)×S; (−1; 1)) with the (discrete) set of linear orderings of S. Together, these homotopy equivalences determine a weak equivalence of the topological operad C1 with the associative operad (see Example 1.1.7). Consequently, an action of the operad C1 can be regarded as a homotopy-theoretic substitute for an associative algebra structure. In other words, assertion (∗) articulates the idea that the loop space ΩX is equipped with a multiplication which associative up to coherent homotopy. If X is a pointed space, then we can consider also the k-fold loop space ΩkX, which we will identify with the space of all maps f :[−1; 1]k ! X which carry the boundary of the cube [−1; 1]k to the base point of X. 3 If k > 0, then we can identify ΩkX with Ω(Ωk−1X), so that ΩkX is equipped with a coherently associative multiplication given by concatenation of loops. However, if k > 1, then the structure of ΩkX is much richer. To investigate this structure, it is convenient to introduce a higher-dimensional version of the topological C1, called the little k-cubes operad. We begin by introducing a bit of terminology. Definition 0.0.1. Let 2k = (−1; 1)k denote an open cube of dimension k. We will say that a map f : 2k ! 2k is a rectilinear embedding if it is given by the formula f(x1; : : : ; xk) = (a1x1 + b1; : : : ; akxk + bk) for some real constants ai and bi, with ai > 0. More generally, if S is a finite set, then we will say that a map 2k × S ! 2k is a rectilinear embedding if it is an open embedding whose restriction to each connected component of 2k × S is rectilinear. Let Rect(2k × S; 2k) denote the collection of all rectitlinear embeddings from 2k × S into 2k. We will regard Rect(2k × S; 2k) as a topological space (it can be identified with an open subset of (R2k)I ). The spaces Rect(2k × S; 2k) determine a topological operad, called the little k-cubes operad; we will (temporarily) denote this operad by Ck. Assertion (∗) has an evident generalization: (∗0) Let X be a pointed topological space. Then the k-fold loop space ΩkX carries an action of the topological operad Ck. Assertion (∗0) admits the following converse, which highlights the importance of the little cubes operad Ck in algebraic topology: Theorem 0.0.2 (May). Let Y be a topological space equipped with an action of the little cubes operad Ck. Suppose that Y is grouplike (Definition 1.3.2). Then Y is weakly homotopy equivalent to ΩkX, for some pointed topological space X. A proof of Theorem 0.0.2 is given in [61] (we will prove another version of this result as Theorem 1.3.16, using a similar argument). Theorem 0.0.2 can be interpreted as saying that, in some sense, the topological operad Ck encodes precisely the structure that a k-fold loop space should be expected to possess. In the case k = 1, the structure consists of a coherently associative multiplication. This structure turns out to be useful and interesting outside the context of topological spaces. For example, we can consider associative algebras in the category of abelian groups (regarded as a symmetric monoidal category with respect to the tensor product of abelian groups) to recover the theory of associative rings. A basic observation in the theory of structured ring spectra is that a similar phenomenon occurs for larger values of k: that is, it is interesting to study algebras over the topological operads Ck in categories other than that of topological spaces. Our goal in this paper is to lay the general foundations for such a study, using the formalism of 1-operads developed in [50]. More precisely, we will define (for each integer k ≥ 0) an 1-operad E[k] of little k-cubes by applying the construction of Notation C.4.3.1 to the topological operad Ck (see Definition 1.1.1). We will study these operads (and algebras over them) in x1. In [49], we develop a general theory of associative algebras and (right or left) modules over them. This theory has a number of applications to the study of E[k]-algebras, which we will describe in x2. For example, we prove a version of the (generalized) Deligne conjecture in x2.5. The description of the 1-operads E[k] in terms of rectilinear embeddings between cubes suggests that the the theory of E[k]-algebras is closely connected with geometry. In x3, we will make this connection more explicit by developing the theories of factorizable (co)sheaves and topological chiral homology, following ideas introduced by Beilinson and Drinfeld in the algebro-geometric context. We conclude this paper with two appendices. The first, xA, reviews a number of ideas from topology which are relevant to the subject of this paper.
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