Higher Algebra - people.math.harvard.edu

1 Higher AlgebraSeptember 18, 2017 Contents1 Stable Foundations .. homotopy Category of a Stable -Category .. Properties of Stable -Categories .. Functors .. Stable -Categories and Homological Algebra .. on Stable -Categories .. Objects and Spectral Sequences .. Dold-Kan Correspondence .. -Categorical Dold-Kan Correspondence .. Homological Algebra and Derived Categories .. of Differential Graded Categories .. -Categories .. Universal Property ofD (A) .. Quasi-Isomorphisms .. Abelian Categories .. Spectra and Stabilization .. Brown Representability Theorem .. Objects .. -Category of Spectra .. Stable -Categories .. 1542 Foundations .. Colored Operads to -Operads .. of -Operads .. Objects .. -Preoperads .. Constructions of -Operads .. ofO-Monoidal -Categories .. -Operads .. of -Operads .. Envelopes .. Products of -Operads.

2 Convolution .. Disintegration and Assembly .. -Operads .. -Operads .. to -Operads .. of -Operads .. Products and Coproducts .. Symmetric Monoidal Structures .. Objects .. Symmetric Monoidal Structures .. Products .. 3043 Algebras and Modules over Free Algebras .. Colimit Diagrams .. Left Kan Extensions .. of Free Algebras .. of Operadic Left Kan Extensions .. Limits and Colimits of Algebras .. Objects and Trivial Algebras .. of Algebras .. of Algebras .. Products of Commutative Algebras .. Modules over -Operads .. -Operads .. Coherence Criterion .. Objects .. General Features of Module -Categories .. Objects of -Categories of Modules .. over Trivial Algebras .. of Modules .. of Modules .. 4354 CONTENTS4 Associative Algebras and Their Associative Algebras .. Associative -Operad .. Objects of -Categories.

3 -Operads andA -Algebras .. and NonunitalAn-Monoids .. toAn+1-Algebras .. Associahedron .. Model Categories .. of Associative Algebras .. Left and Right Modules .. -OperadLM .. Models for Algebras and Modules .. and Colimits of Modules .. Modules .. Bimodules .. -OperadBM .. , Left Modules, and Right Modules .. , Colimits, and Free Bimodules .. The Relative Tensor Product .. Maps .. Products and the Bar Construction .. of the Tensor Product .. Modules over Commutative Algebras .. and Right Modules over Commutative Algebras .. Products over Commutative Algebras .. of Algebra .. of Commutative Algebras .. Duality .. in Monoidal -Categories .. of Bimodules .. Right and Left Actions .. and Proper Algebras .. Algebras .. Monads and the Barr-Beck Theorem .. -Categories .. Simplicial Objects .. Barr-Beck Theorem .. Fibrations.

4 And the Beck-Chevalley Condition .. Tensor Products of -Categories .. Products of -Categories .. Products of Spectra .. and their Module Categories .. of RModA(C) .. of the Functor .. 7435 Little Cubes and Factorizable Definitions and Basic Properties .. Cubes and Configuration Spaces .. Additivity Theorem .. Products ofEk-Modules .. of Tensor Products .. Bar Constructions and Koszul Duality .. Arrow -Categories .. Bar Construction for Associative Algebras .. Bar Constructions .. Pairings .. Duality forEk-Algebras .. Loop Spaces .. Centers and Centralizers .. and Centralizers .. Adjoint Representation .. Products of Free Algebras .. Little Cubes and Manifold Topology .. of Topological Manifolds .. on the Little Cubes Operads .. : Nonunital Associative Algebras and their Modules .. Cubes in a Manifold .. Topological Chiral Homology.

5 Ran Space .. Chiral Homology .. of Topological Chiral Homology .. Cosheaves and Ran Integration .. Duality .. Poincare Duality .. 9986 CONTENTS6 The Calculus of The Calculus of Functors .. Functors .. Taylor Tower .. of Many Variables .. Functors .. from Spaces to Spectra .. Maps .. Differentiation .. of Functors .. of Differentiable Fibrations .. of Functors .. Smash Products .. of -Operads .. of Stabilizations .. The Chain Rule .. Structures .. of Correspondences .. of the Identity Functor .. and Reduction .. of Theorem .. Dual Chain Rule .. 11827 Algebra in the Stable homotopy Structured Ring Spectra .. and Their Modules .. Principles .. of Ring .. over Commutative Rings .. Properties of Rings and Modules .. Resolutions and Spectral Sequences .. and Projective Modules .. and Ore Conditions .. Properties of Rings and Modules.

6 The Cotangent Complex Formalism .. Envelopes and Tangent Bundles .. Adjunctions .. Relative Cotangent Complex .. Bundles to -Categories of Algebras .. Cotangent Complex of anEk- Algebra .. Tangent Correspondence .. Deformation Theory .. Extensions .. Theory ofE -Algebras .. and Finiteness of the Cotangent Complex .. EtaleMorphisms .. EtaleMorphisms ofE1-Rings .. Nonconnective Case .. Morphisms .. EtaleMorphisms ofEk-Rings .. 1395A Constructible Sheaves and Exit Locally Constant Sheaves .. homotopy Invariance .. The Seifert-van Kampen Theorem .. Singular Shape .. Constructible Sheaves .. -Categories of Exit Paths .. A Seifert-van Kampen Theorem for Exit Paths .. Digression: Recollement .. Exit Paths and Constructible Sheaves .. 1456B Categorical Morphisms .. The Model Structure on (Set+ )/P.. Flat Inner Fibrations .. Functoriality.

7 1507 General Index .. 1527 Notation Index .. 15388 CONTENTSLetKdenote the functor ofcomplexK-theory, which associates to every compact HausdorffspaceXthe Grothendieck groupK(X) of isomorphism classes of complex vector bundles onX. ThefunctorX7 K(X) is an example of acohomology theory: that is, one can define more generallya sequence of abelian groups{Kn(X,Y)}n Zfor every inclusion of topological spacesY X, insuch a way that the Eilenberg-Steenrod axioms are satisfied (see [49]). However, the functorKisendowed with even more structure: for every topological spaceX, the abelian groupK(X) has thestructure of a commutative ring (whenXis compact, the multiplication onK(X) is induced bythe operation of tensor product of complex vector bundles). One would like that the ring structureonK(X) is a reflection of the fact thatKitself has a ring structure, in a suitable analyze the problem in greater detail, we observe that the functorX7 K(X) isrepre-sentable.

8 That is, there exists a topological spaceZ=Z BUand a universal class K(Z),such that for every sufficiently nice topological spaceX, the pullback of induces a bijection[X,Z] K(X); here [X,Z] denotes the set of homotopy classes of maps fromXintoZ. Accordingto Yoneda s lemma, this property determines the spaceZup to homotopy equivalence. Moreover,since the functorX7 K(X) takes values in the category of commutative rings, the topologicalspaceZis automatically acommutative ring objectin the homotopy categoryHof topologicalspaces. That is, there exist addition and multiplication mapsZ Z Z, such that all of theusual ring axioms are satisfied up to homotopy . Unfortunately, this observation is not very would like to have a robust generalization of classical Algebra which includes a good theory ofmodules, constructions like localization and completion, and so forth. The homotopy categoryHis too poorly behaved to support such a alternate possibility is to work with commutative ring objects in the category of topologicalspaces itself: that is, to require the ring axioms to hold on the nose and not just up to this does lead to a reasonable generalization of classical commutative Algebra , it notsufficiently general for many purposes.

9 For example, ifZis a topological commutative ring, thenone can always extend the functorX7 [X,Z] to a cohomology theory. However, this cohomologytheory is not very interesting: in degree zero, it simply gives the following variant of classicalcohomology: n 0Hn(X; nZ).In particular, complexK-theory cannot be obtained in this way. In other words, theZ=Z BUfor stable vector bundles cannot be equipped with the structure of a topological commutative reflects the fact that complex vector bundles on a spaceXform acategory, rather than justa set. The direct sum and tensor product operation on complex vector bundles satisfy the ringaxioms, such as the distributive lawE (F F )'(E F) (E F ),but only up to isomorphism. However, althoughZ BU has less structure than a commutative ring,it has more structure than simply a commutative ring object in the homotopy categoryH, becauseCONTENTS9the isomorphism displayed above is actuallycanonicaland satisfies certain coherence conditions(see [92] for a discussion).

10 To describe the kind of structure which exists on the topological spaceZ BU, it is convenientto introduce the language ofcommutative ring spectra, or, as we will call them,E -rings. Roughlyspeaking, anE -ring can be thought of as a spaceZwhich is equipped with an addition and amultiplication for which the axioms for a commutative ring hold not only up to homotopy , butup tocoherenthomotopy. TheE -rings play a role in stable homotopy theory analogous to therole played by commutative rings in ordinary Algebra . As such, they are the fundamental buildingblocks ofderived algebraic of our ultimate goals in this book is to give an exposition of the theory ofE that ordinary commutative ringRcan be viewed as acommutative Algebra objectin thecategory of abelian groups, which we view as endowed with a symmetric monoidal structure givenby tensor product of abelian groups. To obtain the theory ofE -rings we will use the samedefinition, replacing abelian groups byspectra(certain algebro-topological objects which representcohomology theories).