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Contents Introduction 1 1. Equivariant homotopy 2 2. The equivariant stable homotopy category 10 3. Homology and cohomology theories and fixed point spectra 15 4. Change of groups and duality theory 20 5. Mackey functors, K(M, n)'s, and RO(G)-graded cohomology 25 6. Philosophy of localization and completion theorems 30 7. How to prove localization and(More)
Working in the category T of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functors D −→ T for a suitable small topological category D. When D is symmetric monoidal, there is a smash product that gives the category of D-spaces a symmetric monoidal structure. Examples include • Prespectra, as defined classically. •(More)
With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I Definitions and examples of operads and their actions II Partial algebraic structures and conversion theorems III Derived categories from a topological point(More)
  • A D Elmendorf, I Kriz, M A Mandell, J P May, J Dolan
  • 2007
Please send any errors or comments on EKMM to mandell@math. Error: Lemma II.6.1 is false as stated and has incorrect proof. who gave the following counterexample (I have rephrased it, any errors due to me): Let S be the monad on sets that takes a set to its disjoint union with the set {1, 2}. Let T be the monad on the category of (sets under the set {1, 2})(More)
ii Contents Introduction 1 Chapter I. Equivariant Cellular and Homology Theory 13 1. Some basic deenitions and adjunctions 13 2. Analogs for based G-spaces 15 3. G-CW complexes 16 4. Ordinary homology and cohomology theories 19 5. Obstruction theory 22 6. Universal coeecient spectral sequences 23 Chapter II. Postnikov Systems, Localization, and Completion(More)
We explain a fundamental additivity theorem for Euler characteristics and generalized trace maps in triangulated categories. The proof depends on a refined axiomatization of symmetric monoidal categories with a compatible triangulation. The refinement consists of several new axioms relating products and distinguished triangles. The axioms hold in the(More)
We give a quick outline of a bare bones introduction to point set topology. The focus is on basic concepts and definitions rather than on the examples that give substance to the subject. 1. Topological spaces Definition 1.1. A topology on a set X is a set of subsets, called the open sets, which satisfies the following conditions. (i) The empty set ∅ and the(More)
The last few years have seen a revolution in our understanding of the foundations of stable homotopy theory. Many symmetric monoidal model categories of spectra whose homotopy categories are equivalent to the stable homotopy category are now known, whereas no such categories were known before 1993. The most well-known examples are the category of S-modules(More)
Let S be a symmetric monoidal category with product ⊗ and unit object κ. Definition 1. An operad C in S consists of objects C (j), j ≥ 0, a unit map η : κ → C (1), a right action by the symmetric group Σ j on C (j) for each j, and product maps γ : C (k) ⊗ C (j 1) ⊗ · · · ⊗ C (j k) → C (j) for k ≥ 1 and j s ≥ 0, where j s = j. The γ are required to be(More)