E∞ algebras and p-adic homotopy theory

Department of Mathematics, Massachusetts Institute of Technology, Room 2-265, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
Topology (Impact Factor: 0.23). 01/2001; 40(1):43-94. DOI: 10.1016/S0040-9383(99)00053-1


Let denote the field with p elements and its algebraic closure. We show that the singular cochain functor with coefficients in induces a contravariant equivalence between the homotopy category of connected p-complete nilpotent spaces of finite p-type and a full subcategory of the homotopy category of -algebras.

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    • "We are mostly motivated by the applications of our constructions to dg-algebras over the operads P = Com, E , because these categories of dg-algebras define models for the homotopy of spaces (rationally or completed at a prime depending on the context). We mostly refer to [11] [12] for these applications of dg-algebras in homotopy theory. Let A be an object in any of our categories of dg-algebras. "
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    ABSTRACT: We consider the cotriple resolution of algebras over operads in differential graded modules. We focus, to be more precise, on the example of algebras over the differential graded Barratt-Eccles operad and on the example of commutative alegbras. We prove that the geometric realization of the cotriple resolution (in the sense of model categories) gives a cofibrant resolution functor on these categories of differential graded algebras.
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    • "There is also a significant amount of unpublished work due, independently, to Jim Milgram and Ezra Getzler to be acknowledged. These constructions are significant since by [8] [9], under suitable assumptions on Y , an E ∞ -algebra structure on C * (Y ) determines the homotopy type of Y . Perhaps the main theme in this paper is to understand better the relation between chain cooperations (or cochain operations) and the combinatorics of joins. "
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    ABSTRACT: The join operad arises from the combinatorial study of the iterated join of simplices. We study a suitable simplicial version of this operad which includes the symmetries given by permutations of the factors of the join. From this combinatorics we construct an E-infinity operad which coacts naturally on the chains of a simplicial set.
    Preview · Article · Oct 2011 · HOMOLOGY HOMOTOPY AND APPLICATIONS
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    • "The action of the Barratt-Eccles operad, used in this paper, is only defined in [5]. By theorems of M. Mandell [27] [28] (see also [34]), the E ∞ -algebra ¯ N * (X) is sufficient to determine the homotopy type of X provided that X satisfies standard finiteness and completeness assumptions. This result was a first motivation for the constructions of [5] [32]. "
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    ABSTRACT: The goal of this paper is to prove a Koszul duality result for En-operads in differential graded modules over a ring. The case of an E1-operad, which is equivalent to the associative operad, is classical. For n > 1, the homology of an En-operad is identified with the n-Gerstenhaber operad and forms another well-known Koszul operad. Our main theorem asserts that an operadic cobar construction on the dual cooperad of an En-operad En defines a cofibrant model of En. This cofibrant model gives a realization at the chain level of the minimal model of the n-Gerstenhaber operad arising from Koszul duality. Most models of En-operads in differential graded modules come in nested sequences E1⊂E2⊂···⊂E∞ homotopically equivalent to the sequence of the chain operads of little cubes. In our main theorem, we also define a model of the operad embeddings En-1→En at the level of cobar constructions.
    Preview · Article · Jun 2011 · Selecta Mathematica
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