313 research outputs found
Multiplicative structure in equivariant cohomology
We introduce the notion of a strongly homotopy-comultiplicative resolution of
a module coalgebra over a chain Hopf algebra, which we apply to proving a
comultiplicative enrichment of a well-known theorem of Moore concerning the
homology of quotient spaces of group actions. The importance of our enriched
version of Moore's theorem lies in its application to the construction of
useful cochain algebra models for computing multiplicative structure in
equivariant cohomology.
In the special cases of homotopy orbits of circle actions on spaces and of
group actions on simplicial sets, we obtain small, explicit cochain algebra
models that we describe in detail.Comment: 28 pages. Final version (cosmetic changes, slight reorganization), to
appear in JPA
Rational homotopy theory: a brief introduction
These notes contain a brief introduction to rational homotopy theory: its
model category foundations, the Sullivan model and interactions with the theory
of local commutative rings.Comment: A slight revision (some minor errors corrected) of lecture notes from
a minicourse given in the summer school "Interactions between Homotopy Theory
and Algebra," August 2004. (28 pages
A general framework for homotopic descent and codescent
In this paper we elaborate a general homotopy-theoretic framework in which to
study problems of descent and completion and of their duals, codescent and
cocompletion. Our approach to homotopic (co)descent and to derived
(co)completion can be viewed as -category-theoretic, as our framework
is constructed in the universe of simplicially enriched categories, which are a
model for -categories.
We provide general criteria, reminiscent of Mandell's theorem on
-algebra models of -complete spaces, under which homotopic
(co)descent is satisfied. Furthermore, we construct general descent and
codescent spectral sequences, which we interpret in terms of derived
(co)completion and homotopic (co)descent.
We show that a number of very well-known spectral sequences, such as the
unstable and stable Adams spectral sequences, the Adams-Novikov spectral
sequence and the descent spectral sequence of a map, are examples of general
(co)descent spectral sequences. There is also a close relationship between the
Lichtenbaum-Quillen conjecture and homotopic descent along the
Dwyer-Friedlander map from algebraic K-theory to \'etale K-theory. Moreover,
there are intriguing analogies between derived cocompletion (respectively,
completion) and homotopy left (respectively, right) Kan extensions and their
associated assembly (respectively, coassembly) maps.Comment: Discussion of completeness has been refined; statement of the theorem
on assembly has been corrected; numerous small additions and minor
correction
The homotopy theory of coalgebras over a comonad
Let K be a comonad on a model category M. We provide conditions under which
the associated category of K-coalgebras admits a model category structure such
that the forgetful functor to M creates both cofibrations and weak
equivalences.
We provide concrete examples that satisfy our conditions and are relevant in
descent theory and in the theory of Hopf-Galois extensions. These examples are
specific instances of the following categories of comodules over a coring. For
any semihereditary commutative ring R, let A be a dg R-algebra that is
homologically simply connected. Let V be an A-coring that is semifree as a left
A-module on a degreewise R-free, homologically simply connected graded module
of finite type. We show that there is a model category structure on the
category of right A-modules satisfying the conditions of our existence theorem
with respect to the comonad given by tensoring over A with V and conclude that
the category of V-comodules in the category of right A-modules admits a model
category structure of the desired type. Finally, under extra conditions on R,
A, and V, we describe fibrant replacements in this category of comodules in
terms of a generalized cobar construction.Comment: 34 pages, minor corrections. To appear in the Proceedings of the
London Mathematical Societ
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