294 research outputs found
Homotopy Diagrams of Algebras
In [math.AT/9907138] we proved that strongly homotopy algebras are homotopy
invariant concepts in the category of chain complexes. Our arguments were based
on the fact that strongly homotopy algebras are algebras over minimal cofibrant
operads and on the principle that algebras over cofibrant operads are homotopy
invariant. In our approach, algebraic models for colored operads describing
diagrams of homomorphisms played an important role.
The aim of this paper is to give an explicit description of these models. A
possible application is an appropriate formulation of the `ideal' homological
perturbation lemma for chain complexes with algebraic structures. Our results
also provide a conceptual approach to `homotopies through homomorphism' for
strongly homotopy algebras. We also argue that strongly homotopy algebras form
a honest (not only weak Kan) category.
The paper is a continuation of our program to translate the famous book "M.
Boardman, R. Vogt: Homotopy Invariant Algebraic Structures on Topological
Spaces" to algebra.Comment: 24 pages, LaTeX 2.0
Free Loop Spaces and Cyclohedra
In this note we introduce an action of cyclohedra on the free loop space. We
will then discuss how this action can be used for an appropriate recognition
principle in the same manner as the action of Stasheff's associahedra can be
used to recognize based loop spaces. We will also interpret one result of R.L.
Cohen as an approximation theorem, in the spirit of Beck and May, for free loop
spaces.Comment: 7 pages, LaTeX 2.0
Cyclic operads and homology of graph complexes
We will consider P-graph complexes, where P is a cyclic operad. P-graph
complexes are natural generalizations of Kontsevich's graph complexes -- for P
= the operad for associative algebras it is the complex of ribbon graphs, for P
= the operad for commutative associative algebras, the complex of all graphs.
We construct a `universal class' in the cohomology of the graph complex with
coefficients in a theory. The Kontsevich-type invariant is then an evaluation,
on a concrete cyclic algebra, of this class. We also explain some results of M.
Penkava and A. Schwarz on the construction of an invariant from a cyclic
deformation of a cyclic algebra. Our constructions are illustrated by a `toy
model' of tree complexes.Comment: LaTeX 2.09 + article12pt,leqno style, 10 page
Homotopy Algebras via Resolutions of Operads
The aim of this brief note is mainly to advocate our approach to homotopy
algebras based on the minimal model of an operad. Our exposition is motivated
by two examples which we discuss very explicitly - the example of strongly
homotopy associative algebras and the example of strongly homotopy Lie
algebras.
We then indicate what must be proved in order to show that these homotopy
algebraic structures are really `stable under a homotopy.'
The paper is based on a talk given by the author on June 16, 1998, at
University of Osnabrueck, Germany.Comment: LaTeX 2.09, 9 pages; `indecomposables' changed to `decomposables
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