Homotopy diagrams of algebras. (English) Zbl 1024.55012

Slovák, Jan (ed.) et al., The proceedings of the 21th winter school “Geometry and physics”, Srní, Czech Republic, January 13-20, 2001. Palermo: Circolo Matemàtico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 69, 161-180 (2002).
The paper is concerned with homotopy concepts in the category of chain complexes. It is part of the author’s program to translate [J. M. Boardman and R. M. Vogt, Homotopy invariant algebraic structures on topological spaces, Lect. Notes Math. 347, Springer-Verlag (1973; Zbl 0285.55012)] from topology to algebra.
In topology the notion of operad extracts the essential algebraic information contained in the following example (endomorphism operad).
The endomorphism operad \({\mathcal E}_X\) of a based space \(X\) consists of the family \({\mathcal E}_X(j)\) \((j\geq 0)\) of spaces of based maps \(X^j\to X\), together with the collection of continuous maps \[ \gamma:{\mathcal E}_X(k)\times{\mathcal E}_X(j_1)\times\cdots\times{\mathcal E}_X(j_k)\to{\mathcal E}_X(j) \] given by the formula \[ \gamma(f; g_1,\dots, g_k)= f(g_1\times\cdots\times g_k), \] where \(k,j_1,\dots, j_k,j\) are such that \(j= \sum^k_{s=1} j_s\).
Operads have proved to be a convenient tool to investigate, for example, the geometry of iterated loop spaces [cf. J. P. May, The geometry of iterated loop spaces, Lect. Notes Math. 271, Springer-Verlag (1972; Zbl 0244.55009)].
In the present paper the algebraic counterpart of a special type of operad (colored operad) is dealt with. The author gives an explicit description of algebraic models for colored operads describing diagrams of homomorphisms. As an application the homotopy structure on the category of strongly homotopy associative algebra, described in [M. Grandis, J. Pure Appl. Algebra 134, 15-81 (1999; Zbl 0923.18007)], is rediscovered.
For the entire collection see [Zbl 0994.00029].
Reviewer: K.H.Kamps (Hagen)


55U35 Abstract and axiomatic homotopy theory in algebraic topology
18G55 Nonabelian homotopical algebra (MSC2010)
55U15 Chain complexes in algebraic topology
12H05 Differential algebra
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