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The paper is concerned with homotopy concepts in the category of chain complexes. It is part of the author's program to translate [{\it J. M. Boardman} and {\it R. M. Vogt}, Homotopy invariant algebraic structures on topological spaces, Lect. Notes Math. 347, Springer-Verlag (1973; Zbl 0285.55012)] from topology to algebra.\par In topology the notion of operad extracts the essential algebraic information contained in the following example (endomorphism operad).\par The endomorphism operad ${\cal E}_X$ of a based space $X$ consists of the family ${\cal E}_X(j)$ $(j\ge 0)$ of spaces of based maps $X^j\to X$, together with the collection of continuous maps $$\gamma:{\cal E}_X(k)\times{\cal E}_X(j_1)\times\cdots\times{\cal E}_X(j_k)\to{\cal E}_X(j)$$ 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 $j= \sum^k_{s=1} j_s$.\par Operads have proved to be a convenient tool to investigate, for example! (English) |
