On operads (Q2734846)

The authors PhD thesis is a modern reworking of \textit{J. M. Boardman}'s and \textit{R. M. Vogt}'s classical treatise on homotopy invariant algebraic structures [``Homotopy invariant algebraic structures on topological spaces'', Lect. Notes Math. 347 (1973; Zbl 0285.55012)] with some further developments.NEWLINENEWLINENEWLINEIt contains four parts. Part 1 reviews strongly homotopy commutative monoids and has been published [\textit{M. Brinkmeier}, Doc. Math., J. DMV 5, 613-624 (2000; Zbl 0992.55007)]. Part 2 introduces a compactification of the operad of little \(n\)-cubes in order to prove the following generalisation of a result of \textit{G. Dunn} [J. Pure Appl. Algebra 50, No. 3, 237-258 (1988; Zbl 0672.55004)]: The operadic tensor product of the operad of little \(m\)-cubes and of the operad of little \(n\)-cubes is weakly equivalent to the operad of little \((m+n)\)-cubes.NEWLINENEWLINENEWLINEPart 3 studies topologically enriched coloured operads. As main application, the author gives two equivalent descriptions of the homotopy category of \(A_\infty\)-categories. Part 4 has been published [\textit{M. Brinkmeier}, Ann. Inst. Fourier 49, No. 5, 1427-1438 (1999; Zbl 0928.55009)]. It is concerned with the Milgram model of the \(n\)-fold loop space of an \(n\)-fold suspension, cf. \textit{R. J. Milgram} [Ann. Math., II. Ser. 84, 386-403 (1966; Zbl 0145.19901)]. In the reviewers article [\textit{C. Berger}, Ann. Inst. Fourier 46, No. 4, 1133-1166 (1996; Zbl 0853.55007)], the Milgram model has been shown to be the pointed endofunctor associated to a certain preoperad \(J_n\). Unfortunately, the Milgram model only reflects the homotopy type but not the entire multiplicative structure of an \(n\)-fold iterated loop space, as would be the case if the preoperad \(J_n\) was an operad. Brinkmeier shows that the reviewers proposal in loc. cit. of an operad structure on \(J_n\) is not compatible with the degeneracy operators of \(J_n\). This failure of an operad strucure is related to the difficulty of proving the Deligne conjecture on the Hochschild cochains of an associative algebra, cf. \textit{M. Kontsevich} and \textit{Y. Soibelman} [Math. Phys. Stud. 21, 255-307 (2000; Zbl 0972.18005)].