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Lax operad actions and coherence for monoidal \(n\)-categories, \(A_{\infty}\) rings and modules. (English) Zbl 0871.18002

Several coherence results of the following form are proved. Let \(K\) be a 2-category and let \(t\) be a monad on \(K\). Write \(t[K]\) for the 2-category of Eilenberg-Moore \(t\)-algebras and \(t(K)\) for the 2-category of pseudo-\(t\)-algebras. In the cases considered, the inclusion 2-functor \(t[K]\to t(K)\) has a left adjoint for which the unit is an equivalence. The monads here arise from braided operads and the objects of \(K\) are topological categories (that is, categories in the category of topological spaces) sometimes with extra structure (indeed, there is a case where \(K\) is not really a 2-category but merely a Gray-category). At this level there seem to be close links with the work of R. Blackwell, G. M. Kelly and A. J. Power [“Two-dimensional monad theory”, J. Pure Appl. Algebra 59, No. 1, 1-41 (1989; Zbl 0675.18006)]. {The word “lax” in the present paper should be replaced throughout by “pseudo” in the terminology of G. M. Kelly and R. Street [“Review of the elements of 2-categories”, Lect. Notes Math. 420, 75-103 (1974; Zbl 0334.18016)].}
Then higher ring-like structures are introduced: they are symmetric monoidal topological categories equipped with even more operadic structure, and \(K\)-theories can be constructed from them. The coherence results aid in the development, in the present paper, of the theory of bimodules between the rings. The author is preparing a paper to show that Morita equivalent rings have the same \(K\)-theory.

MSC:

18D05 Double categories, \(2\)-categories, bicategories and generalizations (MSC2010)
55P47 Infinite loop spaces
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18C15 Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads
19D23 Symmetric monoidal categories
55U40 Topological categories, foundations of homotopy theory
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