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Homotopy homomorphisms and the Hammock localization. (English) Zbl 0853.55010

In the theory of \(A_\infty\) or \(E_\infty\) monoids, rings and modules, or in the study of diagrams of spaces homomorphisms as structure preserving maps are often too rigid: they are not homotopy invariant. For example, if one changes a homomorphism by a homotopy one obtains a homomorphism up to coherent homotopy, called \(h\)-morphism for short. Although \(h\)-morphisms seem to be the correct notion of morphisms in homotopy coherence theory, they have draw-backs: composition is defined only up to homotopy. Fortunately, composition is homotopy associative with canonical identities. Hence there is a perfectly good homotopy category. The description of naturality properties of constructions such as homotopy limits and colimits of diagrams or topological Hochschild homology or algebraic \(K\)-theory of \(A_\infty\) or \(E_\infty\) rings is rather involved unless one passes to the homotopy category. This passage reduces the spaces of \(h\)-morphisms to their path components, thus depriving homotopy coherence algebra of its appropriate Hom-sets. In view of the work of M. Bökstedt [Topological Hochschild homology, Preprint Univ. Bielefeld], F. Waldhausen [Lect. Notes Math. 1126, 318-419 (1985; Zbl 0579.18006)], and, in particular, A. Robinson [Math. Proc. Camb. Philos. Soc. 101, 249-257 (1986; Zbl 0635.55011)], this is a real loss of information. Hence we are led to analyze the structure of the collection of all \(h\)-morphisms before passage to homotopy. It forms what we will call a \(\Delta\)-category, a category-like structure which can be interpreted as a category up to coherent homotopy. To be precise, any small \(\Delta\)-category can be rectified to a homotopy equivalent honest topological category with discrete space of objects, and the rectification is a “functor” of \(\Delta\)-categories up to coherent homotopy.
In the first part of this paper we develop the necessary theory of \(\Delta\)-categories, give some examples from homotopy coherence theory, and prove the rectification result.
The second part of the paper deals with the relationship between \(h\)-morphisms and hammocks. Up to homotopy they are two sides of the same coin: the space of \(h\)-morphisms is equivalent to the simplicial set of hammocks in the strongest sense one possibly can expect. There is a sequence of maps of \(\Delta\)-categories and genuine simplicial functors of simplicial categories joining the \(\Delta\)-category of \(h\)-morphisms with the simplicial category of hammocks, and each map is a weak homotopy equivalence.

MSC:

55P42 Stable homotopy theory, spectra
55P60 Localization and completion in homotopy theory
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