Strong homotopy theory of cyclic sets (Q1892303)

Let \({\mathbf K} @>\Phi>> {\mathbf L}\), \({\mathbf L} @>\Psi>> {\mathbf K}\) be a pair of adjoint functors, \(\alpha: \Psi \Phi\to 1\), \(\beta: 1\to \Phi\Psi\) the associated natural transformations, then it is a well-known and quite simple fact, that the homotopy categories \({\mathbf K}/\alpha\) and \({\mathbf L}/\beta\) become equivalent, where e.g. \({\mathbf K}/\alpha\) is the quotient category [cf. the reviewer and \textit{J. Dugundji}, Trans. Am. Math. Soc. 140, 239-256 (1969; Zbl 0182.259)]\ meaning that all arrows \(\alpha_X: \Psi\Phi (X)\to X\) are inverted. The standard example is furnished by the geometric realization functor \(|\cdot|\) and the singular complex (= singularization) \(S(\cdot)\). In the present paper the author considers instead of simplicial sets the category \({\mathbf S}^c\) of cyclic sets (= simplicial sets with an extra structure) and replaces topological spaces by the category \(\mathbf {Top}^{S^1}\) of spaces with \(S^1\)-action. He introduces the appropriate adjoint pair \(|\cdot |_c\) and \(S_c (\cdot)= R(\cdot)\) and derives the equivalence of homotopy categories as mentioned above. This is accomplished by endowing \({\mathbf S}^c\) as well as \(\mathbf {Top}^{S^1}\) with the structure of a closed model category (which he calls ``strong'' model structures in order to distinguish them from an earlier closed model structure, introduced by \textit{W. G. Dwyer}, \textit{M. J. Hopkins} and \textit{D. M. Kan} [ibid. 291, 281-289 (1985; Zbl 0594.55020)]).