An \(E^ 2\) model category structure for pointed simplicial spaces
The \(E^ 2\) model category structure in the title is a new closed simplicial model category structure in the sense of \textit{D. G. Quillen} [Homotopical algebra, Lect. Notes Math. 43 (1967; Zbl 0168.209)] on the category \(s{\mathbf T}_ *\), of simplicial objects over the category, \({\mathbf T}_ *\), of pointed topological spaces. Such a structure allows one to define the notion of a (homotopy-theoretic) resolution of a topological space by simplicial spaces and gives rise to the notion of equivalence of resolutions. The \(E^ 2\) model category structure specifies a larger class of weak equivalences than a similar structure due to \textit{C. L. Reedy} [Homotopy theory of model categories] does. This means that greater flexibility in choosing homotopy-theoretic resolutions is available. The \(E^ 2\) terminology is justified by the fact that a map between simplicial objects is a weak equivalence if and only if it induces an isomorphism on \(E^ 2\)-terms of a certain spectral sequence.
simplicial objects over the category of pointed topological spaces
scientific article; zbMATH DE number 500590
scientific article