Derivability structures
The authors introduce and compare different types of ``derivability structures'' for categories \(\mathcal C\) equipped with a class of morphisms \(W\) (which are thought of as weak equivalences). Such structures are given by a category \(\mathcal C_0\) together with weak equivalences \(W_0\) and a functor \(K: \mathcal C_0 \rightarrow \mathcal C\) sending \(W_0\) to \(W\). One often assumes that \(W_0\) contains the identities and is stable under composition. The idea is that \(K\) will allow for the construction of certain resolutions which then permit to establish existence theorems for derived functors. For example in a simplicial derivability structure one requires that the category of \(K\)-resolutions of any object \(c \in \mathcal C\), i.e. the category of weak equivalences from \(c\) to objects of the form \(K(d)\), be \(1\)-connected, that of \(K\)-resolutions of any morphism be connected, and that of \(K\)-resolutions of any pair of composable morphisms be non-empty. The main results are that the derivability structures introduced by \textit{W. G. Dwyer, P. S. Hirschhorn, D. M. Kan}, and \textit{J. H. Smith} [``Homotopy limit functors on model categories and homotopical categories'', Math. Surv. Monogr. 113 (2004; Zbl 1072.18012)], as well as the ``Anderson-Brown-Cisinski'' structures are simplicial, whereas the structures appearing in the work of Radulescu-Banu are derivability structures but probably not of simplicial type.
scientific article; zbMATH DE number 5285714
scientific article