Title: Equipping weak equivalences with algebraic structure,
Abstract: We investigate the extent to which the weak equivalences in a model category
can be equipped with algebraic structure. We prove, for instance, that there
exists a monad T such that a morphism of topological spaces admits T-algebra
structure if and only it is a weak homotopy equivalence. Likewise for
quasi-isomorphisms and many other examples. The basic trick is to consider
injectivity in arrow categories. Using algebraic injectivity and cone
injectivity we obtain general results about the extent to which the weak
equivalences in a combinatorial model category can be equipped with algebraic
structure.