Equipping weak equivalences with algebraic structure

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Authors

BOURKE John Denis

Year of publication 2020
Type Article in Periodical
Magazine / Source Mathematische Zeitschrift
MU Faculty or unit

Faculty of Science

Citation
Web https://doi.org/10.1007/s00209-019-02305-w
Doi http://dx.doi.org/10.1007/s00209-019-02305-w
Keywords Monads; Algebraic injectives; Weak equivalences
Description 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.
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