Equipping weak equivalences with algebraic structure
| Authors | |
|---|---|
| Year of publication | 2020 |
| Type | Article in Periodical |
| Magazine / Source | Mathematische Zeitschrift |
| MU Faculty or unit | |
| 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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