# Equipping weak equivalences with algebraic structure

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## 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.

## Keywords

Monads Algebraic injectives Weak equivalences## Mathematics Subject Classification

Primary 55U35 Secondary 18C35## Notes

### Acknowledgements

The author gratefully acknowledges the support of an Australian Research Council Discovery Grant DP160101519 and the support of the Grant Agency of the Czech Republic under the grant 19-00902S. Particular thanks are due to Emily Riehl whose interest in an algebraic version of Smith’s theorem got me thinking about this topic and to Lukáš Vokřínek who helped me to see the connection between \(Ex_{\infty }\) and the generating cones for simplicial sets. Thanks also to the organisers of the PSSL101 in Leeds for providing the opportunity to present this work, and to the members of the Australian Category Seminar for listening to me speak about it.

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