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Publisher's Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations -
D Ara (2013)
On the homotopy theory of Grothendieck $$\omega $$ ω -groupoidsJ. Pure Appl. Algebra, 217
- Publisher
- Springer Journals
- Copyright
- Copyright © Springer-Verlag GmbH Germany, part of Springer Nature 2019
- Subject
- Mathematics; Mathematics, general
- ISSN
- 0025-5874
- eISSN
- 1432-1823
- DOI
- 10.1007/s00209-019-02305-w
- Publisher site
- See Article on Publisher Site
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.
Journal
Mathematische Zeitschrift – Springer Journals
Published: Apr 5, 2020