Abstract
We provide an extensive study of the homotopy theory of types of algebras with units, for instance unital associative algebras or unital commutative algebras. To this purpose, we endow the Koszul dual category of curved coalgebras, where the notion of quasi-isomorphism barely makes sense, with a model category structure Quillen equivalent to that of unital algebras. To prove such a result, we use recent methods based on presentable categories. This allows us to describe the homotopy properties of unital algebras in a simpler and richer way. Moreover, we endow the various model categories with several enrichments which induce suitable models for the mapping spaces and describe the formal deformations of morphisms of algebras.
| Original language | English |
|---|---|
| Pages (from-to) | 1541-1618 |
| Number of pages | 78 |
| Journal | Algebraic and Geometric Topology |
| Volume | 19 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 21 May 2019 |
Funding
This article is the second part of my PhD thesis. I would like to thank my advisor Bruno Vallette for his precious advice and careful review of this paper. I also would like to thank Damien Calaque and Kathryn Hess for reviewing my thesis, as well as the anonymous referee for his remarks. Also, the Laboratory J A Dieudonné in the University of Nice provided excellent working conditions. Finally, I was supported by the ANR SAT until September 2016 and then by NWO.
Keywords
- operads
- Koszul duality
- bar and cobar constructions