Homotopy Algebras for Operads
| dc.creator | Leinster, Tom | |
| dc.date | 2000-02-22 | |
| dc.date.accessioned | 2026-07-07T04:34:00Z | |
| dc.date.available | 2026-07-07T04:34:00Z | |
| dc.description | We present a definition of homotopy algebra for an operad, and explore its consequences. The paper should be accessible to topologists, category theorists, and anyone acquainted with operads. After a review of operads and monoidal categories, the definition of homotopy algebra is given. Specifically, suppose that M is a monoidal category in which it makes sense to talk about algebras for some operad P. Then our definition says what a homotopy P-algebra in M is, provided only that some of the morphisms in M have been marked out as `homotopy equivalences'. The bulk of the paper consists of examples of homotopy algebras. We show that any loop space is a homotopy monoid, and, in fact, that any n-fold loop space is an n-fold homotopy monoid in an appropriate sense. We try to compare weakened algebraic structures such as A_infinity-spaces, A_infinity-algebras and non-strict monoidal categories to our homotopy algebras, with varying degrees of success. We also prove results on `change of base', e.g. that the classifying space of a homotopy monoidal category is a homotopy topological monoid. Finally, we reflect on the advantages and disadvantages of our definition, and on how the definition really ought to be replaced by a more subtle infinity-categorical version. | |
| dc.description | 100 pages. An introductory paper is available, 8 pages and operad-free: math.QA/9912084 | |
| dc.identifier | https://arxiv.org/abs/math/0002180 | |
| dc.identifier | http://arxiv.org/abs/math/0002180 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/58738 | |
| dc.subject | Quantum Algebra | |
| dc.subject | Algebraic Topology | |
| dc.subject | Category Theory | |
| dc.title | Homotopy Algebras for Operads | |
| dc.type | text |