Algebraic theories in homotopy theory
| dc.creator | Badzioch, Bernard | |
| dc.date | 2001-10-09 | |
| dc.date | 2004-11-22 | |
| dc.date.accessioned | 2026-07-07T04:43:44Z | |
| dc.date.available | 2026-07-07T04:43:44Z | |
| dc.description | An algebraic theory $T$ is a category with objects $t_0,t_2...$ such that for each $n$ the object $t_n$ is an $n$-fold categorical product of $t_1$. A strict $T$-algebra is a product preserving functor $A: T\to Spaces$. Lawvere showed that for a suitable choice of T giving such an algebra amounts to providing the space $A(t_1)$ with a familiar structure of a monoid group, ring, Lie algebra... Given a functor $X: T\to Spaces$ which preserves products up to a weak equivalence we show that $X$ is more or less canonically weakly equivalent to a strict $T$-algebra $LX$. Thus any `homotopy' algebraic structure on the space $X(t_1)$ can be rigidified to a strict algebraic structure on a space weakly equivalent to $X(t_1)$. This fact can be interpreted as a generalization of the results establishing equivalence of homotopy theories of loop spaces and simplicial groups, products of Eilenberg-Mac Lane spaces and abelian monoids etc. | |
| dc.description | 19 pages, published version | |
| dc.identifier | https://arxiv.org/abs/math/0110101 | |
| dc.identifier | http://arxiv.org/abs/math/0110101 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/62358 | |
| dc.subject | Algebraic Topology | |
| dc.subject | 55P48 (Primary); 18G55 (Secondary) | |
| dc.title | Algebraic theories in homotopy theory | |
| dc.type | text |