Algebraic theories in homotopy theory

Details Algebraic theories in homotopy theory Algebraic theories in homotopy theory

2001-10-09

arxiv.org/abs/math/0110101v2

Mathematics

Algebraic Topology

19 pages, published version

Scientific paper

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.

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