A P-local Delooping Machine

TL;DR

This paper constructs a Lawvere theory for certain spaces, showing that spaces weakly equivalent to mapping spaces out of a localized sphere can be characterized algebraically, extending the delooping machine concept.

Contribution

It introduces a new Lawvere theory $T_A$ that characterizes spaces as $T_A$-algebras, linking algebraic structures to localized sphere mapping spaces.

Findings

Spaces satisfying a smallness condition are equivalent to $T_A$-algebras.

Localized spheres at a set of primes meet the smallness condition.

Provides an algebraic framework for understanding certain mapping spaces.

Abstract

We show that for spaces $A$ that satisfy a certain smallness condition, there is a Lawvere theory $T_A$ so that a space $X$ has the structure of a $T_A$-algebra if and only if $X$ is weakly equivalent to a mapping space out of $A$. In particular, spheres localized at a set of primes satisfy this condition.