Isomorphic Loday functors of non-homeomorphic spaces

TL;DR

This paper demonstrates that different topological spaces, specifically the M"obius strip and the cylinder, can have non-isomorphic algebras but still produce isomorphic Loday functors, revealing limitations in the functor's distinguishing power.

Contribution

It constructs explicit examples of non-homeomorphic spaces with isomorphic Loday functors, highlighting a new phenomenon in the representation theory of commutative algebras.

Findings

The algebras of continuous functions on the M"obius strip and the cylinder have isomorphic Loday functors.

The Loday functor does not always distinguish between non-homeomorphic spaces.

Provides explicit counterexamples in the context of algebraic topology and functor theory.

Abstract

Each commutative algebra $A$ gives rise to a representation $\mathcal{L}_A$, which we call the Loday functor of $A$, of the category $\Omega$ of finite sets and surjective maps. In this paper we present two (infinite-dimensional) non-isomorphic algebras over $\mathbb{C}$ with isomorphic Loday functors -- the algebras of continuous functions on the M\"obius strip and on the cylinder.