Frobenius and the derived centers of algebraic theories

TL;DR

This paper demonstrates that the derived center of simplicial algebras over any algebraic theory is homotopically discrete, linking it to classical centers and illustrating the role of Frobenius in characteristic p.

Contribution

It establishes the homotopical discreteness of derived centers for simplicial algebras and connects these centers to classical algebraic structures, including Frobenius.

Findings

Derived centers are homotopically discrete

Center of commutative algebras in characteristic p is generated by Frobenius

Calculations involve homotopy coherent centers and Bousfield localizations

Abstract

We show that the derived center of the category of simplicial algebras over every algebraic theory is homotopically discrete, with the abelian monoid of components isomorphic to the center of the category of discrete algebras. For example, in the case of commutative algebras in characteristic $p$, this center is freely generated by Frobenius. Our proof involves the calculation of homotopy coherent centers of categories of simplicial presheaves as well as of Bousfield localizations. Numerous other classes of examples are discussed.