Double factorization systems in equivariant topology and topos theory

TL;DR

This paper extends double factorization systems from certain topological categories to their Eilenberg-Moore algebra categories, linking them to LT-topologies and topos of coalgebras.

Contribution

It demonstrates the extension of double factorization systems to Eilenberg-Moore categories and explores their connection to LT-topologies and coalgebra toposes.

Findings

Double factorization systems extend to Eilenberg-Moore algebra categories.

Connection established between cartesian double factorization systems and LT-topologies.

Provides conditions for extending these systems to topos of coalgebras.

Abstract

In the present work, we investigate the extension of double factorization systems to the categories of Eilenberg-Moore (co)algebras. We show that the double factorization systems $(\texttt{ExEpi},\texttt{Bim},\texttt{ExMono})$ in the categories $\mathbf{Tych}$, $\mathbf{Unif}$ and $\mathbf{Comp}$ extend to the same double factorization systems in the corresponding categories of Eilenberg-Moore algebras $\mathbf{Tych}^{\mathbb{H}^t}$, $\mathbf{Unif}^{\mathbb{H}^u}$ and $\mathbf{Comp}^{\mathbb{H}^c}$. We establish a connection between cartesian double factorization systems and LT-topologies. We provide sufficient conditions for the extension of cartesian double factorization systems to the topos of coalgebras.