Double factorization systems in equivariant topology and topos theory

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 $(\mathtt{\text{ExEpi}},\mathtt{\text{Bim}},\mathtt{\text{ExMono}})$ in the categories $\mathbf{T}\mathbf{y}\mathbf{c}\mathbf{h}$, $\mathbf{U}\mathbf{n}\mathbf{i}\mathbf{f}$ and $\mathbf{C}\mathbf{o}\mathbf{m}\mathbf{p}$ extend to the same double factorization systems in the corresponding categories of Eilenberg-Moore algebras ${\mathbf{T}\mathbf{y}\mathbf{c}\mathbf{h}}^{{\mathbb{H}}^t}$, ${\mathbf{U}\mathbf{n}\mathbf{i}\mathbf{f}}^{{\mathbb{H}}^u}$ and ${\mathbf{C}\mathbf{o}\mathbf{m}\mathbf{p}}^{{\mathbb{H}}^c}$. 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.