Cellular generation revisited
Cellular generation, which generalises cofibrant generation, is an important categorical smallness condition on a class of morphisms. A general challenge is to determine whether a given class of morphisms $\mathcal{M}$ is cellularly generated, in which $\mathcal{M}$-effective squares are often useful. These are commuting squares consisting of morphisms in $\mathcal{M}$, so that the induced morphism from the pushout square is also in $\mathcal{M}$. When we drop the requirement that the vertical morphisms in the square are in $\mathcal{M}$ we obtain the weaker notion of $\mathcal{M}$-quasieffective square. We prove that, in a locally presentable category, $\mathcal{M}$ is cellularly generated if and only if $\mathcal{M}$ is almost everywhere quasieffective. The latter is a set-theoretic condition stating that for almost every partial elementary set-theoretic subuniverse $\mathfrak{N}$, we have that restricting any morphism in $\mathcal{M}$ to $\mathfrak{N}$ yields an $\mathcal{M}$-quasieffective square. For locally finitely presentable categories this yields an additional categorical characterisation in terms of filtrations of $\mathcal{M}$-quasieffective squares. If we additionally assume that $\mathcal{M}$ is continuous (i.e., the corresponding wide subcategory is closed under directed colimits) then we obtain a stronger characterisation of cellular generation in terms of accessibility of the category of $\mathcal{M}$-effective squares. This improves on a theorem by Lieberman, Vasey, and the third author.