nLab filtrality

Idea

For an object $X$ of a coherent category $C$, consider $\mathcal{B}(X)$, the Boolean algebra of complemented subobjects of $X$, and then $\mathcal{F}(\mathcal{B}(X))$, the set of all non-empty filters of $\mathcal{B}(X)$, partially ordered by reverse inclusion.

There is a canonical (bounded) lattice homomorphism

$$\phi_X: Sub(X) \to \mathcal{F}(\mathcal{B}(X))$$

taking a subobject $A \hookrightarrow X$ to the filter of complemented subobjects over it. Then $X$ is said to be filtral if $\phi_X$ is an isomorphism.

Furthermore, the category $C$ is said to be filtral if each of its objects is covered by a filtral one, that is, for every $Y$ in $C$ there is a regular epimorphism with $X$ filtral (Marra-Reggio 18, Sec. 4).

Examples

• The category of compacta

(Presumably because free compact Hausdorff spaces $\beta X$ are filtral.)

References

• Vincenzo Marra, Luca Reggio, A characterisation of the category of compact Hausdorff spaces, (arXiv:1808.09738)