nLab
filtrality

Contents

Contents

Idea

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

There is a canonical (bounded) lattice homomorphism

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

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

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

Examples

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

References

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

Last revised on December 24, 2020 at 14:01:41. See the history of this page for a list of all contributions to it.