nLab profinite completion of a space

(Beware there are two possible interpretations of this term. One is handled in the entry on profinite completion of a group, being profinite completion of the homotopy type of a space. The entry here treats another more purely topological concept.)

Idea

The profinite completion functor (on topological spaces) is the left adjoint to the inclusion of the category of profinite spaces into that of all topological spaces. It is particularly useful when applied to discrete topological spaces (i.e. really: sets!).

Definition:

Let $X$ be a topological space. A profinite completion of $X$ is a profinite space, $\hat{X}$, together with a continuous map, $\eta_X : X \to \hat{X}$, such that, if given any profinite space, $Y$, and a continuous map, $g : X \to Y$, there is a unique continuous map $\psi : \hat{X}\to Y$ with $\psi \eta_X = g$

Related concepts

• localic reflection

References

• Abolfazl Tarizadeh, On the category of profinite spaces as a reflective subcategory (arXiv:1207.5963)