nLab concretification

Idea

In a local topos there is a notion of concrete objects. These form a reflective subcategory. The corresponding reflector is the concretification map which universally approximates any object by a concrete object.

Definition

A local topos is a topos equipped with a sharp modality $\sharp$.

Definition

For $X$ any object of the topos, the image projection of the unit $\eta^{\sharp}_X \colon X \to \sharp X$ is the concretification of $X$

(1)$$(X \to \sharp_1 X) \coloneqq \big( X \twoheadrightarrow im(\eta^\sharp_X) \big) \,.$$

Properties

Lemma

Given a morphism $f \colon X \longrightarrow Y$ into a concrete object $Y$, in that $Y \overset{\sim}{\twoheadrightarrow} \sharp_1 Y$, it factors uniquely through the concretification unit $\eta^\sharp_X$, so that we have a natural bijection of hom-sets

$$Y\;\text{concrete} \;\;\;\;\;\;\;\;\;\; \Rightarrow \;\;\;\;\;\;\;\;\;\; Hom\big( \sharp_1 X ,\, Y \big) \xrightarrow[\sim]{\phantom{--} \big( X \twoheadrightarrow \sharp_1 X \big)^\ast \phantom{--}} Hom\big( X ,\, Y \big) \mathrlap{\,.}$$

Proof

This is a special case of the functoriality of image factorization:

Consider the following diagram of given solid arrows, which commutes by naturality of the $\sharp$-unit and where we show the (epi,mono)-factorization of the vertical maps through their images, hence through the concretifications (1):

By construction, the top left morphism is thus an epimorphism and the bottom right is a morphism, as shown. Therefore the orthogonality of the (epi,mono) factorization system implies that there exists a unique dashed lift as shown

References

General

(…)

Of smooth sets

On concretification in the cohesive topos of smooth sets, taking values in diffeological spaces:

• Urs Schreiber, Def. 1.2.60 in: Differential cohomology in a cohesive $\infty$-topos [arXiv:1310.7930]