nLab topos of pointed objects

Idea

The category of pointed objects $1\backslash \mathcal{E}$ of a topos $\mathcal{E}$ has zero objects hence can be the degenerate topos at best. By altering the notion of morphism it is nevertheless possible to obtain a topos $^\bullet\mathcal{E}$ with objects $1\to X$, called the topos of pointed objects.

Definition

Let $\mathcal{E}$ be a topos. The topos $^\bullet\mathcal{E}$ of pointed objects has objects the morphisms $1\to X$ and morphisms pullback squares:

$$\begin{array}{cccc}1& \to & X \\ \downarrow & & \downarrow \\ 1 & \to & Y \end{array}$$

Properties

• Foremost, $^\bullet\mathcal{E}$ is a topos (cf. Freyd 1987).

Related entries

• pointed object

• point

• fibration of points

Reference

• Peter Freyd, Choice and Well-ordering , APAL 35 (1987) pp.149-166. (section 5)