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).