Boolean topos

Definition

A boolean topos is a topos $\mathcal{C}$ where for every object $X$, the subobjects form a Boolean algebra:

$$(\mathrm{Sub}(X),\subseteq )$$

This is equivalent to each of the following:

• $(\mathrm{Sub}(\Omega ),\subseteq )$ is a Boolean algebra
• $\mathrm{true}:1\to \Omega$ has a complement in $\mathrm{Sub}(\Omega )$.
• $\mathcal{C}$ is classical: $[\mathrm{true},\mathrm{false}]:1+1\simeq \Omega$

References

• goldblatt2006-topoi