Elementary Topos

Definition

An elementary topos is a category $\mathcal{C}$ that is:

• $\mathcal{C}$ is finitely complete
• $\mathcal{C}$ is finitely cocomplete
• $\mathcal{C}$ has exponentiation
• $\mathcal{C}$ has a subobject classifier

Properties

If $\mathcal{C}$ is a topos then any full subcategory is also a topos.

Examples

Set is a topos in the presence of a Power Set axiom. FinSet is also a topos. FinOrd is naturally a topos since it is a .