Topos

Idea

A topos is a class of category that behaves like sheaves of sets on a site.

Kinds of Topoi

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 .

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