Topos

Remarks

An equivalent definition asks for finite limits and power objects: for each object $A$, an object $PA$ with a natural bijection between subobjects of $B \times A$ and morphisms $B \to PA$; then $PA = \Omega^A$. Every topos also has finite colimits, and its subobjects of any object form a Heyting algebra. Example: $\mathbf{Set}$ is a topos: it has finite limits, the exponential $B^A$ is the set of functions $A \to B$, and $\set{0, 1}$ classifies subsets. Example: for a group $G$, the category of $G$-sets is a topos with $\Omega = \set{0, 1}$ and trivial action. Non-example: the category of real vector spaces fails clause 3. Its terminal object is $0$, so the pullback of $\top$ along $\chi \colon X \to \Omega$ is $\ker \chi$. If $\chi \colon \R \to \Omega$ classifies the zero subspace, then $\chi$ is injective, so $2\chi \ne \chi$, yet $\ker(2\chi) = 0$ too, and uniqueness fails. Non-example: the category $\mathbf{Top}$ of topological spaces has finite limits but is not Cartesian closed, so it fails clause 2.