PlanetPhysics/Topos Axioms

The two axioms that define an elementary topos, or a standard topos , as a special category $$\tau$$ are:


 * {\mathbf i.} $$\tau$$ has finite limits
 * {\mathbf ii.} $$\tau$$ has power objects $$\Omega(A)$$ for objects $$A$$ in $$\tau$$.

To complete the axiomatic definition of topoi, one needs to add the ETAC axioms which allow one to define a category as an interpretation of ETAC. The above axioms imply that any topos has finite colimits, a subobject classifier (such as a Heyting logic algebra), as well as several other properties.

Alternative definitions of topoi have also been proposed, such as:

A topos is a category $$\tau$$ subject to the following axioms:


 * {\mathbf $$\mathbb{T}_1$$}. $$\tau$$ is cartesian closed
 * {\mathbf $$\mathbb{T}_2$$}. $$\tau$$ has a subobject classifier.

One can show that axioms i. and ii.  also imply axioms $$\mathbb{T}_1$$ and $$\mathbb{T}_2$$; one notes that property $$\mathbb{T}_2$$ can also be expressed as the existence of a representable subobject functor.