PlanetPhysics/Topos Axioms

The two axioms that define an elementary topos , or a standard topos , as a special [[../Cod/|category]] $\tau$ are:

• {\mathbf i.} $\tau$ has finite limits
• {\mathbf ii.} $\tau$ has [[../Power/|power]] [[../TrivialGroupoid/|objects]] $\mathrm{\Omega }(A)$ for objects $A$ in $\tau$.

To complete the axiomatic definition of topoi, one needs to add the [[../Formula/|ETAC axioms]] which allow one to define a category as an interpretation of [[../Formula/|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 ${𝕋}_1$}. $\tau$ is cartesian closed
• {\mathbf ${𝕋}_2$}. $\tau$ has a subobject classifier.

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