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Hometopos axioms
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topos axioms
Definition 0.1.
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:
Definition 0.2.
A topos is a category $\tau$ subject to the following axioms:

$\mathbb{T}_{1}$. $\tau$ is cartesian closed

$\mathbb{T}_{2}$. $\tau$ has a subobject classifier.
One can readily 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.
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