A pretopos is a coherent category which is both extensive and exact. (See familial regularity and exactness for why extensivity and exactness deserve to be considered together.)
Frequently one is especially interested in pretoposes having additional properties, such as:
A Heyting pretopos, is a pretopos which is also a Heyting category; a Boolean pretopos is a pretopos which is also a Boolean category. These are suitable as frameworks for predicative mathematics, respectively with intuitionistic or classical logic.
A -pretopos is a pretopos which is also a locally cartesian closed category. A -pretopos is automatically a Heyting pretopos.
A -pretopos is a pretopos which has (locally) inductive objects (initial algebras for polynomial endofunctor?s), most famously a natural numbers object.
Mike Shulman: In a non- pretopos, does “polynomial endofunctor” just mean the more naive sense of a functor that “looks like a polynomial” constructed from and ? If that’s so, then it’s not clear to me that a -pretopos which is also a -pretopos is in fact a --pretopos, since the latter requires initial algebras for polynomial endofunctors in the “internal” sense using dependent products. But I don’t know how to define any more general notion of polynomial endofunctor without having dependent products.
A --pretopos is of course a pretopos that has both and ; these are widely studied as frameworks for predicative mathematics in the weaker constructive sense.
A topos is a pretopos that has power objects. A topos is automatically a -pretopos; conversely, a -pretopos is a topos iff it has a subobject classifier, and a Boolean -pretopos is always a topos.
A -topos is of course a topos that has ; it is sufficient that the topos have a natural numbers object (see van den Berg & Moerdijk), so this is often called a topos with NNO. These are widely studied as frameworks for (non-predicative) constructive mathematics, while a Boolean -topos is a framework for ordinary classical mathematics without the axiom of choice.
A boolean -topos with the axiom of choice is a framework for classical mathematics including the axiom of choice; a well-pointed boolean -topos with the axiom of choice is precisely a model of ETCS.
Like any coherent (or Heyting) category, a (Heyting) pretopos has an internal logic. Extensivity and exactness make a Heyting pretopos a very set-like category. One can say imprecisely that it has “all the good first-order properties of a topos”, meaning not that it has those properties that can be expressed in elementary terms (which is false) but that it has those properties that (unlike exponential and power objects) correspond to first-order reasoning in ordinary mathematics. Therefore, pretoposes (especially Heyting, , and/or ones) are related to predicative constructive mathematics in a way similar to how toposes are related to non-predicative constructive mathematics.
A pretopos is necessarily balanced, but while it has coproducts and coequalizers of equivalence relations, it need not have all finite colimits. However, if it has countable pullback-stable unions of subobjects, then any internal binary relation generates an equivalence relation and therefore has a quotient, so we can construct arbitrary coequalizers and thus arbitrary finite colimits. And we can perform an “internal” version of this argument in a -pretopos with a NNO, such as a --pretopos.
A pretopos, being a coherent category, admits a subcanonical Grothendieck topology called the coherent topology. In a pretopos, this topology is generated by finite jointly epimorphic families. Since the canonical topology on a Grothendieck topos consists of all jointly epimorphic families, the coherent topology on a pretopos is sometimes called the precanonical topology.
The codomain fibration of a pretopos is always a stack for its precanonical topology. Being a pretopos is stronger than necessary for this condition to hold in a coherent category, however; see coherent category for the necessary and sufficient conditions.
An infinitary pretopos is an infinitary coherent category which is both infinitary extensive and exact. Giraud’s theorem says that infinitary pretoposes with small generating sets are the same as Grothendieck toposes, and in particular are toposes.