This definition is somewhat vague by design. Rather than canonize a fixed set of principles, the nLab adopts a ‘pluralist’ point of view which recognizes different needs and foundational assumptions among mathematicians who use set theory. Thus, there are various axes to consider when formulating categorical properties one thinks should satisfy, including
First-order vs. higher-order logic
Impredicative vs. predicative mathematics
‘Variability’ vs. ‘constancy’ (Lawvere)
Classical logic vs. intuitionistic logic
Replacement or collection principles
just to name a few. Quite a bit of axiomatic fine-tuning can enter when one considers the panoply of hypotheses that might appeal to one or another school of intuitionism or constructivism, or various combinatorial or cardinal hypotheses one might attach to ZFC, etc.
The category has many marvelous properties, which make it a common choice for serving as a ‘foundation’ of mathematics. For instance:
So in particular it is locally cartesian closed.
It is locally small.
At least assuming classical logic, these properties suffice to characterize uniquely up to equivalence among all categories; see cocomplete well-pointed topos. Note, however, that the definitions of “locally small” and “(co)complete” presuppose a notion of small and therefore a knowledge of what a set (as opposed to a proper class) is.
As a topos, is also characterized by the fact that
It is usually assumed that satisfies the axiom of choice and has a natural numbers object. In Lawvere’s theory ETCS, which can serve as a foundation for much of mathematics, is asserted to be a well-pointed topos that satisfies the axiom of choice and has a natural numbers object. It follows that it is automatically “locally small” and “complete and cocomplete” relative to the notion of “smallness” defined in terms of itself (actually, this is true for any topos).
Conversely, in constructive mathematics cannot satisfy the axiom of choice (since this implies excluded middle), although constructivists might accept COSHEP (that has enough projectives). In predicative mathematics, is not even a topos, although most predicativists would still agree that it is a pretopos, and predicativists of the constructive school would even agree that it is a locally cartesian closed pretopos.
Above we considered to be the category of all sets, so that in particular itself is a large category. Authors who assume a Grothendieck universe as part of their foundations often define to be the category of small sets (those contained in the universe). One often then writes for the category of large sets, which is the universe enlargement of .
See for instance van Oosten, theorem 2.4
In constructive mathematics, defines an equivalence of with the opposite category of that of complete atomic Heyting algebras. In fact, for any elementary topos , the power object functor defines an equivalence of with the opposite category of that of internal complete atomic Heyting algebras. (This phrase can be interpreted using the internal language of .)
In higher category theory the role of is played for instance by