nLab
Sandbox/Set

Definition

$\mathrm{Set}$ is the (or a) category with sets as objects and functions between sets as morphisms.

Properties

This category has many marvelous properties, which make it a common choice for serving as a ‘foundation’ of mathematics. For instance:

• It is a well-pointed topos,

  So in particular it is locally cartesian closed.

• It is locally small.

• It is complete and cocomplete, and therefore $\mathrm{\infty }$-extensive (as is any cocomplete topos).

At least assuming classical logic, these properties suffice to characterize $\mathrm{Set}$ 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, $\mathrm{Set}$ is also characterized by the fact that

• $\mathrm{Set}$ is the terminal object in the category of Grothendieck toposes and geometric morphisms. The terminal morphism $\Gamma :E\to \mathrm{Set}$ from any other topos $E$ is the global sections functor.

It is usually assumed that $\mathrm{Set}$ 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, $\mathrm{Set}$ 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, $Set$ in constructive mathematics cannot satisfy the axiom of choice (since this implies excluded middle), although constructivists might accept COSHEP (that $\mathrm{Set}$ has enough projectives). In predicative mathematics, $Set$ 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.

Remarks

• Above we considered $\mathrm{Set}$ to be the category of all sets, so that in particular $\mathrm{Set}$ itself is a large category. Authors who assume a Grothendieck universe as part of their foundations often define $\mathrm{Set}$ to be the category of small sets (those contained in the universe). One often then writes $\mathrm{SET}$ for the category of large sets, which is the universe enlargement of $\mathrm{Set}$.

Meta-Testing

The monomorphisms in Set are exactly the Injections.

The category is complete but not cocomplete (that is wrong, but for testing).