nLab
Set

Context

Category theory

Foundations

Categories of categories

The category of sets

Definition

SetSet is the (or a) category with sets as objects and functions between sets as morphisms.

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 SetSet should satisfy, including

  • First-order vs. higher-order logic

  • Impredicative vs. predicative mathematics

  • ‘Variability’ vs. ‘constancy’ (Lawvere)

  • Classical logic vs. intuitionistic logic

  • Choice principles

  • Replacement or collection principles

  • Smallness structures

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.

Properties

Characterization

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

At least assuming classical logic, these properties suffice to characterize SetSet 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, SetSet is also characterized by the fact that

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

Size

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

Opposite category and Boolean algebras

Proposition

The power set-functor

𝒫:SetBool op \mathcal{P} \;\colon\; Set \to Bool^{op}

is a faithful functor which in its (eso+full, faithful) factorization induces an equivalence of categories between SetSet and the opposite category of that of complete atomic Boolean algebras.

See for instance van Oosten, theorem 2.4

Remark

Restricted to FinSet this equivalence restricts to an equivalence with finite Boolean algebras. See at Stone duality for more on this.

Remark

In constructive mathematics, 𝒫\mathcal{P} defines an equivalence of Set\Set with the opposite category of that of complete atomic Heyting algebras. In fact, for any elementary topos \mathcal{E}, the power object functor defines an equivalence of \mathcal{E} with the opposite category of that of internal complete atomic Heyting algebras. (This phrase can be interpreted using the internal language of \mathcal{E}.)

A variant of SetSet where functions are generalized to relations is Rel.

In higher category theory the role of SetSet is played for instance by

References

category: category

Revised on April 11, 2014 08:04:42 by Ingo Blechschmidt (137.250.162.16)