A natural numbers object (NNO) in a topos is an object that behaves in that topos like the set of natural numbers does in Set; thus it provides a formulation of the “axiom of infinity” in structural set theory (such as ETCS). The definition is due to William Lawvere (1963).
an object in
for every other diagram
there is a unique morphism such that
All this may be summed up by saying that a natural numbers object is an initial algebra for the endofunctor (the functor underlying the “maybe monad”). Equivalently, it is an initial algebra for the endo-profunctor .
By the universal property, the natural numbers object is unique up to isomorphism.
Note that this definition actually makes sense in any category having finite products. However, if is not cartesian closed, then it is better to explicitly assume a stronger version of this definition “with parameters” (which follows automatically when is cartesian closed, such as when is a topos). What this amounts to is demanding that not only be a natural numbers object (in the above, unparametrized sense) in , but that also, for each object , this is preserved by the cofree coalgebra functor into the Kleisli category of the comonad (which may be thought of as the category of maps parametrized by ). (Put another way, the finite product structure of gives rise to a canonical self-indexing, and we are demanding the existence of an (unparametrized) NNO within this indexed category, rather than just within the base ).
To be explicit:
In a category with finite products, a parametrized natural numbers object is an object together with maps , such that given any objects , and maps , , there is a unique map making the following diagram commute:
The functions which are constructable out of the structure of a category with finite products and such a “parametrized NNO” are precisely the primitive recursive ones. Specifically, the unique structure-preserving functor from the free such category into Set yields a bijection between and the actual natural numbers, as well as surjections from onto the primitive recursive functions of arity for each finite . With cartesian closure, however, this identification no longer holds, since non-primitive recursive functions (such as the Ackermann function) become definable as well.
In a topos, a triple is a natural numbers object if and only if
The necessity of the first condition holds in any category with binary coproducts and a terminal object, and the necessity of the second holds in any category whatsoever.
For a category with binary coproducts and 1, the natural numbers object can be equivalently described as an initial algebra structure for the endofunctor defined on . Then condition 1 is a special case of Lambek's theorem, that the algebra structure map of an initial algebra is an isomorphism.
As for condition 2, given such that , the claim is that factors as
for some unique , in fact for . Uniqueness is clear since , being a retraction for , is epic. On the other hand, substituting either or for in the diagram
this diagram commutes, so that by the uniqueness clause in the universal property for .
Here we just give an outline, referring to (Johnstone), section D.5.1, for full details. Let be an object satisfying the two colimit conditions of Freyd. First one shows (see the lemma 1 below) that has no nontrivial -subalgebras. Next, let be any -algebra, and let be the intersection of all -subalgebras of . One shows that is an (-algebra) isomorphism. Thus we have an -algebra map . If is any -algebra map, then the equalizer of and is an -subalgebra of , and therefore itself, which means .
Let be the endofunctor . If satisfies Freyd’s colimit conditions, then any -subalgebra of is the entirety of .
Following (Johnstone), we may as well show that the smallest -subalgebra of (the internal intersection of all -subalgebras) is all of . Let be the union of the relation and its opposite, so that is a symmetric relation. Working in the Mitchell-Bénabou language, one may check directly that the following formula is satisfied:
Let us say a term of type is -closed if the formula
is satisfied. Now define a relation on by the subobject
Observe that is an equivalence relation that contains and therefore . It therefore contains the kernel pair of the coequalizer of and ; since this coequalizer is by assumption , the kernel pair is all of . Also observe that since is -closed by definition, it is -closed as well, and we now conclude
so that, putting , we conclude that , i.e., that is all of .
A slightly alternative proof of sufficiency uses the theory of well-founded coalgebras, as given here. If is a fixpoint of the functor , regarded as an -coalgebra, then the internal union of well-founded subcoalgebras of is a natural numbers object . Then the subobject can also be regarded as a subalgebra; by the lemma, it is all of . Thus is a natural numbers object.
In topos theory the existence of a natural numbers object (NNO) has a couple of far-reaching consequences.
Firstly, it is a theorem is due to C. J. Mikkelsen that the existence of a NNO in a topos is equivalent to the existence of free monoids in :
Let be a topos and its category of internal monoids. Then has a NNO precisely if the forgetful functor has a left adjoint.
For a proof see Johnstone (1977,p.190).
So from a different perspective, in a topos the existence of free objects over various gadgets like e.g. algebraic theories or geometric theories (often) hinge on the existence of free monoids, the intuition being that the free monoids permit to construct a free model syntactically by providing for the (syntactic) building blocks needed for this process.
Notice that algebraic theories can nevertheless have free algebras even if the ambient topos lacks a NNO. This may happen for algebraic theories that have the property that the free algebra on a finite set of generators has a finite carrier e.g. in the topos of finite sets free graphic monoids exist.
There are interesting cases in which such sheaf toposes contain objects that look like they ought to be natural numbers objects but do not satisfy the above axioms: for instance some of the models described at Models for Smooth Infinitesimal Analysis are sheaf toposes that contain besides the standard natural number object a larger object of smooth natural numbers that has generalized elements which are “infinite natural numbers” in the sense of nonstandard analysis.
Natural number objects are preserved by inverse images:
with the evident successor and constant , is the natural nunbers object in .
Given a natural numbers object in a pretopos, we can construct an integers object as follows. Let be the kernel pair of the addition map , and let be the product projections. We define to be the coequalizer of the congruence . A similar construction yields a rational numbers object .
Jean Bénabou, Some Remarks on Free Monoids in a Topos , pp.20-29 in LNM 1488 Springer Heidelberg 1991.
Andreas Blass, Classifying topoi and the axiom of infinity , Algebra Universalis 26 (1989) pp.341-345.
Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint Minneola 2014, chap. 6)