Recall that a topos is a category that behaves likes the category Set of sets.
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).
A natural numbers object in a topos (or any cartesian closed category) with terminal object is
an object in
equipped with
a morphism from the terminal object ;
a morphism (successor);
such that 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.
Let be cartesian closed categories, and suppose has a natural numbers object . If is a left adjoint that preserves the terminal object, then is a natural numbers object in .
The proof is straightforward. It follows for example that the left adjoint part of a geometric morphism between toposes with natural numbers objects preserves the natural numbers object, and also that a Grothendieck quasitopos presented by a site has a natural numbers object, since the reflection functor preserves finite products and the terminal object in particular.
One could generalize the above definition of a natural numbers object to any closed symmetric monoidal category: pointed objects in a symmetric monoidal category are represented by morphisms out of the tensor unit. Thus, a natural numbers object in a closed symmetric monoidal category with tensor unit is
an object in
equipped with
a morphism from the tensor unit ;
a morphism (successor);
such that for every other diagram there is a unique morphism such that
Note that this definition actually makes sense in any category having finite products, such as a pretopos. 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:
In the internal language, commutativity of this diagram says that and .
It may seem odd that only appears in the domain of and not . However, this simplification is inessential, and indeed we are free to add to the domain of as well:
If is a parametrized natural numbers object in a category with finite products, then for any objects , with maps , , there is a unique map such that, in the internal language, and for all .
Let , and define by and by
Then the assumption gives such that
The composite satisfies
Thus, by the uniqueness assumption, we have for all . By a similar argument, we have for all . Therefore, , and hence the composite is the desired .
The functions which are constructible 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 some non-primitive recursive functions (such as the Ackermann function) become definable as well.
In this context an important class is the class of pretoposes with a parametrized NNO - the so called arithmetic pretoposes.
One could generalize the above definition of a parameterised natural numbers object to any symmetric monoidal category: pointed objects in a symmetric monoidal category are represented by morphisms out of the tensor unit. Thus,
In a symmetric monoidal category with tensor unit and tensor product , 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:
where is the right unitor of the symmetric monoidal category for object .
For the moment see at inductive type the section Examples - Natural numbers
In a topos, the natural numbers object is uniquely characterized by the following colimit conditions due to Peter Freyd:
In a topos, a triple is a natural numbers object if and only if
The morphism is an isomorphism;
The diagram
is a coequalizer.
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 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, the existence of a NNO in a topos is equivalent to the existence of free unary systems in , unary systems being objects with an endomorphism in .
Let be a topos and its category of internal unary systems. Then has a NNO precisely if the forgetful functor has a left adjoint.
Let be a left adjoint to the forgetful functor , and let be a terminal object in . Then is a NNO in , by definition of an NNO.
Secondly, it is a theorem 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).
It then is a theorem due to Andreas Blass (1989) that has a NNO precisely if has an object classifier .
A consequence of this, discussed in sec. B4.2 of (Johnstone 2002,I p.431), is that classifying toposes for geometric theories over exist precisely if has a NNO.
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 unary systems or monoids, the intuition being that the free unary systems and 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 many examples of natural numbers objects.
The natural numbers are the natural numbers object in the closed symmetric monoidal category Set.
In classical mathematics, the extended natural numbers are the natural numbers object in the closed symmetric monoidal category of pointed sets , with taking the boolean true to and false to and taking natural numbers to its successor and to . In constructive mathematics, the extended natural numbers and the disjoint union are no longer the same; it is which remains the natural numbers object in .
The underlying abelian group of the polynomial ring is the natural numbers object in the closed symmetric monoidal category Ab, with taking integers to constant polynomials and multiplying polynomials by the indeterminant .
More generally, given a commutative ring , the underlying -module of the polynomial ring is the natural numbers object in the closed symmetric monoidal category RMod, with taking scalars to constant polynomials and multiplying polynomials by the indeterminant .
In any Grothendieck topos the natural numbers object is given by the constant sheaf on the set of ordinary natural numbers, i.e. by the sheafification of the presheaf that is constant on the set .
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:
let be a geometric morphism of toposes. If is a natural numbers object, then its inverse image is a natural numbers object in .
(Johnstone, lemma A.4.1.14). Of course, by the finite colimit characterization, we need only the fact that inverse images preserve finite colimits and the terminal object.
If is a sheaf topos, then there is a unique geometric morphism , the global section geometric morphism, with the inverse image being the locally constant sheaf functor, it follows that
with the evident successor and constant , is the natural nunbers object in .
If is a topos and an object, then the slice topos sits by an etale geometric morphism over
where the inverse image form the product with . Hence for a natural numbers object, the projection is a natural numbers object in .
The initial rig object in a category with finite products, ( with suitable commutative diagrams expressing the rig axioms and initiality, has the structure of a natural numbers object given by the triple .
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 .
For a real numbers object, rather more care is needed; see real numbers object.
F. William Lawvere, Functorial Semantics of Algebraic Theories, Ph.D. thesis Columbia University 1963. (Published with an author’s comment as: TAC Reprint no.5 (2004) pp 1-121. (abstract)
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)
Peter Johnstone, Sketches of an Elephant, Oxford UP 2002.
Natural numbers as a classifying object for dinatural numbers:
Last revised on July 18, 2024 at 13:28:02. See the history of this page for a list of all contributions to it.