nLab natural numbers object

Idea

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 $\mathbb{N}$ 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).

Definition

In a topos or cartesian closed category

A natural numbers object in a topos (or any cartesian closed category) $E$ with terminal object $1$ is

• an object $\mathbb{N}$ in $E$

• equipped with

  • a morphism $z :1 \to \mathbb{N}$ from the terminal object $1$;

  • a morphism $s : \mathbb{N} \to \mathbb{N}$ (successor);

• such that for every other diagram $1 \stackrel{q}{\to}A \stackrel{f}{\to} A$ there is a unique morphism $u : \mathbb{N} \to A$ such that

All this may be summed up by saying that a natural numbers object is an initial algebra for the endofunctor $X \mapsto 1 + X$ (the functor underlying the “maybe monad”). Equivalently, it is an initial algebra for the endo-profunctor $Hom_E(1,=) \times Hom_E(-,=)$.

By the universal property, the natural numbers object is unique up to isomorphism.

Proposition

Let $C, D$ be cartesian closed categories, and suppose $D$ has a natural numbers object $(\mathbb{N}, z: 1 \to \mathbb{N}, s: \mathbb{N} \to \mathbb{N})$. If $L: D \to C$ is a left adjoint that preserves the terminal object, then $(L \mathbb{N}, L z, L s)$ is a natural numbers object in $C$.

The proof is straightforward. It follows for example that the left adjoint part $f^\ast$ of a geometric morphism $f^\ast \dashv f_\ast: E \to F$ between toposes with natural numbers objects preserves the natural numbers object, and also that a Grothendieck quasitopos $Q$ presented by a site $(C, J)$ has a natural numbers object, since the reflection functor $L: Set^{C^{op}} \to Q$ preserves finite products and the terminal object in particular.

In a general category with finite products

Note that this definition actually makes sense in any category $E$ having finite products, such as a pretopos. However, if $E$ is not cartesian closed, then it is better to explicitly assume a stronger version of this definition “with parameters” (which follows automatically when $E$ is cartesian closed, such as when $E$ is a topos). What this amounts to is demanding that $(\mathbb{N}, z, s)$ not only be a natural numbers object (in the above, unparametrized sense) in $E$, but that also, for each object $A$, this is preserved by the cofree coalgebra functor into the Kleisli category of the comonad $X \mapsto A \times X$ (which may be thought of as the category of maps parametrized by $A$). (Put another way, the finite product structure of $E$ 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 $E$).

To be explicit:

Definition

In a category with finite products, a parametrized natural numbers object is an object $N$ together with maps $z: 1 \to N$, $s: N \to N$ such that given any objects $A$, $X$ and maps $f: A \to X$, $g: X \to X$, there is a unique map $\phi_{f, g}: A \times N \to X$ making the following diagram commute:

$$\array{ A & \stackrel{\langle 1_A, z \circ !\rangle}{\to} & A \times N & \stackrel{1_A \times s}{\leftarrow} & A \times N \\ & \mathllap{f} \searrow & \downarrow \mathrlap{\phi_{f, g}} & & \downarrow \mathrlap{\phi_{f, g}} \\ & & X & \underset{g}{\leftarrow} & X }$$

In the internal language, commutativity of this diagram says that $\phi_{f,g}(a,z) = f(a)$ and $\phi_{f,g}(a,s(n)) = g(\phi_{f,g}(a,n))$.

It may seem odd that $A$ only appears in the domain of $f$ and not $g$. However, this simplification is inessential, and indeed we are free to add $N$ to the domain of $g$ as well:

Proposition

If $(N,z,s)$ is a parametrized natural numbers object in a category with finite products, then for any objects $A$, $X$ with maps $f:A\to X$, $g:A\times N\times X \to X$, there is a unique map $\phi_{f,g} : A\times N \to X$ such that, in the internal language, $\phi_{f,g}(a,z) = f(a)$ and $\phi_{f,g}(a,s(n)) = g(a,n,\phi_{f,g}(a,n))$ for all $n:N$.

Proof

Let $X' = A\times N\times X$, and define $f':A\to X'$ by $f'(a) = (a,z,f(a))$ and $g':X'\to X'$ by

$$g'(a,n,x) = (a,n,g(a,n,x)).$$

Then the assumption gives $\phi': A\times N \to X'$ such that

$$\phi'(a,z) = f'(a) = (a,z,f(a)), \qquad \phi'(a,s(n)) = g'(\phi'(a,n)).$$

The composite $A \times N \xrightarrow{\phi'} X' \xrightarrow{\pi_1} A$ satisfies

$$\pi_1 \phi'(a,z) = \pi_1(a,z,f(a)) = a, \qquad \pi_1\phi'(a,s(n)) = \pi_1 g'(\phi'(a,n)) = \pi_1 \phi'(a,n).$$

Thus, by the uniqueness assumption, we have $\pi_1 \phi'(a,n) = a$ for all $a,n$. By a similar argument, we have $\pi_2 \phi'(a,n) = n$ for all $a,n$. Therefore, $\phi'(a,s(n)) = (a,n,g(a,n,\phi'(a,n)))$, and hence the composite $A\times N \xrightarrow{\phi'} X' \xrightarrow{\pi_3} X$ is the desired $\phi_{f,g}$.

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 $F$ into Set yields a bijection between $Hom_F(1, \mathbb{N})$ and the actual natural numbers, as well as surjections from $Hom_F(\mathbb{N}^m, \mathbb{N})$ onto the primitive recursive functions of arity $m$ for each finite $m$. 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.

In type theory

For the moment see at inductive type the section Examples - Natural numbers

Finite colimit characterization

In a topos, the natural numbers object $\mathbb{N}$ is uniquely characterized by the following colimit conditions due to Peter Freyd:

Theorem

In a topos, a triple $(\mathbb{N}, 0: 1 \to \mathbb{N}, s: \mathbb{N} \to \mathbb{N})$ is a natural numbers object if and only if

1. The morphism $(0, s): 1 + \mathbb{N} \to \mathbb{N}$ is an isomorphism;

2. The diagram

   $$\mathbb{N} \stackrel{\overset{s}{\to}}{\underset{1}{\to}} \mathbb{N} \to 1$$

   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.

Proof of theorem : necessity

For a category $C$ with binary coproducts and 1, the natural numbers object can be equivalently described as an initial algebra structure $(0, s): 1 + \mathbb{N} \to \mathbb{N}$ for the endofunctor $F(c) = 1 + c$ defined on $C$. 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 $f: \mathbb{N} \to X$ such that $f = f \circ s$, the claim is that $f$ factors as

$$\mathbb{N} \overset{!}{\to} 1 \overset{x}{\to} X$$

for some unique $x$, in fact for $x = f(0)$. Uniqueness is clear since $!: \mathbb{N} \to 1$, being a retraction for $0: 1 \to \mathbb{N}$, is epic. On the other hand, substituting either $f$ or $f(0) \circ !$ for $g$ in the diagram

$$\array{ 1 & \overset{0}{\to} & \mathbb{N} & \overset{s}{\to} & \mathbb{N} \\ & ^\mathllap{f(0)} \searrow & \downarrow ^g & & \downarrow ^g \\ & & X & \underset{1_X}{\to} & X }$$

this diagram commutes, so that $f = f(0) \circ !$ by the uniqueness clause in the universal property for $\mathbb{N}$.

Proof of theorem : sufficiency

Here we just give an outline, referring to (Johnstone), section D.5.1, for full details. Let $N$ be an object satisfying the two colimit conditions of Freyd. First one shows (see the lemma below) that $N$ has no nontrivial $F$-subalgebras. Next, let $A$ be any $F$-algebra, and let $i: B \to N \times A$ be the intersection of all $F$-subalgebras of $N \times A$. One shows that $\pi_1 \circ i: B \to N$ is an ($F$-algebra) isomorphism. Thus we have an $F$-algebra map $f: N \to A$. If $g: N \to A$ is any $F$-algebra map, then the equalizer of $f$ and $g$ is an $F$-subalgebra of $N$, and therefore $N$ itself, which means $f = g$.

Lemma

Let $F$ be the endofunctor $F(X) = 1+X$. If $N$ satisfies Freyd’s colimit conditions, then any $F$-subalgebra of $N$ is the entirety of $N$.

Proof

Following (Johnstone), we may as well show that the smallest $F$-subalgebra $N'$ of $N$ (the internal intersection of all $F$-subalgebras) is all of $N$. Let $S \hookrightarrow N \times N$ be the union of the relation $R = \langle 1, s \rangle: N \to N \times N$ and its opposite, so that $S$ is a symmetric relation. Working in the Mitchell-Bénabou language, one may check directly that the following formula is satisfied:

$$\forall x, y: N (\langle x, y \rangle \in S) \Rightarrow ((x \in N') \Leftrightarrow (y \in N'))$$

Let us say a term $w$ of type $P N$ is $S$-closed if the formula

$$\forall x, y: N (\langle x, y \rangle \in S) \Rightarrow ((x \in [w]) \Leftrightarrow (y \in [w]))$$

is satisfied. Now define a relation $T$ on $N$ by the subobject

$$\{\langle x, y \rangle \in N \times N: \forall w: P N (w \; is \; S closed) \Rightarrow ((x \in [w]) \Leftrightarrow (y \in [w])).$$

Observe that $T$ is an equivalence relation that contains $S$ and therefore $R$. It therefore contains the kernel pair of the coequalizer of $1$ and $s$; since this coequalizer is by assumption $N \to 1$, the kernel pair is all of $N \times N$. Also observe that since $N'$ is $S$-closed by definition, it is $T$-closed as well, and we now conclude

$$\forall x, y \in N: (x \in N') \Leftrightarrow (y \in N')$$

so that, putting $y = 0: 1 \to N'$, we conclude that $x \in N \Rightarrow x \in N'$, i.e., that $N'$ is all of $N$.

Remark

A slightly alternative proof of sufficiency uses the theory of well-founded coalgebras, as given here. If $N$ is a fixpoint of the functor $F(X) = 1+X$, regarded as an $F$-coalgebra, then the internal union of well-founded subcoalgebras of $N$ is a natural numbers object $\mathbb{N}$. Then the subobject $\mathbb{N} \hookrightarrow N$ can also be regarded as a subalgebra; by the lemma, it is all of $N$. Thus $N$ is a natural numbers object.

Free constructions in a topos

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 $\mathcal{S}$ is equivalent to the existence of free unary systems in $\mathcal{S}$, unary systems being objects with an endomorphism in $\mathcal{S}$.

Proposition

Let $\mathcal{S}$ be a topos and $\mathbf{unsys}(\mathcal{S})$ its category of internal unary systems. Then $\mathcal{S}$ has a NNO precisely if the forgetful functor $U:\mathbf{unsys}(\mathcal{S})\to \mathcal{S}$ has a left adjoint.

Let $F:\mathcal{S}\to\mathbf{unsys}(\mathcal{S})$ be a left adjoint to the forgetful functor $U$, and let $1$ be a terminal object in $\mathcal{S}$. Then $F(1)$ is a NNO in $\mathcal{S}$, by definition of an NNO.

Secondly, it is a theorem due to C. J. Mikkelsen that the existence of a NNO in a topos $\mathcal{S}$ is equivalent to the existence of free monoids in $\mathcal{S}$:

Proposition

Let $\mathcal{S}$ be a topos and $\mathbf{mon}(\mathcal{S})$ its category of internal monoids. Then $\mathcal{S}$ has a NNO precisely if the forgetful functor $U:\mathbf{mon}(\mathcal{S})\to \mathcal{S}$ has a left adjoint.

For a proof see Johnstone (1977,p.190).

It then is a theorem due to Andreas Blass (1989) that $\mathcal{S}$ has a NNO precisely if $\mathcal{S}$ has an object classifier $\mathcal{S}[\mathbb{O}]$.

A consequence of this, discussed in sec. B4.2 of (Johnstone 2002,I p.431), is that classifying toposes for geometric theories over $\mathcal{S}$ exist precisely if $\mathcal{S}$ 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 $FinSet$ of finite sets free graphic monoids exist.

Examples

In a sheaf topos

In any Grothendieck topos $E = Sh(C)$ 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 $C^{op} \to Set$ that is constant on the set $\mathbb{N}$.

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.

Properties

Transfer along inverse images

Proposition

Natural number objects are preserved by inverse images:

let $f = (f^* \dashv f_*) : \mathcal{E} \underoverset{f_*}{f^*}{\leftrightarrows} \mathcal{F}$ be a geometric morphism of toposes. If $\mathbb{N} \in \mathcal{F}$ is a natural numbers object, then its inverse image $f^* \mathbb{N}$ is a natural numbers object in $\mathcal{E}$.

(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.

Example

If $\mathcal{E}$ is a sheaf topos, then there is a unique geometric morphism $(\Delta \dashv \Gamma): \mathcal{E} \underoverset{\Gamma}{\Delta}{\leftrightarrows} Set$, the global section geometric morphism, with the inverse image being the locally constant sheaf functor, it follows that

$$\Delta(\mathbb{N}) \cong \Delta\left(\sum_{n: \mathbb{N}} 1\right) \cong \sum_{n: N} \Delta 1 \cong \sum_{n: \mathbb{N}} 1,$$

with the evident successor and constant $0$, is the natural nunbers object in $\mathcal{E}$.

Example

If $\mathcal{E}$ is a topos and $X \in \mathcal{E}$ an object, then the slice topos sits by an etale geometric morphism over $\mathcal{E}$

$$\mathcal{E}_{/X} \stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}} \mathcal{E} \,,$$

where the inverse image form the product with $X$. Hence for $\mathbb{N} \in \mathcal{E}$ a natural numbers object, the projection $(X \times \mathbb{N} \to X)$ is a natural numbers object in $\mathcal{E}_{/X}$.

Relation to rig objects

The initial rig object in a category with finite products, ($\mathbb{N},0:1\rightarrow\mathbb{N},1:1\rightarrow\mathbb{N},+:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N},\times:\mathbb{N}\times\mathbb{N}\rightarrow\mathbb{N})$ with suitable commutative diagrams expressing the rig axioms and initiality, has the structure of a natural numbers object given by the triple $(\mathbb{N}, 0:1\rightarrow\mathbb{N}, x+1:\mathbb{N}\rightarrow\mathbb{N})$.

Relation to object of integers

Given a natural numbers object $\mathbb{N}$ in a pretopos, we can construct an integers object as follows. Let $a, b\colon E \to \mathbb{N} \times \mathbb{N}$ be the kernel pair of the addition map ${+}\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$, and let $\pi_1, \pi_2\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ be the product projections. We define $\mathbb{Z}$ to be the coequalizer of the congruence $(\pi_1 \circ a, \pi_2 \circ b), (\pi_1 \circ b, \pi_2 \circ a)\colon E \to \mathbb{N} \times \mathbb{N}$. A similar construction yields a rational numbers object $\mathbb{Q}$.

For a real numbers object, rather more care is needed; see real numbers object.

Related concepts

• natural numbers

• natural numbers type

• Peano arithmetic

• free monoid

• arithmetic pretopos

• integers object

References

• 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.

In elementary $(\infty,1)$-toposes:

• Nima Rasekh, Every Elementary Higher Topos has a Natural Number Object, Theory and Applications of Categories 37 13 (2021) 337-377. (arXiv:1809.01734, tac:37-13)