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).
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.
Note that this definition actually makes sense in any category $E$ having finite products. 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:
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:
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 $F$ 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 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.
For the moment see at inductive type the section Examples - Natural numbers
In a topos, the natural numbers object $\mathbb{N}$ is uniquely characterized by the following colimit conditions due to Peter Freyd:
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
The morphism $(0, s): 1 + \mathbb{N} \to \mathbb{N}$ 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 $C$ 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
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
this diagram commutes, so that $f = f(0) \circ !$ by the uniqueness clause in the universal property for $\mathbb{N}$.
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 1 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 $f = g$.
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 $N$.
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:
Let us say a term $w$ of type $P N$ is $S$-closed if the formula
is satisfied. Now define a relation $T$ on $N$ by the subobject
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
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 $N$.
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.
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 $\mathcal{S}$ is equivalent to the existence of free monoids in $\mathcal{S}$:
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).
Secondly, 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 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 $FinSet$ of finite sets free graphic monoids exist.
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.
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, being left adjoints, preserve finite colimits.
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
with the evident successor and constant $0$, is the natural nunbers object in $\mathcal{E}$.
If $\mathcal{E}$ is a topos and $X \in \mathcal{E}$ an object, then the slice topos sits by an etale geometric morphism over $\mathcal{E}$
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 $\mathcal{E}_{/X}$.
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 $(\pi_1 \circ a, \pi_2 \circ b), (\pi_2 \circ a, \pi_1 \circ b)\colon E \to \mathbb{N} \times \mathbb{N}$. A similar construction yields a rational numbers object $\mathbb{Q}$.
For a real numbers object, rather more care is needed; see real numbers object.
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.