Editing Natural numbers object (section)

== Properties ==

* Every NNO is an initial object of the category of [[Diagram (category theory)|diagrams]] of the form
::<math displaystyle="block">1 \xrightarrow{~ \quad q \quad ~} A \xrightarrow{~ \quad f \quad ~} A</math>

* If a cartesian closed category has weak NNOs, then every [[slice category|slice]] of it also has a weak NNO.
* NNOs can be used for [[non-standard model]]s of [[type theory]] in a way analogous to non-standard models of analysis. Such categories (or topoi) tend to have "infinitely many" non-standard natural numbers.{{clarify|date=December 2017}} (Like always, there are simple ways to get non-standard NNOs; for example, if ''z'' = ''s z'', in which case the category or topos '''E''' is trivial.)
* [[Peter Freyd|Freyd]] showed that ''z'' and ''s'' form a [[coproduct]] diagram for NNOs; also, !<sub>''N''</sub> : ''N'' → 1 is a [[coequalizer]] of ''s'' and 1<sub>''N''</sub>, ''i.e.'', every pair of global elements of ''N'' are connected by means of ''s''; furthermore, this pair of facts characterize all NNOs.