So, here are some definitions about Natural Numbers Object (nno), that is a key concept in category theory related to Computer Science. They are given in Lambek and Scott (LS) in the following form:
- A (simple) nno is a group of three (N,z,s) where N is an object and z (as zero) and s (as successor) morphisms in a category and the diagram $1\overset{z}{\longrightarrow}N\overset{s}{\longrightarrow}N$ is initial among all the diagrams in the form $1\overset{f}{\longrightarrow}Y\overset{g}{\longrightarrow}Y$
- A parametrized nno is a group of three as the one above where the diagram $X\overset{zX}{\longrightarrow}NX\overset{sX}{\longrightarrow}NX$$X\overset{(z,1_X)}{\longrightarrow}N \times X\overset{s \times 1_X}{\longrightarrow}N \times X$ is initial among all the diagrams in the form $X\overset{f}{\longrightarrow}Y\overset{g}{\longrightarrow}Y$
Barr and Wells propose to say the latter simply a nno because is the only of our interest. It is closely related to a descrption of the primitive recursive functions. A nno makes an object of a category behave as the natural numbers.
With all of these my questions are: while LS demonstrate the existence of a parametrized nno when we are in a cartesian closed category (CCC) with a (simple) nno...in which cases can it be done in the context of a CC (not closed)? Could it be done only with numerals in the sense of arrows $1\longrightarrow N^{k}$ standard (built up in terms of z and s morphisms)?
To ask about weak and strong nno's I would need more definitions, so I let it here for the moment. Thank you in advance and sorry for my english.
Ximo.
