This is probably well-known to many people, but you don't need quotients or pushouts to construct Z from N. Coproducts suffice. If (N,0,$) is an nno, let <$ | 0> : N+1 -> N be the initial (+1)-algebra and # : N -> N+1 be its inverse (Lambek's theorem). Now (Z, z, s, p) is an integers object, where Z = N + 1 + N z : 1 -> Z "picks out the 1" s : Z -> Z is assembled from # : N -> N+1 and <0 | $> : 1+N -> N p : Z -> Z is assembled from <$ | 0> and the inverse of <0 | $> (composing # with the swap). Obviously, p=s^-1. The universal property of the nno ensures that there is a unique map from Z to any pointed automorphism, making the required diagram commute. Best, Andrew On Sun, Jun 28, 2026 at 3:10 PM David Roberts via Categories < categories-list@categories.org.au> wrote:
Hi Steve,
I think my example answer
I'd like to hear from someone from pre-1980 who can say "I recall discussing this definition at the time and it was so obvious then that no one wrote it down"
....if not outright "here's a CT paper from 1981 or earlier that literally wrote this property down and called it an axiomatisation of the internal integers object under weak assumptions", is fairly self-explanatory. That it was published verbatim in a paper in 1985 means it wasn't so obvious as to not be worth writing down, in that setting.
Please note that "constructed in the usual way" cannot be just asserted to exist when you have eg a parameterised NNO in a finite product category. So any argument that says "if you have constructed the integers, then of course they are the free group on one generator" is not helpful, no. That is like saying "well, in ZFC you can construct \omega, the first infinite von Neumann ordinal, and this satisfies the NNO axioms in a topos. Does this help you in trying to find who first wrote down the definition in print?" ^_^
(I believe that one can in fact construct Z in a finite product category with parameterised NNO |N by very very carefully splitting an idempotent on |N^2, so I have to be careful in my phrasing and not say "Z constructed in the usual way from |N doesn't exist in a finite product category", at least the quotient construction *does* in fact exist, but the method is very different from just applying the internal logic and it working out ok.)
I will check out the references, thanks for pointing them out.
All the best, David -- Dr David Roberts http://ncatlab.org/nlab/show/David+Roberts
Adjunct Associate Lecturer School of Mathematical Sciences Adelaide University — Tirkangkaku SA 5005 AUSTRALIA
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------------------------------ *From:* Steven Vickers <s.j.vickers.1@bham.ac.uk> *Sent:* Sunday, 28 June 2026 9:15 PM *To:* David Roberts via Categories <categories-list@categories.org.au> *Cc:* David Roberts <droberts.65537@gmail.com>; David Roberts < david.roberts@adelaide.edu.au> *Subject:* Re: Did you see this characterisation of Z in the 70s?
* CAUTION: External email. Only click on links or open attachments from trusted senders. * ------------------------------ David - I'm not quite sure what kind of answer you're looking for, so apologies if I tell you stuff you already know.
The 1978 Johnstone-Wraith paper, "Algebraic theories in toposes", showed that in a topos with nno any finitary algebraic theory (eg that of a single sort with automorphism) has free algebras: so the general results for toposes were certainly known by 1980.
You asked specifically about Z as the free automorphism algebra on one generator. I think it would have been obvious that Z "constructed from N in the usual way" (using constructions not generally available in CCCs, either as free group over N as additive monoid, or more concretely) has that property.
Was that ever actually suggested? I don't remember. But I used free algebras a lot in my thesis (completed 1979) and learned much of the ideas from Manes's 1976 book "Algebraic theories". It might be worth checking that if you haven't already. (I don't have a copy to hand.)
Steve. ------------------------------ *From:* David Roberts via Categories <categories-list@categories.org.au> *Sent:* Sunday, June 28, 2026 11:31 AM *To:* categories-list@categories.org.au <categories-list@categories.org.au
*Cc:* David Roberts <droberts.65537@gmail.com>; David Roberts < david.roberts@adelaide.edu.au> *Subject:* [categories] Did you see this characterisation of Z in the 70s?
*CAUTION:* This email originated from outside the organisation. Do not click links or open attachments unless you recognise the sender and know the content is safe. Hi all,
I'm requiring the brains trust of the older generation to let me know if the following definition (as exactly like this as possible) was in the air or written down pre-1980. I will assume C is a cartesian closed category, but perhaps it was stated for C a topos at the time.
Definition: Let C be a cartesian closed category. An _integers object_ is a pointed object 0 : 1 --> Z with the data s : Z ---> Z and p : Z ---> Z such that p = s^{-1}, and is the initial among pointed objects with an automorphism.
I have found this definition published in a computer science paper:
- Turner, D.A. (1985). Miranda: A non-strict functional language with polymorphic types. In: Jouannaud, JP. (eds) Functional Programming Languages and Computer Architecture. FPCA 1985. Lecture Notes in Computer Science, vol 201. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-15975-4_26 <https://urldefense.com/v3/__https://doi.org/10.1007/3-540-15975-4_26__;!!CF15FET90Tp8!ERcFYxo6DHAuVzm806FMznwb-kSb88GcUvUB426xkUteo3_BLS6kx_qAm-_OIiC496FRq3qbLInT41lpNg65Y3EP99ulsiMzD2BiMmM$>
and something very similar in papers in 1982 and 1980 coming from the ADJ school of computer science (heavily influenced by category theory). There it's stated more in terms of giving the definition of a algebraic data type in some inductive means. But I can't find it in the category theory literature proper. I don't know if it came from some old logic/set theory paper, or was from CT, or was just first written down by computer scientists, but it was so incredibly obvious no one wrote it down in mathematics.
Please note that "well the definition is obvious" _now_ is not really sufficient, I'd like to hear from someone from pre-1980 who can say "I recall discussing this definition at the time and it was so obvious then that no one wrote it down" if this is the type of answer I'm going to get. Moreover, since I'm interested in the case where I can't assume I have colimits, any early mention _constructing_ an object of integers from the NNO in a topos in "the usual way" also isn't useful (what colimits does a CCC have for free? Certainly not quotients or pushouts)
More recently (in the last decade) this characteristion has been used in MLTT/Homotopy Type Theory, and this being written on the nLab in a purely categorical way is what alerted me to the characterisation. I'm hoping to get a reference to add there, aside from the computer science ones which strike me as very likely influenced by something a category theorist could have written down.
Thanks in advance, David
-- Dr David Roberts http://ncatlab.org/nlab/show/David+Roberts <https://urldefense.com/v3/__http://ncatlab.org/nlab/show/David*Roberts__;Kw!!CF15FET90Tp8!ERcFYxo6DHAuVzm806FMznwb-kSb88GcUvUB426xkUteo3_BLS6kx_qAm-_OIiC496FRq3qbLInT41lpNg65Y3EP99ulsiMz8Zv52GA$>
Adjunct Associate Lecturer School of Mathematical Sciences Adelaide University — Tirkangkaku SA 5005 AUSTRALIA
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