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I've completely rewritten this question, in accordance with this advice.

As a motivating example, suppose we're working in ETCS. Let $\bar{1}$ denote the canonical singleton set, and assert that by $x \in X$ what we really mean is that $x : \bar{1} \rightarrow X$ is a function.

Now let $f : X \rightarrow Y$ denote a function and suppose $x \in X$. Then $f \circ x \in Y$.

That's all well and good, but there's - not a problem, exactly, but more like an inconvenience.

Suppose the symbols $0,1,2,...$ are defined to be elements of the set $\mathbb{N}$. So for example, we have that $2 \in \mathbb{N}.$ In other words, we have that $2 : \bar{1} \rightarrow \mathbb{N}$. Suppose also that we define $f : \mathbb{R} \rightarrow \mathbb{R}$ by asserting that $f \circ x = x^2$. The inconvenience is that $f \circ 2$ is ill-defined, because $2$ has codomain $\mathbb{N}$, while $f$ has domain $\mathbb{R}$.

The solution, of course, is to view $2$ as an element of $\mathbb{R}$, rather than $\mathbb{N}$. Technically what we've done is chosen a "canonical" injection $\chi : \mathbb{N} \rightarrow \mathbb{R}$. Thus $2$ can be used as an abuse of notation for $\chi \circ 2$, and therefore $f \circ 2$ can be viewed as an abuse of notation for $f \circ \chi \circ 2$.

That was a motivating example, but the inconvenience is more general.

For instance, a metric space does not have open sets. However, a topological space has open sets, and thankfully there is a canonical way of obtaining a topological space from any metric space. Thus we may speak of "the open sets of $(X,d)$" rather than "the open sets of $f(X,d)$," where $f$ is a functor $\mathsf{Met} \rightarrow \mathsf{Top}$.

So what I'm looking for is a systematic way of being rigorous about these kinds of abuses of notation. A reference would be nice.

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It's not the point, but your $f(n)$ is usually denoted $\pi(n)$ and I assume you meant $g(x)=x^2$? - –  Hagen von Eitzen Feb 10 '13 at 8:37
Thanks, I changed the $g$, no reason to change the $f$ as the details of the function aren't of the essence. –  user18921 Feb 10 '13 at 8:39
It looks like what you are asking for is the concept of Universal Property. –  Damien L Feb 10 '13 at 8:39
There is a unique map $\mathbb{N} \to \mathbb{R}$ such that $0$ goes to $0$, $1$ goes to $1$, and $x + y$ goes to $x + y$. That's all you have to say. –  Zhen Lin Feb 10 '13 at 9:17
No, I'm not talking about a Universal Property. I tried rewriting the question in order to make that clear. –  user18921 Feb 10 '13 at 10:28
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3 Answers

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In programming, this is the notion of an implicit conversion; we set up a system of rules for converting between types, and if we have an object of type $X$ that we want to use as type $Y$, we can if one of our conversion rules includes an arrow $X \to Y$. Of course, we better not have two such arrows $X \to Y$!

In other words, as part of the same notational convention we use to give meaning to the strings of symbols "$\mathbb{N}$" and "$\bar{2}$", we've selected a subgraph $G$ of our category. If we have an element $x \in X$ and there is a (unique!) arrow $f:X \to Y$ in $G$, then we're allowed to use $x$ wherever our notation would require an expression of type $Y$, and it is interpreted as $f(x)$ (i.e. as $f \circ x$).

This applies not just to global elements as you talk about but generalized elements as well.

As an aside you may enjoy reading Lattices of Compatibly Embedded Finite Fields which documents how magma handles implicit conversions between finite fields.

Another thing to note is that $\mathbb{N}$ isn't just an object in the topos. Instead, it (along with $\overline{1}$, $z$ and $s$) is a part of some natural number object $\mathcal{N}$. Furthermore, there's no reason $\mathcal{N}$ needs to be a specific natural number object: it could be an indeterminate natural number object!

So we can set things up so that when we define $2 = ssz : \overline{1} \to \mathbb{N}$, we're talking about all natural number objects at once. Unwinding the notation, we're essentially defining $2$ as a functor from the groupoid of natural numer objects in our topos to the arrow category of our topos.

(generalized elements are great!)

More generally, in the back of our heads, all of the named objects and arrows we are interested in are arranged in a sketch (definitions at wikipedia and nLab) which remembers that $\mathbb{N}$ is supposed to be arranged in a natural number object and that $\mathbb{R}$ (along with its attendant structure) is the Dedekind completion of the fraction field of the ring completion of the canonical semiring structure on the natural number object $\mathcal{N}$.

It is actually from this sketch that we select our canonical arrows. And for any model of this sketch that picks out specific objects of our topos for $\mathbb{N}$ and $\mathbb{R}$ and everything else, the model selects the corresponding canonical arrows as well.

If we're ambitious, we can even assume that we're working with an indeterminate model, so that everything we do is actually being carried out in all models at once.

(actually, we need something stronger than a sketch: the construction of $\mathbb{Z}$ from $\mathbb{N}$ can be sketched. But I don't think the condition stating that $1 \to \mathbb{N} \to \mathbb{N}$ is a NNO can be sketched. So there are additional restrictions upon which models are allowed)

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I get the notion of implicit conversion - this is exactly what I was looking for. However the second part about sketches is a little over my head. For instance, suppose I write down the (second order) Peano Axioms (PA2) and the (second order) Complete Ordered Field Axioms (COF2). For every model of the former and every model of the latter, there is a unique mapping that, in most instances, is the most useful "canonical" mapping. How does the notion of a sketch (a category together with a set of limit cones and a set of colimit cones) relate to this? –  user18921 Feb 11 '13 at 21:53
Awesome answer, by the way. –  user18921 Feb 11 '13 at 21:54
@user18921: I'm probably being more complicated than I need to be (this is the first time I've thought this through). The point is, you can arrange all of your hypotheses, conventions, and assumptions into a diagram together with requirements on functors from that diagram to the topos. Limit and colimit cones can express many of those requirements: e.g. if the diagram has an object named "$A \times B$", I want a model to actually map that to a product of the interpretations of $A$ and of $B$, which is expressed as saying the functor must turn the limit cone into an actual limit. –  Hurkyl Feb 12 '13 at 3:29
Im interested in getting things right, making the implicit explicit etc. So i dont mind if its a little complicated. Where can i get more information? I am currently reading Toposes, Triples and Theories, is this a good place to start? –  user18921 Feb 12 '13 at 6:43
Also. Can you give concrete examples of the diagrams/functors/topos you speak of? Where would the theory of arithmetic sit? Where would models of the theory reside? Where would the canonical mapping $\mathbb{N} \rightarrow \mathbb{R}$ reside? –  user18921 Feb 12 '13 at 6:46
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It sounds like the "system of canonical arrows" would be a "subcategory that is a poset (or a preorder?) and whose arrows are all mono in the original category".

However, that concept doesn't sit quite well with category theory in general, because it doesn't respect isomorphisms -- and neither does your motivating example, because there are monoids (??, what is your category, actually?) that are isomorphic to $\mathbb N$, but are not subsets of $\mathbb R$ (or even different subsets of $\mathbb R$). So you can't necessarily transfer your system of canonical arrows to an equivalent category.

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Can you flesh out the –  user18921 Feb 10 '13 at 23:06
the second parahraph? –  user18921 Feb 11 '13 at 1:48
$\mathbb{N}$ and $\mathbb{R}$ aren't really objects of the topos: they're objects of a sketch. The chosen canonical arrow is taken from the sketch, and thus yields an appropriate arrow for every model of the sketch in the topos. –  Hurkyl Feb 11 '13 at 3:04
sketch at nLab and sketch at wikipedia –  Hurkyl Feb 11 '13 at 3:09
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The problem is of course that we'd like to talk about numbers as some kind of "urelements", but doubtful as we require a construction from $\omega$ and $\mathbb N$ via $\mathbb Z$ to $\mathbb R$ and $\mathbb C$ (and to other objects such as $\mathbb C[X]$). Note that there are different ways to perform these constructions, for example, elements of $\mathbb R$ can be equivalence classes of Cauchy sequences modulo zero sequences, or they could be Dedekind cuts. One then shows that this does not matter. And in fact as the construction really does not matter, one might implement such extensions from $A$ to $B$ as follows: You make a set theoretical construction of some $B'$ (say, $B'$ is the set of Dedekind cuts of $A$), note that there is a canonical inclusion map $\iota\colon A\to B'$ and let $B=(B'\setminus\iota(A))\cup A$. If you are worried that we might have $A\cup B'\ne\emptyset$, you might carefully prefer $B=((B'\setminus\iota(A))\times\{A\})$. It is always clear that this can be done, so nobody seems to really care.

This concept usually boils down to the category theory concept of universal properties. For example: Among all unitary ring homomorphisms $\mathbb Z\to Q$ where $Q$ happens to be a field and not just a ring, there exist some special ones: Whenever $\mathbb Z\to Q'$ is another unitary ring homomorphism to a field $Q'$, there is exactly one field homomorphism $Q\to Q'$. One example can be obtained by letting $Q=\mathbb Q$ (and taking the "inclusion" map). By the universal property itself, it follows that any other $\mathbb Z\to Q$ with this property is isomorphic to the specific example $\mathbb Z\to \mathbb Q$, in fact it is canonically isomorphic because there is a unique isomorphism. We may need some set-theoretical construction to show the existence of such a universal guy, but after that we need not care what its intestines look like. In many cases, it turns out that the homomorphism in question (i.e. here $\mathbb Z\to Q$) is injective (as far as this makes sense, i.e. homomorphisms are in fact maps between sets - to be precise we should talk about the forgetful functor to the category of sets here ...). In this case it makes sense to actually view one object as subibject of the other and view the homomorphism as the inclusion. It's because we "know" the universal object only upto canonical isomorphism anyway.

Nevertheless, the matter should be definitely addressed in some way or the other in a textbook.

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