What would be an “arrows-only” defintion of a product in a category?

Suppose we are working in an "arrows-only" definition of a category such as given in Mac Lane's "Categories of the Working Mathematician" (1998) p.279 or on nlab. How can we formulate the definition of a product in such a category? I do not see how to do this without implicitly referring to objects (identity arrows).

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So you want a universal property of the morphism $f \times g : x \times x' \to y \times y'$ of two morphisms $f : x \to y, g : x' \to y'$? – Martin Brandenburg Jul 27 '10 at 16:40
mb, If avoids explicit reference to objects, that may help. -mg – Mike Gass Jul 28 '10 at 13:17

Using the notation of nlab, the following is a fibered product: if $x, y$ are arrows with $t(x) = t(y)$, then their fiber product is the pair of arrows $u, v$ with $t(u) = s(x)$, $t(v) = s(y)$, and $s(u) = s(v)$ such that for any pair of arrows $a, b$ with $s(a) = s(b)$, $t(a) = t(u) \, (= s(x))$ and $t(b) = t(v) \, (= s(y))$ and such that $x \circ a = y \circ b$, there is a unique arrow $c$ having $s(c) = s(a) = s(b)$, $t(c) = s(u) = s(v)$, and $a = u \circ c$, $b = v \circ c$.

To define a plain product, suppose the category has a final object (that is, of course, that there exists an arrow $f$ such that for any arrow $x$ there exists a unique arrow $x'$ with $s(x') = s(x)$ and $t(x') = f$) and replace $x$ and $y$ by $x'$ and $y'$ in the above. If it doesn't have one, of course you can just add one.

Can't get any farther away from identity arrows than that; you need to be able to specify sources and targets to define composition.

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Yes, this may be what I need. Thanks for the help. Mike Gass – Mike Gass Jul 28 '10 at 13:17

Exactly the same as in the objects-and-arrows presentation of category!

Working with the “arrows-only” definition of a category doesn't mean you can't talk about objects, it just means that they're themselves a defined notion. Some constructions can be very nicely given in purely arrows-only language, but for many things — and I think product is one — it seems most natural to define “objects” and then to use them in the statements of further definitions.

You can certainly then unfold the definition to give it in a way that doesn't mention objects. But I think it's important to note that you don't need to do this, and a priori, no obvious big reasons one would want to!

Edit: Reading Martin B's comment, I realise I may well have misunderstood the intent of your question. I'm leaving this answer, though, as I think it's still a point worth making!

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