Welcome to MathOverflow

Question

1

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).

flag	asked Jul 27 2010 at 16:22 Mike Gass 33●3

	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 2010 at 16:40
	mb, If avoids explicit reference to objects, that may help. -mg – Mike Gass Jul 28 2010 at 13:17

Ryan Reich · Accepted Answer · 2010-07-27 16:47:42Z UTC

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.

Peter LeFanu Lumsdaine · Answer 2 · 2010-07-27 18:29:26Z UTC

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!

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

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.