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As I study category theory, I'm finding the use of universal mapping properties in defining basic concepts to be both simple and clever. Yet, the idea seems non-obvious enough that I expect quite a bit of mathematics had been done before the discovery of the technique.

What is the chronologically earliest abstract definition given by a universal mapping property?

Note that this question is not intended to be restricted to category theory.

Thank you!

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There are some notes on the subject in Mac Lane: books.google.com/… –  user2734 May 2 '10 at 12:06
Thank you for the citation! To clarify, I'm asking not about when the idea of a UMP was itself generalized, but rather the first instance of the (I expect) many cases from which the generalization was eventually drawn. –  Matthew Willis May 2 '10 at 12:16
Does the least element of a poset count? –  Steven Gubkin May 2 '10 at 15:54
I think this is hard to answer: with hindsight, we see UMPs in many classical constructions. Whether, at what point and to what extent a given construction was viewed in terms of UMPs seems much more nebulous and surjective. For instance, the approach to combinatorial group theory via generators and relations seems to be predicated on the UMP of free groups. But is that how the turn of the century founders of this subject viewed things? Probably not -- I doubt homomorphisms were even that important to them. For the next 50 years, the point of view steadily became more functorial... –  Pete L. Clark May 2 '10 at 16:23
"surjective" |-> "subjective". A Freudian slip only a mathematician could make. :) –  Pete L. Clark May 4 '10 at 22:09

2 Answers 2

up vote 8 down vote accepted

I very much agree with Pete's comment that we will find incipient instances of the universal mapping property among many classical constructions in mathematics, even if the original users of those concepts would not describe the idea in those terms. Indeed, I believe that these instances will stretch back through the whole of mathematics, and for this reason, there may be no definitive answer to the question.

But let me anyway introduce a very early classical construction that we might agree has the hallmarks of a universal mapping property. To my mind, one of the fundamental essences of the UMP definitions is that they specify an object $U$ that relates to given objects $A,B,\ldots$ in a certain way, such that any other object $V$ relating to $A,B,\ldots$ in that same way then stands in a certain relation to $U$. This is the sense in which $U$ is universal with that property, and the particular details of the relations determine the nature of the universal property.

My proposal is to consider the ancient idea of commensurability of line segments, appearing in Euclid's Elements and used earlier. Commensurability is of course intimately connected with the concept of greatest common divisor arising in the Euclidean algorithm, appearing in Books VII and X of Euclid's Elements.

Specifically, two line segments $K$ and $L$ are comensurable when there is another line segment $R$ such that $K$ and $L$ are common multiples of $R$. The ancients knew not only that there was a largest such common measure $R$, but also that this largest common measure has a universal property: if $S$ is any other common measure of $K$ and $L$, then $R$ is a multiple of $S$. Thus, the largest common measure of commensurable line segments is characterized by a universal property, known in antiquity.

This fact is of course related to the fact, also known to the ancients, that the greatest common divisor $d$ of two natural numbers $a$ and $b$ is not only the greatest common divisor of $a$ and $b$, but has the universal property that any other common divisor of $a$ and $b$ is a divisor of $d$. Thus, the gcd of two integers is characterized by a universal property, also known in antiquity.

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This is, of course, a case of "sup" (or "inf", depending on your preference) in a certain lattice. –  Pete L. Clark May 5 '10 at 2:08
Very very Nice! –  some guy on the street May 5 '10 at 2:20
Pete, yes, but I don't think Euclid (or anyone at that time) had that general concept. But they did have the particular universal concept of greatest common divisor and largest common measure. –  Joel David Hamkins May 5 '10 at 2:22
@Pete: it's arguably much much older –  some guy on the street May 5 '10 at 2:22
I believe this is what wikipedia calls "original research" (OR) –  Victor Protsak May 5 '10 at 8:11

I'm betting on Supremum.

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What does that mean? There is no need to conceive of a supremum by a universal mapping property, whatever that might be, to think about and work with it. –  KConrad May 4 '10 at 18:56
The idea --- and it's pretty standard in introducing Categories --- is that any partial order is a (free) category with at most one morphism between any two objects, called $\leq$. The supremum of a set $S$, if it exists, is an object $X$ with $s\leq X$ (a specified structure map $s\to X$) for $s\in S$ such that for any given object $Y$ and maps $s\leq Y$ there exists a unique map $X\leq Y$ such that $s\leq X\leq Y$ is $s\leq Y$. In this poset case, existence and uniqueness both follow from the at-most-one property. –  some guy on the street May 4 '10 at 20:31
eep! I should say uniqueness, existence would mean $X$ is a supremum, and in any poset suprema might not exist... this is why colimits and limits "are" limits, btw. –  some guy on the street May 4 '10 at 20:32
OK some guy, but how does this relate to the question seeking a concept that was abstractly defined chronologically earliest by a universal mapping property? I'd think an answer to the original question would be an example with a lot more going on than a supremum on a poset, i.e., an example where the universal mapping property point of view provides an essential way to understand what the concept "really means". –  KConrad May 4 '10 at 21:14
I'm really not clear on what you're getting at. True, the categorical way of thinking about posets (or even the abstract notion of a poset) come much later than the definition of a category or a universal object; all I mean by what I've said, is that the Supremum as originally defined is literally an object with a specific universal mapping property, despite the fact Cauchy/Dirichlet/Dedekind/Riemann weren't thinking in those terms when whoever first wrote the definition. BTW, if the Questioner accepts my answer, I'll delete it; this question shouldn't have a "final answer". –  some guy on the street May 4 '10 at 23:39

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