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What (exactly) is the difference between defining a mathematical object by it's axioms and by a universal construction ? Please take my 3 opinions into consideration, as they also contain more specific (implicitly posed) questions and an answer telling me where my perception of universal constructions conflicts with the what they really are, is infinitely more enlightening than just slamming a definition of them, which would leave me wondering if (and how) my views/intuition are not mainstream.

(Please also note, that I yet don't have any background in category theory whatsoever, so the only place I came in contact with universal constructions was during an intro course in abstract algebra. So I know how to characterize the field of fractions for example with a universal construction.)

  • To me axioms and universal constructions seems almost identical in the sense that they only specify what the property such an object would have to fulfil (please bare with me, that I may be totally wrong, since I know so little about universal property), without usually saying anything about how to construct such objects and whether they are unique (to revert to the field of fractions example: this is to be understood in the sense that there doesn't exist an absolute field of fractions; different integral domains give rise to different fields of fractions although for one integral domain there can only be one (up to isomorphism); although I don't know if the uniqueness).
  • These definitions (via axioms/via universal constructions) only differ, as far as I can see, in how they're stated: Axioms can be formulated (if we are very precise) in first-order logic, which is a fairly flexible setting to make all kinds of statements, whereas for universal properties one has to describe the object of choice by setting it in relation to other, previously described objects, and using maps (i.e. diagrams) with certain properties, between those objects, to define the object of choice.
  • (This would imply that one has to use axioms at one point to describe some abstract objects, since universal constructions rely on previously define objects, so one can't define everything via universal constructions. Or do you know of some attempts to formalize foundations using something like universal constructions ?)
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I would say that universal constructions are distinguished by absolutely requiring diagrams. I.e. they are arrow theoretic in nature. So we are not merely constructing the field of fractions, just as important is the map which embeds the integral domain in its field of fractions. As for the uniqueness concern, note that the various objects (e.g. the fields) given by a universal construction are all isomorphic. Finally, the object was already there before the universal construction was provided, which is a point that especially severs the purported link with axioms defining objects. – peoplepower Feb 28 '13 at 16:29
Can you be more specific? I don't use axioms and universal constructions to construct the same kinds of things. What's an example of a mathematical object that you construct both ways? – Qiaochu Yuan Feb 28 '13 at 18:08
@QiaochuYuan That's a good question: I don't know, since after thinking about a bit I came to the conclusion that I know too little about universal constructions. But it inspired me to ask a separate question, asking exactly this. – temo Mar 1 '13 at 11:29
I suggest that you learn more about universal constructions. – Qiaochu Yuan Mar 1 '13 at 19:17

I will try to tell you how I think of universal objects, and why in my opinion they are quite different by objects defined via axioms. But first, be aware that I have no background in logic, so I might use the word "axiom" in a too naive way. Also, what follows is just my personal (philosophical) interpretation.

So, what is the typical shape of a universal property? It always goes like this: an object is universal for a certain property when the following (informal!) condition is satisfied:

whenever an "extraneous" object attempts to imitate the property, then it acts in a manner that must "factor" through the universal object in a unique way.

(This usually translates in suitable commutative diagrams, as what I called "object" above is in fact a "thing + some arrows").

This implies that objects defined by a universal property are always unique up to a unique isomorphism. For instance, if you start with an integral domain, you can define its field of fractions via a universal property (and then check it exists).

But consider this example: start with a set $X$, and suppose you want a topology on it. A topology is something defined by axioms (the axioms of the "open sets"). In this case, the object you get (after you have chosen a collection of subsets satisfying the axioms) cannot be called universal in any sense, because you might have many different topologies on $X$. Whatever you chose as open sets, your choice is not better than another, so... universal with respect to what? The difference is, then, that you do not have the notion of factor through.

Let me explain this better (but still informally). If you fix a topology on a set $X$, and you ask yourself: "is this topology universal?", what does it mean? Here is the important point, the reason why the notion factor through cannot exist: a topology does not DO anything, it is just a static definition of something. Instead, a universal object actually DOES something (what is encoded in the property): it comes with at least one arrow (for instance, the morphism $A\to \textrm{Frac }A$) and we have a concrete way to compare it to other objects acting like it. But this is not the case for things defined by axioms: they are "inert", static.

So I think the main issue regarding universal objects is really that they are

\begin{equation} \textrm{unique up to a unique isomorphism, and} \end{equation}

\begin{equation} \textrm{dynamic, while axioms transmit static nature.} \end{equation}

Also, there is another (aside) issue: the universal object often tells us something about non-universal objects (those trying to "imitate" it). It has this power thanks to its dynamical nature. Let me give you this example. Suppose you have a fine moduli space (a universal solution for a given moduli problem). Then properties of this object thanslate into properties of the other objects that the moduli problem intended to classify. You really have these properties for free! they pass to all families of non-universal objects through the universal family.

Summing up: in my opinion, universal objects often encode more profound information, because they are not static. And this is because a universal property is something which is formulated in the powerful language of categories, where all that matters is how things move (arrows!).

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