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 ?)