I don't really understand why there are so many universal properties in math or why they all need to be highlighted.

For example, I'm studying some Algebra right now. I have found three universal properties that are all basically saying the same thing, although the details are different:

Universal property 1: If $R, S$ are rings and $\theta: R \to S$ is a ring map, then for each $s \in S$, there is a unique map $\hat{\theta_{s}} : R[x] \to S$ such that if $i: R \to R[x]$ is the inclusion map, we get $\theta = \hat{\theta_{s}} \circ i$.

Universal property 2: If $D$ is an integral domain and $F$ is a field with $\phi : D \to F$ a one-to-one ring map, then there is a unique map $\hat{\phi} : Q(D) \to F$ such that $\hat{\phi} \circ \pi = \phi$, where $\pi : D \to Q(D)$ sends $a$ to $\frac{a}{1}$ ($Q(D)$ the fractional field of $D$).

Universal property two was used to prove that in a field of characteristic $0$, the rationals are a subfield, and in a field of characteristic $p$ ($p$ prime), $\mathbb{Z}_{p}$ is a subfield.

Universal property 3: If $R, S$ are rings, $\phi: R \to S$ is a ring map, and $I$ is an ideal such that $I \subseteq \text{ker}(\phi)$, then there is a unique map $\overline{\phi} : R/I \to S$ such that $\phi = \overline{\phi} \circ i$ where $i: R \to R/I$ maps $a$ to $\overline{a}$.

It is really hard for me to keep track of all of these universal properties, especially when they are all usually referenced by the single name "universal property". Is there a point to all of these universal properties?

Honestly, I don't even know if my question is clear, or how to ask a better question in this regard.

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    $\begingroup$ Its a universal property of category theory. Sometimes the truth is bland. $\endgroup$ – copper.hat Oct 12 '14 at 20:09
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    $\begingroup$ Maybe you want category theory for a broader, unifying view? Or do I mistake your question? $\endgroup$ – mvw Oct 12 '14 at 20:10
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    $\begingroup$ I second this question, I'm also being introduced to universal properties but without the context of category theory and it is very confusing/hard to grasp what a universal property even is. $\endgroup$ – Matthew Levy Oct 12 '14 at 20:12
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    $\begingroup$ What you refer to as Universal Property 1, 2 and 3 is usually referred to as Universal property of polynomial rings, universal property of fraction fields, universal property of quotients respectively. It will be difficult for you to remember these universal properties if you just label them as universal properties 1, 2 and 3. The motivation for this comes from category theory. $\endgroup$ – Rankeya Oct 12 '14 at 20:16
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    $\begingroup$ @user46944: I wouldn't worry about universal properties too much if this is your first time studying abstract algebra. I thought of $R/I$ as a collection of cosets when I first learned about rings. But with time, as I read more, universal properties and constructions made sense. $\endgroup$ – Rankeya Oct 12 '14 at 20:27

A universal property of some object $A$ tells you something about the functor $\hom(A,-)$ (or $\hom(-,A)$, but this is just dual). For example, $\hom(R[x],S) \cong |S| \times \hom(R,S)$ is the universal property of the polynomial ring (where $|S|$ denotes the underlying set of $S$). Conversely, we may consider the functor which takes a commutative ring $S$ to $|S| \times \hom(R,S)$ and say that it is a representable functor, represented by $R[x]$. This can be also interpreted as the statement that $R[x]$ is the free commutative $R$-algebra on one generator, see free object for categorical generalizations. Roughly, representing a functor means to give a universal example of, or to classify, the things which the functor describes. This happens all the time in mathematics. Conversely, whenever you have an object $A$, it is interesting to ask what it classifies, i.e. to look at $\hom(A,-)$ and give a more concise description of it. The Yoneda Lemma tells you that all information of $A$ is already encoded in $\hom(A,-)$.

Also, one of the main insights of category theory is that it is very useful to work with morphisms instead of elements. For example, what the quotient ring $R/I$ does for us is not really that we can compute with cosets, but rather that it is the universal solution to the problem to enlarge $R$ somehow to kill (the elements of) $I$. In other words, $\hom(R/I,S) \cong \{f \in \hom(R,S) : f|_I = 0\}$. This makes things like $(R/I)/(J/I) = R/J$ for $I \subseteq J \subseteq R$ really trivial: On the left side, we first kill $I$ and then $J$, which is the same as to kill $J$ directly, which happens on the right hand side. No element calculations are necessary. (On math.stackexchange, I have posted lots of examples for this kind of reasoning.) Quotient rings, quotient vector spaces, quotient spaces etc. are all special cases of colimits.

The universal property of the field of fractions states that $\hom(Q(D),F) \cong \hom(D,F)$, where on the right hand side we mean injective homomorphisms. This says that $Q(-)$ is left adjoint to the forgetful functor from fields to integral domains (in each case with injective homomorphisms as morphisms). This is a special case of localizations. Adjunctions are ubiquitous in modern mathematics. They allow us to "approximate" objects of a category by objects of another category.

So far I have only mentioned some patterns of universal properties, but not answered the actual "philosophical" question "Why are there so many universal properties in math?" in the title. Well first of all, they are useful, as explained above. Also notice that many objects of interest turn out to be quotients of universal objects. For example, every finitely generated $k$-algebra is a quotient of a polynomial algebra $k[x_1,\dotsc,x_n]$. Thus, if we understand this polynomial algebra and its properties, we may gain some information about all finitely generated $k$-algebras. A specific example of this type is Hilbert's Basis Theorem, which implies that finitely generated algebras over fields are noetherian. Perhaps one can say: Universal objects are there because we have invented them in order to study all objects.

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    $\begingroup$ I'd add that 95% of "universal properties" in mathematics can be interpreted as adjoint functors, and the remaining ones are generally Kan extensions. There is a chapter in Categories for the Working Mathematician called "All Concepts Are Kan Extensions," which gives some idea of how fanatically one can pursue this viewpoint. $\endgroup$ – Slade Oct 12 '14 at 22:59
  • $\begingroup$ I don't know any category theory yet, so I can't say this answer opened my eyes and made me understand. But maybe when I do learn category theory, I can come back to this for its insight. You must have given some good information to receive 5 upvotes. $\endgroup$ – layman Oct 13 '14 at 2:07
  • $\begingroup$ @user46944: Sorry for that. Feel free to ask some specific questions here in the comments. Perhaps there is a way to reformulate all this without any "technical" language from category theory. $\endgroup$ – Martin Brandenburg Oct 13 '14 at 7:43

Any time $X$ satisfies a universal property, it means that the inventor of $X$ chose well (rather than arbitrarily) how to define $X$.

So I guess the literal answer would be "Because people are telling you about good mathematics".

  • $\begingroup$ I don't mean that universality is the only good property an $X$ can have (for example $X$ could be creative, relevant to key questions, or be useful in industrial applications). But universality is a good property to have. $\endgroup$ – isomorphismes Apr 30 '15 at 16:13

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