Unique isomorphisms and universal properties

Question

Having not studied category theory, I'm trying to piece together without using category theory what is meant when an algebraic structure is said to possess a universal property or be unique up to unique isomorphism.

I don't understand Qiaochu Yuan's answer since it uses the notion of a universal property, which I'm having trouble finding a definition for that isn't worded in terms of category theory and/or the phrase "unique up to unique isomorphism." For example, on the Universal Property Wikipedia page, the only formal definition that seems not to be worded in terms of category theory says "Universal properties define objects uniquely up to a unique isomorphism."

Here's something I do understand: In my abstract algebra class, we said that

given an integral domain $D$, a field of quotients of $D$ is a field $Q$ such that there is an embedding $i:D\hookrightarrow Q$ where $\forall q\in Q$, $\exists a,b\in D: q = i(a)(i(b))^{-1}$.

We showed that we can construct a field of quotients of an arbitrary integral domain $D$ using equivalence classes of $D\times D\setminus\{0\}$. We then proved the following theorem:

Let $D$ be an integral domain and let $Q$ be a field of quotients of $D$ with the embedding $i:D\hookrightarrow Q$. If $F$ is a field such that there is an embedding $f:D\hookrightarrow F$, then there is a unique embedding $\overline f:Q \hookrightarrow F$ such that $\overline f\circ i = f$.

Is this a universal property? If so, is it because of the following corollary?

Let $Q$ be a field of quotients of $D$ with the embedding $i:D\hookrightarrow Q$, and let $Q'$ be another field of quotients of $D$ with the embedding $i':D\hookrightarrow Q'$. Then there is a unique isomorphism $\overline{i'}: Q\hookrightarrow Q'$ such that $\overline{i'}\circ i = i'$.

Is this when we would say $Q$ is unique up to unique isomorphism?

Answer 1

Products in many categories are said to possess a universal mapping property- formally, a product of two objects (say $A$ and $B$), is another object (say $P$) along with two arrows: $p_1:P \to A$ and $p_2:P \to B$ such that if $C$ is any other object (in our category), along with any pair of arrows, $f_1:C \to A, f_2: C \to B$, then there exists a UNIQUE arrow $\phi:C \to P$ such that:

$p_1\circ\phi = f_1\\p_2\circ\phi = f_2.$

The "baby example" of this construction is the cartesian product in the category $\mathbf{Set}$. Indeed, if we set, for two sets $A,B$:

$P = \{(a,b): a \in A, b \in B\}$, and take:

$p_1((a,b)) = a\\p_2((a,b)) = b,$

Then given any pair of maps $f_1:C \to A, f_2:C \to B$ for an abitrary set $C$, we can let:

$\phi(c) = (f_1(c),f_2(c))$, for any $c \in C$, and it's clear we have:

$(p_1\circ\phi)(c) = p_1(\phi(c)) = p_1(f_1(c),f_2(c)) = f_1(c)$ for ALL $c \in C$, and similarly:

$(p_2\circ \phi)(c) = f_2(c)$.

So certainly the map $\phi$ exists (for "this product"). On the other hand, if we have a map $\psi: C \to A \times B$ such that:

$p_1 \circ \psi = f_1$ and $p_2\circ \psi = f_2$, then from $p_1\circ \psi = f_1$ we know that:

$\psi(c) = (f_1(c),-)$, and similarly we know that $\psi(c) = (-,f_2(c))$, that is: $\psi$ has to be $\phi$, so $\phi$ is unique.

Now it should be pretty clear that this isn't "the only product" we can make: for example, we could take:

$P' = B \times A$ with $p_1:P' \to A$ given by $p_1'((b,a)) = a$, and similarly for $p_2'$.

In this case, we have two unique maps $\phi: P \to P'$, and $\phi': P' \to P$, and since $\phi((a,b)) = (b,a)$ and $\phi'((b,a)) = (a,b)$ fulfil the requirements, they must be the maps the universal property describes. Here, it's clear these are isomorphisms (in $\mathbf{Set}$, isomorphisms are just bijections), as these maps are inverses of each other.

The universal property invoked here is often paraphrased as: "the product map (what we are calling $\phi$, and is often written $f_1 \times f_2$) factors through the projections (what we are calling $p_1,p_2$)".

Now, it's "intuitively clearer" to regard the cartesian product of sets to be "the product" of $A$ and $B$. Note how the emphasis is on the elements in each ordered pair. In the construction above, the emphasis is on the mapping $\phi$, and the projection maps $p_1,p_2$, we don't even "look inside" the sets to see who lives there.

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There is a similar factorization happening in your field of fractions example: any embedding $f$ of $D$ in any field $F$ factors through the embedding $i: D \to Q$. This captures (in this case) the sense of $Q$ being the "smallest" field we can insert $D$ into.