Trying to understand the definition of a universal property

Question

Here's the definition of a universal property in Wikipedia:

(where $U:D\to C$ is a functor and $X$ is an object in $C$)

A terminal morphism from $U$ to $X$ is a final object in the category $(U\downarrow X)$ of morphisms from $U$ to $X$, i.e. consists of a pair $(A,\Phi)$ where $A$ is an object of $D$ and $\Phi: U(A) \to X$ is a morphism in $C$, such that the following terminal property is satisfied:

• Whenever $Y$ is an object of $D$ and $f: U(Y) \to X$ is a morphism in $C$, then there exists a unique morphism $g: Y \to A$ such that the following diagram commutes:

$\ \ \ \ \ \ \ \ \ $

So I'm trying to "unpack" this definition and figure out what each of the things here "means". E.g. what does it become in the case of a limit, or something.

• A limit is an example of a terminal morphism, right? And a colimit an initial morphism?
• Does $U$ usually represent a diagram? In the case of a limit, does it represent the diagram we want to take the limit of?
• Whatever is $X$? I honestly have no clue here. What's the analog in the case of a limit?
• What does a morphism from $U\to X$ even mean? What does it mean in the case of a limit? I've seen morphisms from a diagram to an object in colimits.
• In the case of a limit, is $(U\downarrow X)$ the category of cones? But how can each cone be a morphism from $U$ to something (I thought it was a morphism from something to $U$)?
• $A$ (or $U(A)$) corresponds to the actual thing we construct, like the source of a limit or the target of a colimit? But what is $\Phi$? In the construction of a limit, there's a morphism from the limit to the diagram, this seems wrong.

My guess is that $X$ represents some sort of "subsetting" of the candidates for the object so you don't have to quantify over everything like you do with cones and the limit. Is that right?

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Edit: So as a short summary -- it turns out $X$ represents (in the case of limits and colimits) the diagram we're trying to take the limit of, while $A$ represents the actual limit object (with its morphism $\Phi$). $U$ is the diagonal functor, because the limit is constructed here as an object in the category of diagrams of shape at most that of $X$.

Answer 1

First of all, that wikipedia definition is not great. You should however go back to it once your wrap your head around all this stuff.

When one first encounters the definition of a universal construction, it's a bit weird. The true way to understand this concept is with examples: just keep looking at examples until it makes sense.

Let's start with a simple example. Let $X, Y$ be sets. Then (one possible way) I can define the disjoint union is $$ X \amalg Y = \{(x, 0), (y, 1) \mid x \in X, y \in Y \} $$ and we can define injection morphisms $$i_X: X \to X \amalg Y \qquad i_X(x) = (x, 0)\\ i_Y: Y \to X \amalg Y \qquad i_Y(y) = (y, 1) $$ Now suppose I have a function $f: X \to Z$ and $g: Y \to Z$. Then I can construct a unique (this is really key here) map $h: X \amalg Y \to Z$ where $$ h(x, 0) = f(x) \text{ and } h(y, 1) = g(y) $$ Why is it unique? Because the definition depends directly on $f$ and $g$. So what this really gives me is this diagram

Given the arrows $f:X \to Z, g: Y \to Z$, we can get a unique arrow $h: X \amalg Y \to Z$. The forced existence of $h$ is indicated by the dashed arrow.

This is an example of a universal construction. Actually, this is an example of a colimit, which I will explain. This happens in so many places in math that there's a name for it, and thats what this universal arrow stuff is about.

So suppose $\mathcal{C}$ and $\mathcal{D}$ are categories with a functor $F: \mathcal{C} \to \mathcal{D}$. A universal morphism from an object $D$ to the functor $F$ is a pair $(C, u: D \to F(C))$ such that, for any $f: D \to F(C')$, the following diagram holds.

So if you have a morphism $f: D \to F(C')$, you automatically get a morphism $h: C \to C'$.

A colimit is an example of this construction. To understand this, you first need to understand the diagonal functor $$ \Delta: \mathcal{C} \to \mathcal{C}^J. $$ Here, $\mathcal{C}^J$ is the functor category of functors $F: J \to C$, with morphisms as natural transformations. This functor $\Delta$ takes each object $C$ to the functor $F_C: J \to C$, where for each $j \in J$ $$ F(j) = C. $$ So it sends it to a constant valued functor.

Now when we speak of a "colimit" in a category $\mathcal{C}$ it's with respect to some functor $F: J \to C$. This is an element of the functor category $\mathcal{C}^J$. So, we define a colimit to be a universal morphism from $F$ to $\Delta$. That is, $$ (\text{Colim }F, u: F \to \Delta(\text{Colim} F). $$ Note that $u$ is natural transformation; as I said earlier $\Delta$ sends objects to functors. So, you get the diagram

However, this isn't very intuitive. A better way to look at this is to realize that if you have a natural transformation $u: F \to \Delta(\text{Colim } F)$, then you have a family of morphisms. How so? For each object $i \in J$, our natural transformation should give us a morphism $$ u_i: F(i) \to \Delta(\text{Colim } F)(i). $$ But $\Delta(\text{Colim } F)$ is a constant valued functor. So this really becomes a family of morphisms $$ u_i: F(i) \to \text{Colim }F. $$ This is usually how people define colimits, but since you asked how they're related to universal constructions, this is how. Anyways, one way to imagine the above diagram is to picture the one below. This isn't exactly precise, but it's a good way to imagine the colimit.

Compare this diagram with the one in the beginning with the coproduct. If you understand that, then you can understand the concept of limits because limits are the same exact story, just with the arrows reversed.