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From section 22 of Munkres:

Theorem 22.2. Let $p : X \rightarrow Y$ be a quotient map. Let $Z$ be a space and let $g : X \rightarrow Z$ be a map that is constant on each set $p^{-1}({y})$, for $y \in Y$. Then $g$ induces a map $f : Y \rightarrow Z$ such that $f \circ p = g$. The induced map $f$ is continuous if and only if $g$ is continuous; $f$ is a quotient map if and only if $g$ is a quotient map.

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I don't understand the purpose or point of "commutative diagram" theorems like this. The pattern of ones I've seen is usually to: 1) start with two functions that are sides of a triangle, and then 2) say that the third side/function exists, or has some property that one of the others exist.

But then it basically comes down to defining the "missing side" as just the composition of the other two (sometimes with an inverse instead). And... I don't understand the point of these. It almost seems to just be rephrasing that you can compose functions and usual properties (like continuity) follow.

I would understand the motivation if it was unexpected in some way for a missing edge of the diagram to exist, but it usually seems it exists precisely just because it's the composition of the other ones.

Is there some deeper motivation for these theorems?

skymonkey
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    in general the notion of these commutative diagrams comes from the "universal property" in category theory, e.g., https://math.stackexchange.com/questions/1493993/ – SummerAtlas Nov 19 '23 at 19:13
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    Universal properties are a formal way of saying this thing we found is actually a limit or co-limit (depending on the direction of the arrows). I think at least nowadays, people aren't fleshing out unexpected morphisms, but rather just listing various common underlying diagrams. – I Zuka I Nov 19 '23 at 20:05
  • I'm only dimly familiar with the basics of category theory, but I think I see the argument there, thank you – skymonkey Nov 20 '23 at 15:37

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