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I'm looking at this and stuck immediately first paragraph, page 24:

This can be made precise by giving a better definition of product, in terms of a universal property. Given two sets $M$ and $N$, a product is a set $P$, along with maps $µ : P → M$ and $ν : P → N$, such that for any set $P′$ with maps $µ′ : P′ → M$ and $ν′ : P′ → N$, these maps must factor uniquely through $P$:

What got me was the last line "these maps must factor uniquely through $P$". I realize this is precise language, but what is meant by factoring uniquely through $P$? Then just a few lines later it says

This definition agrees with the traditional definition, with one twist: there isn’t just a single product; but any two products come with a unique isomorphism between them. In other words, the product is unique up to unique isomorphism.

If we have two products $(a_1,b_1)$ and $(a_2,b_2)$, what is meant by they come with a unique isomorphism between them? Then he talks about the diagram communting. Again, not sure what that means. Any help appreciated.

147pm
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1 Answers1

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these maps must factor uniquely through $P$

He means there exists a unique $h : P' \to P$ such that $\mu' = \mu \circ h$ and $\nu' = \nu \circ h$.

what is meant by ... a unique isomorphism between them?

If we have two different products $(P, \mu, \nu)$ and $(P', \mu', \nu')$ of $M$ and $N$, there exists a unique isomorphism $h : P' \to P$ such that $\mu' = \mu \circ h$ and $\nu' = \nu \circ h$.

Then he talks about diagrams commuting

A diagram drawn on a page is a "commuting diagram" if, for any two object points $A$ and $B$ in the graph and any two arrow-paths from $A$ to $B$, the arrow paths compose to give the same morphism.

In other words, there's at most one way to get from $A$ to $B$ through the diagram.

Mark Saving
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  • First, I'd like to hear an explanation of why he's saying "factor uniquely through $P$". How does this terminology relate to your explanation of the compositions $\mu' = \mu \circ h$ and $\nu' = \nu \circ h$? IOW, with three words he's said what you did in more words and I can't follow his words. – 147pm Aug 18 '21 at 02:51
  • @147pm $\mu'$ factors through the morphism $\mu$ because it can be written as $\mu \circ h$; similarly, $\nu'$ factors through $\nu$ because it can be written as $\nu \circ h$. "Factoring" is used because there is an analogy between $\circ$ and multiplication. We want $\mu'/\nu'$ to factor through $\mu/\nu$ using the same $h : P' \to P$, and we want this $h$ to be unique. – Mark Saving Aug 18 '21 at 03:14
  • I know of "factors" as primes and factoring as taking a number apart down to the product of its primes. So where could I learn about this usage of the idea of factoring? This analogy between the composition operator $\circ$ and multiplication (operator) is unknown to me. Suggest a text, treatment that explains this expanded idea of factoring for this beginner? – 147pm Aug 18 '21 at 03:25
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    @147pm There isn't anything profound here. If you can write $a = b \cdot c$, then $b$ is a factor of $a$, and therefore $a$ "factors through" $b$. Similarly, if we can write $\mu' = \mu \circ h$, then $\mu$ is a "factor" of $\mu'$, and hence we can say $\mu'$ "factors through" $\mu$. There isn't any notion of getting the "prime factorization" of a morphism in a category (at least as far as I know). If you want a general introduction to category theory, MacLane's Category Theory for the Working Mathematician is pretty decent. – Mark Saving Aug 18 '21 at 03:56
  • Just found this: https://math.stackexchange.com/questions/21932/what-does-it-mean-to-say-a-map-factors-through-a-set So I'm not alone in my confusion.... Thanks. It just stymied me for a while. Also here https://en.wikipedia.org/wiki/List_of_mathematical_jargon – 147pm Aug 18 '21 at 04:15