During my time in grad school, I have often wondered what I would talk about if I were invited to give a talk to an undergraduate audience. Like the math club here, or where I went for undergrad. I've generally had a good time with categories, and am pretty convinced something like the following would work as a fun talk. Perhaps my definition of fun needs some work.

For starters, let's just think about sets for a while - and I'll have finite sets in mind, but mostly that shouldn't matter. If $S$ and $T$ are sets, the Cartesian product is the set $S\times T=\{(s,t)|s\in S,t\in T\}$. What makes this new set useful? Why should we care about it? I suppose it depends on who you ask, but since I've got categories in mind, I want to think about functions (=maps) between sets. The product comes with 2 maps, I'll call them $\pi_S$ and $\pi_T$. The map $\pi_S:S\times T\to S$ is defined by $(s,t)\mapsto s$, and the map $\pi_T$ is similar. Now, if I have a set $A$, and a map $f:A\to S\times T$, then I can compose $f$ with either $\pi_S$ or $\pi_T$ and obtain maps $\pi_S\circ f:A\to S$ and $\pi_T\circ f:A\to T$. That is, I can tease out the component pieces of the map $f$. On the other hand, if I started with a set $B$, and maps $g:B\to S$ and $h:B\to T$, I can define a map $(g,h):B\to S\times T$ via $b\mapsto (g(b),h(b))$.

So when I think about the product of two sets, I notice that maps to the product are essentially the same as individual maps to the sets I started with. This is how I want to think about the product in other contexts. If I start with two things (objects in some category) $x$ and $y$, I want their product $x\times y$ to be another thing (object) with the property that whenever I have component maps (a map to $x$ and a map to $y$), then I get a map to the product. Additionally, my product will come with two maps, $\pi_x$ and $\pi_y$, which I think of as the projections, following the example above.

What I'd like to do now is pick a fun context (category) and figure out what the product is in that category. In a category I'm supposed to have some collection of objects, and some collection of arrows between objects (satisfying various properties). So the category of finite sets, that I used above, just had finite sets as objects and functions as arrows. My new category will have as objects the positive integers. There will be a single arrow from $n$ to $m$ precisely when $n$ divides $m$.

So what is the product of $n$ and $m$ in this category? For now, let me call it $p$. Remember that $p$ comes with maps to $n$ and $m$ - which is to say, $p$ divides $n$ and $m$. There's a word for that - $p$ is a common divisor of $n$ and $m$. Which common divisor? Well, suppose $a$ is some other divisor. That means $a$ divides $n$ and $m$, or, in arrow notation, $a\to n$ and $a\to m$. But part of being the product was that whenever I have maps to $n$ and $m$, I get a map to the product, $p$. So $a\to p$, or, in other words, $a$ divides $p$. So, we've found that if $a$ is any other divisor of $n$ and $m$, then since $p$ is the product, $a$ must divide $p$. That makes $p$ the greatest common divisor.

Wasn't that fun?

These things can be 'dualized', and what you get is supposed to be called the sum. 'Dualizing' means flipping all the arrows around. Let's go back to finite sets for a minute. The product $S\times T$ came with maps $S\times T\to S$ and $S\times T\to T$. The dual of this idea (call it the coproduct, or sum), is an object, which I'll denote $S\coprod T$ and it has arrows $S\to S\coprod T$ and $T\to S\coprod T$. Notice how the arrows are going the 'other way' than they did for the product? For the product we mapped from the product to the components, whereas for the sum we map from the components to the sum. The sum also has the property that if $S\to A$ and $T\to A$, then $S\coprod T\to A$ (again, notice how this is flipped from the product case, where $A\to S$ and $A\to T$ gave $A\to S\times T$). The object, in sets, that is the coproduct is more commonly known as the disjoint union.

Exercise for the reader: what is the coproduct in the other category above, the 'divides' category?

You might ask why I decided that what was important, back in the beginning, was functions (maps). It's a good question. I've read somewhere (though I forget where) that thinking about objects by themselves is ok, but thinking about how they related to other objects (via maps) is even better. For a while I wasn't really sure why this should be, but I'm starting to put it together. If you go back and think about set cardinalities, you'll quickly notice that you are really asking about maps between sets, as a way to compare their sizes. If there is an injective or surjective map, you know one of the objects is at least as big as the other. If you've got a bijection, the two sets are the same size. It's hard to think about the size of a set, once you get past finite sets (but even there, really), without comparing it to other sets. And the way you compare sets is via maps between them. For an somewhat more advanced example, algebraic topology is wholly dedicated to maps between spaces. If you've got some space $X$, you decide that you'd really like to know about maps from various dimensional spheres into your space $X$. But perhaps that's a subject for another day.

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