The Unapologetic Mathematician

Mathematics for the interested outsider

You cannot run from Carnival!

It’s up and running at MathNotations. And even running is a kind of dance!

June 15, 2007 Posted by | Uncategorized | Leave a comment

Multiple products, coproducts, equalizers, coequalizers, pullbacks, pushouts…

We’ve got products and coproducts of two objects at a time, equalizers and coequalizers of two morphisms at a time, and pushouts and pullbacks of two objects at a time over a third. We can generalize all of these.

First let’s do equalizers. If we have three arrows f_1, f_2, and f_3 from an object A to an object B, the equalizer of all three will be an arrow e:E\rightarrow A so that f_1\circ e=f_2\circ e=f_3\circ e. If we have regular equalizers we can build something to satisfy this property.

Let e_{1,2}:E_{1,2}\rightarrow A be the equalizer of f_1 and f_2. That is, f_1\circ e_{1,2}=f_2\circ e_{1,2}. Now we have f_1\circ e_{1,2} and f_3\circ e_{1,2} going from E_{1,2} to B, so we can take their equalizer e_{1,2,3}:E_{1,2,3}\rightarrow E_{1,2}. Now f_1\circ e_{1,2}\circ e_{1,2,3}=f_3\circ e_{1,2}\circ e_{1,2,3}. And clearly f_1\circ e_{1,2}\circ e_{1,2,3}=f_2\circ e_{1,2}\circ e_{1,2,3}. So we have e_{1,2}\circ e_{1,2,3}:E_{1,2,3}\rightarrow A as the equalizer of all three morphisms.

Of course we didn’t have to start with f_1 and f_2. We could have started with f_1 and f_3, taken their equalizer, and so on to get another equalizer e_{1,3}\circ e_{1,3,2}:E_{1,3,2}\rightarrow A. It’s important to note here that these two equalizers are not the same in general. Whatever category we’re working in will have some construction to give binary equalizers, and when we apply it twice in two different ways we’ll usually get two different results. But the two results are isomorphic, since each is a couniversal object in the category of arrows that equalize the three arrows we started with, and couniversal objects are unique up to isomorphism. Speaking roughly it’s not too much of a problem to talk about “the” equalizer, but it’s useful to keep in the back of your mind that we’re really talking about an isomorphism class of such objects.

We can similarly define equalizers of any finite number of parallel arrows, and if we have equalizers of pairs we have all of them. We can even define the equalizer of an infinite family of parallel arrows, though now we can’t show that they exist using only pairwise equalizers. And, of course, all this goes the same for coequalizers.

Now we’ll do multiple products. Pullbacks are a kind of product so they’ll come along for the ride, and pushouts and coproducts are dual.

If we have three objects A_1, A_2, and A_3, their product P=\prod\limits_{i=1}^3A_i will be an object with arrows \pi_i:P\rightarrow A_i that is couniversal among such objects. Again, if we have binary products we can define the product of three objects: (A_1\times A_2)\times A_3 has arrows \pi_{A_1}\circ\pi_{A_1\times A_2}, \pi_{A_2}\circ\pi_{A_1\times A_2}, and \pi_{A_3} that go from the product to the three objects, and it’s straightforward to verify that it satisfies the universal property required. Again, we can also take the product A_1\times(A_2\times A_3) with the obvious arrows, and this also satisfies the universal property. And again, these two may not be the same, but the universal property guarantees that they’re isomorphic.

We can do the same thing to define the product of any finite number of objects, or even an infinite family (though now we can’t build them from finite products). One interesting case is when we take the product of no objects. This is just an object T, since we don’t need any projection arrows out of it to the factors. For any other object A (with no particular arrows out) there is a unique arrow from A to T. That is, the product of no objects is a terminal object. Similarly, the coproduct of no objects is an initial object.

June 15, 2007 Posted by | Category theory | 1 Comment

   

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