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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 , , and from an object to an object , the equalizer of all three will be an arrow so that . If we have regular equalizers we can build something to satisfy this property.
Let be the equalizer of and . That is, . Now we have and going from to , so we can take their equalizer . Now . And clearly . So we have as the equalizer of all three morphisms.
Of course we didn’t have to start with and . We could have started with and , taken their equalizer, and so on to get another equalizer . 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 , , and , their product will be an object with arrows that is couniversal among such objects. Again, if we have binary products we can define the product of three objects: has arrows , , and 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 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 , since we don’t need any projection arrows out of it to the factors. For any other object (with no particular arrows out) there is a unique arrow from to . That is, the product of no objects is a terminal object. Similarly, the coproduct of no objects is an initial object.