Turning math inside-out

Here’s one of the things about category theory that takes a while to get used to.

Mathematical objects are usually defined internally. For example, the Cartesian product P of two sets A and B is defined to be the set of all ordered pairs (ab) where a comes from A and b comes from B. The definition of P depends on the elements of A and B but it does not depend on any other sets.

Category theory turns this inside-out. Operations such as taking products are not defined in terms of elements of objects. Category theory makes no use of elements or subobjects [1]. It defines things by how they act, not their inner workings. People often stress what category theory does not depend on, but they less often stress what it does depend on. The definition of the product of two objects in any category depends on all objects in that category: The definition of the product of objects A and B contains the phrase “such that for any other object X …” [More on categorical products].

The payoff for this inside-out approach to products is that you can say something simultaneously about everything that acts like a product, whether it’s products of sets, products of fields (i.e. that they don’t exist), products of groups, etc. You can’t say something valid across multiple categories if you depend on details unique to one categories.

This isn’t unique to products. Universal properties are everywhere. That is, you see definitions containing “such that for any other object X …” all the time. In this sense, category theory is extremely non-local. The definition of a widget often depends on all widgets.

There’s a symmetry here. Traditional definitions depend on the internal workings of objects, but only on the objects themselves. There are no third parties involved in the definition. Categorical definitions have zero dependence on internal workings, but depend on the behavior of everything in the category. There are an infinite number of third parties involved! [2] You can have a definition that requires complete internal knowledge but zero external knowledge, or a definition that requires zero internal knowledge and an infinite amount of external knowledge.

Related: Applied category theory

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[1] Category theory does have notions analogous to elements and subsets, but they are defined the same way everything else is in category theory, in terms of objects and morphisms, not by appealing to the inner structure of objects.

[2] You can have a category with a finite number of objects, but usually categories are infinite. In fact, they are usually so large that they are “classes” of objects rather than sets.

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