In real life, i.e., when dealing with the physical world, there are usually many ways to generalize any given thing or phenomenon.
For example, a tomato is a fruit, but it’s also a vegetable; that is, it belongs to a botanical grouping, but also to a culinary grouping. Neither classification is more ‘real’ or ‘true’ than the other[1]; and indeed there are many other possible categories within which we can put tomatoes (red things, throwable things, round things, soft things, etc.).
Is this also the case in category theory? That is: for anything which we might be tempted to generalize with the aid of category theory, are there multiple ways to generalize it, dictated only by convenience and preference? Or, is there necessary some single canonical generalization for any given mathematical… thing? If the former: how and by what criteria are generalizations selected? If the latter: what pitfalls does this create when using real-world-based analogies to understand category theory?
Recall that taxonomic classifications aren’t written in the heavens somewhere, but are merely a useful way for humans to classify organisms (namely, by putting them into groups arranged by common descent). This is useful for various reasons, but by no means unambiguous or necessary, nor dictated by reality—as “in truth there are only atoms and the void.”
Math certainly has ambiguous generalizations. As the image hints, these are also studied in category theory. Usually, when you must select one, the one of interest is the least general one that holds for each of your objects of study. In the image, this is always unique. I’m guessing that’s why bicentric has a name. I’ll pass on the question of how often this turns out unique in general.
Maybe a more direct answer is that in the very next post in the series, we’ll see that sets can be considered the objects of the category of sets and functions, and also the objects of the category of sets and binary relations. Functions are binary relations, so that’s not a perfect answer, but yes, you can think of an individual category as a context of sorts through which you view the objects, like how you can view a tomato as a fruit or vegetable depending on the context.
Conceptual question:
In real life, i.e., when dealing with the physical world, there are usually many ways to generalize any given thing or phenomenon.
For example, a tomato is a fruit, but it’s also a vegetable; that is, it belongs to a botanical grouping, but also to a culinary grouping. Neither classification is more ‘real’ or ‘true’ than the other[1]; and indeed there are many other possible categories within which we can put tomatoes (red things, throwable things, round things, soft things, etc.).
Is this also the case in category theory? That is: for anything which we might be tempted to generalize with the aid of category theory, are there multiple ways to generalize it, dictated only by convenience and preference? Or, is there necessary some single canonical generalization for any given mathematical… thing? If the former: how and by what criteria are generalizations selected? If the latter: what pitfalls does this create when using real-world-based analogies to understand category theory?
Recall that taxonomic classifications aren’t written in the heavens somewhere, but are merely a useful way for humans to classify organisms (namely, by putting them into groups arranged by common descent). This is useful for various reasons, but by no means unambiguous or necessary, nor dictated by reality—as “in truth there are only atoms and the void.”
Math certainly has ambiguous generalizations. As the image hints, these are also studied in category theory. Usually, when you must select one, the one of interest is the least general one that holds for each of your objects of study. In the image, this is always unique. I’m guessing that’s why bicentric has a name. I’ll pass on the question of how often this turns out unique in general.
One of the reasons for my own interest in category theory is my interest in the question you raise. I’m hoping that we’ll explore the idea that universal properties offer an “objective” way of defining “subjective” categories.
Maybe a more direct answer is that in the very next post in the series, we’ll see that sets can be considered the objects of the category of sets and functions, and also the objects of the category of sets and binary relations. Functions are binary relations, so that’s not a perfect answer, but yes, you can think of an individual category as a context of sorts through which you view the objects, like how you can view a tomato as a fruit or vegetable depending on the context.