I'm reading about category theory. A question I don't see answered anywhere is about the choice of the word "category" in this context. Why this concept was named "category"?
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3$\begingroup$ They're just supposed to be different categories of mathematical objects in the colloquial sense, e.g. groups, rings, topological spaces. I don't think it's particularly deep. $\endgroup$– Qiaochu YuanFeb 18 at 20:28
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2$\begingroup$ @QiaochuYuan The answer from Geoff suggests that the etymology is in fact pretty interesting, somewhat deep. $\endgroup$– Ethan BolkerFeb 18 at 21:02
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$\begingroup$ @QiaochuYuan Maybe you're right. Groups are another example. The colloquial concept of "group" anticipates nothing about its meaning in mathematics, I think. $\endgroup$– a06eFeb 18 at 21:30
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The term ``category'' came from the Kantian perspective on a category being the most general form of thinking, although I may be mistaken on the finer points of the actual philosophy there. This is discussed in this philosophy.stackexchange post here (see the accepted answer in particular) which speaks far more eloquently on the subject than I could hope to mimic.
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1$\begingroup$ Thanks, that link is a very interesting discussion! $\endgroup$– a06eFeb 18 at 21:33
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$\begingroup$ I find this quote from the linked OP particularly illuminating: ... my guess is that categories in mathematics don't usually mention individual elements of a set explicitly. They only define properties of elements via how various sets relate to each other. This in turn defines the elements. $\endgroup$– a06eFeb 18 at 21:35