You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.
Matteo Capucci (he/him) (Jul 22 2024 at 12:50):someone might be able to answer this SE question:
The category-theoretical concept called "category" seems quite dissimilar to what we consider a "category" in everyday life or colloquial speech and even other fields of science, where the term usually refers to a class in a categorization / classification, or more generally, of a taxonomy. So what was the motivation to use "category" as the term for that central concept of category theory?
Jean-Baptiste Vienney (Jul 22 2024 at 12:56):I think it comes from the text "Categories" by Aristotle?
Jean-Baptiste Vienney (Jul 22 2024 at 13:00):Maybe it's just my supposition. I've always thought that since the name "monad" in category theory is inspired from Leibniz's monads, another major concept of a great philosopher.
Jean-Baptiste Vienney (Jul 22 2024 at 13:09):I haven't read Aristotle's Categories, but on Wikipedia they say that the text "enumerates all the possible kinds of things that can be the subject or the predicate of a proposition."
Matteo Capucci (he/him) (Jul 22 2024 at 13:27):The OP question is philological in nature so can only be answered by finding Eilenberg or MacLane's account of how the name was chosen. I posted it here because I remembered such an account exist but I don't remember where.
Jean-Baptiste Vienney (Jul 22 2024 at 13:37):I've answered on MathSE a bit more correctly that what I've just said by citing the entry "Category Theory" of the Stanford Encyclopedia of Philosophy.
Jean-Baptiste Vienney (Jul 22 2024 at 13:42):lol someone quoted almost the same paragraph in a comment before but I haven’t read before posting my answer.
John Baez (Jul 22 2024 at 13:54):The Stanford Encyclopedia of Philosophy's explanation is quite dull compared to Mac Lane's actual story of why he and Eilenberg used "category" and "functor". I'll quote the actual story.
John Baez (Jul 22 2024 at 14:06):Here is Saunders Mac Lane's own explanation of why he and Eilenberg chose the words "category", "functor" and "natural transformation":
A direct treatment of categories in their own right appeared in Eilenberg–Mac Lane [1945]. Now the discovery of ideas as general as these is chiefly the willingness to make a brash or speculative abstraction, in this case supported by the pleasure of purloining words from the philosophers: “Category” from Aristotle and Kant, “Functor” from Carnap (Logische Syntax der Sprache), and “natural transformation” from then current informal parlance.
This is from the notes at the end of Chapter I of Mac Lane's famous book Categories for the Working Mathematician. This book is worth looking at.
fosco (Jul 22 2024 at 14:52):see also Kromer's "Tool and object"
fosco (Jul 22 2024 at 14:53):
fosco (Jul 22 2024 at 14:53):
fosco (Jul 22 2024 at 14:53):
fosco (Jul 22 2024 at 14:54):
fosco (Jul 22 2024 at 14:55):(I disagree with the last passage a bit, for etymological reasons, but still it's a valid point to make)
John Onstead (Jul 22 2024 at 18:48):I've been aware of the Aristotle connection for a while and in fact have tried to construct category theory categories corresponding to each of Aristotle's categories. However they are more philosophical concepts than mathematical ones so it didn't really work out. Instead I went off on a tangent trying to analyze all the points of disagreement between Aristotle's categories and modern physics!
Though, I actually disagree with the OP in that a mathematical category is dissimilar to the plain language category. I do think of category theory as being a way of categorizing mathematical objects by forming their respective categories (of course category theory is a lot more than just that as well!). But that could just be me!
Madeleine Birchfield (Jul 22 2024 at 19:41):John Onstead said:
I do think of category theory as being a way of categorizing mathematical objects by forming their respective categories (of course category theory is a lot more than just that as well!). But that could just be me!
IMO that's really only the case when the categories come with a faithful functor into Set.
Eric M Downes (Jul 22 2024 at 21:29):Madeleine Birchfield said:
John Onstead said:
I do think of category theory as being a way of categorizing mathematical objects by forming their respective categories (of course category theory is a lot more than just that as well!). But that could just be me!
IMO that's really only the case when the categories come with a faithful functor into Set.
Could you elaborate?
I would expect "mathematical objects" to include homotopies, which are rather famously quite often not concrete, yet they have categories, have been categorized in both the mathematical and common senses of that word.
Jules Hedges (Jul 23 2024 at 11:18):Besides that Mac Lane enjoyed trolling philosophers, I always assumed it was part of an existing tradition of naming algebraic structures after natural language words for collections of things: group, ring (which afaik was done first in German)
fosco (Jul 23 2024 at 12:21):Besides that Mac Lane enjoyed trolling philosophers
who doesn't!? :heart:
Matteo Capucci (he/him) (Jul 24 2024 at 06:37):fosco said:
(I disagree with the last passage a bit, for etymological reasons, but still it's a valid point to make)
Thanks for the great quotes Fosco. What are the etymological reasons that make you disagree?
David Corfield (Jul 24 2024 at 10:52):Jules Hedges said:
Besides that Mac Lane enjoyed trolling philosophers
Does that include trolling himself? Colin McLarty, The last mathematician from Hilbert's göttingen: Saunders Mac Lane as philosopher of mathematics
Peva Blanchard (Jul 24 2024 at 11:34):From what I recall from my reading of Aristotle, his "category" is not really a collection of thing. The greek verb "kategorein" means "to state or bring as a charge against a person, accuse of it". I think Aristotles was interested in studying the different ways to assign an attribute to a subject (I think we would call his a "predicate" today, although I'm not sure).
fosco (Jul 24 2024 at 12:47):I think we would call his a "predicate" today, although I'm not sure
Yes, afaik before Aristotle a katégoros was a prosecutor and A. took the term to indicate a predicate/attribute of an object, or an "attribute of Being" https://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.04.0057:entry=kathgore/w
I hope someone more experienced than me with ancient Greek can make a stronger point :grinning: