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From Function To Category (Prof. Barr)

June 13, 2012

[cm] From Function To Category

Michael Barr: barr at math.mcgill.ca, Feb 28, 2011

I have been away and not reading email much but I have a few, if belated, comments on this one.

I think it is fair to say that originally the abstract notion of function was pretty much restricted to real or complex valued functions.  Galois’s automorphism group was not, but was not really considered as the same kind of thing.  This does not lead to categories at all.

What about group homomorphisms?  Well until sometime after 1940 (which is to say, until after the E-M invention of category) a group homomorphism was invariably considered to be surjective.  In fact, in the 1940s people wrote “homomorphism into” (and even “isomorphism into”) for what we now call homomorphism (resp. injective homomorphism).  And they clearly meant the actual quotient object itself, not the mapping.  I think it was algebraic topology that really became the source of abstract functions. Incidentally, it is apparently not known who first used the arrow notation for functions.

There is an interesting story in this connection.  When “General Theory of Natural Equivalence” first appeared, Paul Smith told Sammy (Eilenberg) that he had never seem a more trivial paper in his life.  Steenrod told Sammy that no paper had influenced his thinking more.  And Sammy, in recounting these reactions added that both comments were valid.  It was trivial in the sense of no new theorems.  But Steenrod had been searching for an axiomatization of homology theory for years.  Algebraic topologists had known that continuous maps of spaces induced homomorphisms (“into”) of the homology groups, but it never occurred to Steenrod to use this fact.  This may have had something to do with the fact that general homomorphisms of groups were not usually looked at.  In other words, general functions were not part of methematical consciousness in those days.  In fact, don’t forget that even homology groups were not really used until Emmy Noether did in the 1930s.  They used Betti numbers and torsion numbers, being the rank of the torsion-free part and the order of the generators of the torsion subgroup of the (finitely generated) homology groups.  According to an interview with Vietoris when he was in his 90s (he died at around 110 in the early years of this millennium) they knew there were groups, but it was not the style of the time to use them.  It is hard to understand the mind-set of the times, but to answer the implicit questions raised here you must.

This brings up a speculation that I have been wondering about for quite a while.  At some point, someone (Mac Lane?) mentioned that with the emphasis on sub and quotient structures in the early part of the last century, Birkhoff thought that lattice theory would revolutionize mathematical thinking.  My speculation is that perhaps if general homomorphisms had been considered important, rather than the sub and quotient lattices, might Birkhoff have invented categories instead?  I should mention that when I tried this idea out on Mac Lane, he didn’t think so.

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On Tue, 15 Feb 2011, Posina Venkata Rayudu wrote:

Dear Professors,

In preparing for the class in which I planned to introduce the notion of CATEGORY, I was studying MacLane’s Duality for Groups, and was drawn to a comment on p. 495, which says that, paraphrasing, one obtains the notion of group in considering the properties of automorphisms, and that one obtains the notion of category in considering the properties of functions in general. What puzzled me the most is how the notion of CATEGORY could be held in abeyance all the years since the conception of FUNCTION. My impression is that once one has the definition of FUNCTION, and based on that definition has a list of properties that each and every function satisfies it seems only natural to name the collection of properties and see if any other mathematical entity satisfies the collection of properties by way of replacing function with an undefined arrow. It’s easy to imagine many mathematicians scribbling the notion of (collection of properties of functions) CATEGORY, and not bothering to publish simply because they might have thought of the notion of CATEGORY too trivial or uninteresting. It’s hard to imagine that no mathematician has thought of the notion of CATEGORY (as an abstraction of the properties satisfied by function) until MacLane and Eilenberg had to define it to define functors as part of a theory of natural equivalences, all the while studying the notion of group obtained in considering the properties of automorphisms or groupoids obtained in considering the properties of isomorphisms not to mention the study of function spaces.

Taking a cue from MacLane’s observation that the notion of category can be obtained by considering the properties of functions, I thought of introducing the notion of CATEGORY as the structural quintessence of functions or as the structure formed of the collection of properties satisfied by each and every one of the functions (please see attached note titled ‘A Study Of Function’). Am I mistaken in thinking that it would be natural, from the perspective of pedagogy, to begin with functions and introduce the notion of category as that which is obtained from the properties of functions, and go on to groupoids as that which we get if we restrict the functions (arrows of the category) to isomorphisms, and finally groups as that which we get if we further restrict the arrows to automorphisms. Needless to say I am apprehensive that I might have committed a blunder by missing something thanks to my mathematical naivety. I’d appreciate very much your corrections, critique and suggestions.

On a related note, given that we have a concept CATEGORY and a list of properties:

1. Arrow f: A –> B has a domain A and a codomain B, which are identity arrows 1A: A –> A and 1B: B –> B, respectively

2. Given two arrows f: A –> B and g: C –> D composition of f and g is defined if B = C and the composite h: A –> C is given by h = gf

3. Composite of an arrow f: A –> B with identities 1A: A –> A and 1B: B –> B satisfies: f1A = f = 1Bf

4. Given three arrows f: A –> B, g: C –> D, and h: E –> F, the triple composite hgf: A –> F is defined if the pair-wise composites gf: A –> D and hg: C –> F are defined, or in other words hgf is defined if B = C and D = E and is given by h(gf) = (hg)f = hgf

from which the concept of CATEGORY is obtained, can we exploit this particular case to formalize the “more” in the empirical observation that there is more to a concept than a mere listing of properties. One simplistic illustration is, say, the concept FACE. Listing of the properties such as: 1. has nose, 2. has two eyes, 3. has one mouth doesn’t suffice to conceptualize, leave alone, picture a face; we also know that in addition to the list of properties, there is also the structure or organization of nose, eyes, and mouth that goes into the conceptualization of FACE. However, in cases such as these, it’s not easy to formalize the organization of properties as distinct from the collection of properties that go into the making of a concept. I was hoping that the notion of CATEGORY, along with the collection of properties of functions that went into the making of the category might help formalize (and generalize with an eye on simplicity) organization of properties-sans-properties. In other words, is there a formal entity that corresponds to organization or structure in its most abstract sense in the sense of not involving any specification of any particular properties? I must admit that I am not sure that the list of properties {1, 2, 3, 4} is all properties; 3 and 4 (identity and associative laws) might very well be the formalization of the structure that organizes the properties 1 and 2 (arrows and composition of arrows).

On a final note, elements and set theory are associated with Cantor and Dedekind, but when it comes to functions and composition of functions I can’t think of any mathematician as the one who introduced those concepts, though I remember reading functions were invented and re-invented many times and that the arrow notation of function is a relatively recent development. Would you be kind enough to point me to any literature that might address these issues, especially the issue of the legitimacy of conceptualizing CATEGORY as a collection of properties of function?

I eagerly look forward to your suggestions and guidance.

Thanking you,

Yours sincerely,

posina

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