FOM: Re: Structures prior to homomorphisms

[Colin Mclarty]

Reply to message from sf at Csli.Stanford.EDU of Sun, 23 Nov
>
>The following quote is from Peter Freyd's book "Abelian Categories", 1964,
>p.1:  "It is not too misleading, at least historically, to say that
>categories are what one must define in order to define functors, and that
>functors are what one must define in order to define natural
>transformations."
>
>Not only is it "not too misleading", what other order could these
>definitions be put in?


	I know what Peter had in mind: These notions were created 
(and actually published, between 1942 and 1945) in the opposite order. 
Eilenberg and Mac Lane discovered the now current sense of "natural 
transformation" before they worked out what it is that these transform
--namely functors. And only after this thought came to them did they 
see what it must be that functors act on--namely categories. Of course 
foundations need not follow this kind of historically important 
thinking--but you could come closer to it, if you wanted to, using 
Vaughan's favored axioms for the 2-category of categories.


>The relation of categories to functors is just a special case of the
>general relationship of algebraic structures to homomorphisms.  You can't
>define the latter without specifying the former (at least, not
>mathematically).  

	If you mean it is impossible to give *logically rigorous*
definitions of the latter before the former, then I'd like to see
some particulars--say on my post on linear transformations or
publications by Lawvere and others. And notice this issue is quite
separate from categorical versus set theoretic foundations: Let us
take ZF or your theory of operations and collections as our 
foundation. We may still choose to define linearity by the Abelian
category axioms and define linear spaces after linear transformations.
Do you claim to have an argument that this cannot be *logically
rigorous*?

	If you mean that defining the morphisms before the 
structures must always be *psychologically impossible* then you 
are speaking for yourself but not for the actual creators of 
category theory.

	You may mean something else, I know, but I can't see what it
would be myself. It seems odd to me, to claim that modern homology
theory rests on a "mathematical impossibility".

best, Colin McLarty