FOM: McLarty has not defined "category"
Colin Mclarty
cxm7 at po.cwru.edu
Thu Nov 20 18:44:50 EST 1997
Reply to message from pratt at cs.Stanford.EDU of Thu, 20 Nov
>
>
>There is an issue here that we have been talking around and around
>without ever connecting up on, namely
>
> Has McLarty defined 'morphism' or 'category'?
>
>Here I'm thinking of a theory as serving to characterize the
>individuals it is a theory of, as opposed to characterizing the
>universe of those individuals itself. This is in agreement with the
>idea that ZF defines the notion of "set."
I would say that on this list I have given a very weak
axiomatization of "morphism" by my axioms C1-3 (compare the way
the nullset axiom and the extensionality axiom by themselves give a
very weak axiomatization of "set"). I have not defined "category" at
all (though I've made informal remarks about categories) just as the
ZF axioms do not define "universe of sets" (or use that term in
any way).
I have used two definitions of "category" in print (this is
no hypocrisy, these two ways have different uses). There is the
one Feferman prefers, where a category is a set (or other collection)
of objects and one of arrows and some operations. And there is the
categorical one, where "categories" and "functors" are defined
simultaneously as forming a universe of objects and arrows meeting
certain conditions. The conditions are given in Lawvere's "Category
of Categories as Foundations" and altered somewhat in my "Axiomatizing
a Category of Categories" (I don't have citations with me.)
>
>what would get at the difference more directly would be to address
>Sol's (and Torkel's?) specific concern about the axioms seeming to be
>useful only for axiomatizing a single category, as opposed to category
>theory.
I am not sure this would get at Sol's concern. But I am sure
I have not yet understood his concern very clearly.
Anyway, the axioms C1-3 do axiomatize a single category,
very incompletely so that they have many quite different extensions
to more specifically axiomatize a category. And one family of those
extensions axiomatizes a Category of categories and functors. Here I
capitalize the first word "Category" to indicate that it is used
quite informally. We who plan the axiomatization know it is categorical
in method, but all the axioms actually speak of are the small "c"
categories and functors. The axioms that have been given are logically
adequate, but I think a lot remains to do in exploring them and related
ones--such as for a 2-Category of categories, functors, and natural
transformations.
Colin McLarty
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