Leopold Schlicht <schlicht.leopold@gmail.com> quotes, in part:
Notation 1.1: The arrow notation f : A -> B just means the domain of f is A and the codomain of f is B, and we write dom(f) = A and cod(f) = B.
and continues:
Here, the authors talk about the equality of two sets (dom(f) = A).
No. The symbolatry dom(f) = A is NOT expressing the equality of two sets, dom(f), and A, it is an abbreviation for the assertion that the map f has domain A, which is itself a gloss on (a part of) the significance we may attach to the arrow notation f: A -> B. The remaining part of that significance is expressed in the symbolatry cod(f) = B. There again it's NOT an expression of the equality of two sets, cod(f), and B. Rather, it's as if we split f: A -> B into two chunks, f: A -> _ (abbreviated dom(f) = A) and f: _ -> B (abbreviated cod(f) = B). (In the above, please treat the simple underscore _ as just more white space: my usa.net mail service will not permit a string of 3 or 4 space characters to pass through without condensing them into a single space character.) Taking a fresh breath of air, now, and asking about objects of a category: does one not have a right to suppose that, should you and I each focus our respective attentions on one object in a category, you on A, say, and I on B, it should be possible to determine when we have chosen to focus on ONE AND THE SAME object, i.e., to determine when your A is my B, i.e., when A = B ? No question on elementhood or membership enters into that question. And we NEVER actually say that TWO sets are equal -- only that ONE set is equal to itself. If you and I have focused on sets A and B, respectively, and if it happens that A = B, it means exactly that we were NOT focused on TWO distinct sets, but only on ONE. Cheers, -- Fred Linton [For admin and other information see: http://www.mta.ca/~cat-dist/ ]