Re: abstraction of notation from sets.

[Michael Shulman <shulman@uchicago.edu>]

<peasthope <at> shaw.ca> writes:
> When S is a set, the notation "a \epsilon S" is familiar.
> Is this ever extended to CT?  All the texts I recall use
> natural language such as "A is an object of C".  What if
> a more symbolic notation is required?

As has been pointed out, if category theory is formalized within set
theory, so that a category has a set of objects and a set of arrows,
then one cannot write "a \in C" to mean that a is an object of the
category C, at least as long as \in is restricted to its precise
set-theoretic meaning.  However, in my experience it is fairly common to
write "a \in C" with this meaning, although perhaps not so common in
formal mathematical writing.

I regard this as precisely analogous to writing "g \in G" when G is a
group, since after all when formalized within set theory, a group is not
just the set of its elements, but a triple (G,m,e) of a set, a
multiplication, and an identity (or some other equivalent encoding).
One can refer to this sort of thing perjoratively as an "abuse of
notation," but one can also regard it as a perfectly legitimate part of
informal mathematical language which is not captured by the
set-theoretic encoding.  One could also formalize it by regarding the
symbol "\in" as "overloaded" in a precise sense analogous to programming
languages.

Mike


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