Re: abstraction of notation from sets.

[Toby Bartels <toby+categories@ugcs.caltech.edu>]

In a context where "C" is a term for a category and "a" is a new variable,
I would interpret "a \in C" to declare that a is an object of C,
but I wouldn't be surprised if the author really meant to declare
that a is a map in C (with notation for source and target derived).
Using scripts can make things more clear, as peasthope@shaw.ca wrote:

>"$a \in C_0$" = "a is an object in C"
>"$f \in C_1$" = "f is a map in C"

This generalises immediately to higher categories.

If "a" is not a new variable but instead a term with an established meaning,
then "a \in C" might mean either that a is an object or a map of C,
but the meaning should be clear from context, particularly if "C" is
a term for a subcategory of a category D in which a is an object or map.

Even in set theory, there can be a logical difference between
declaring that a new variable stands for an element of a given set
and stating that a given element belongs to a given subset,
and people using structural or type-theoretic approaches to set theory
sometimes even use different notation to distinguish these.
One possibility (not the only one) is to use a colon for the former;
then if "a" is a new variable and "S" is a term for a set,
then "a: S" would declare that a is an element of S;
while if "a" is a term for an element of some set T
and "S" is a term for a subset of T, then "a \in S"
would denote the proposition that a belongs to S.
The former ("a: S") only makes sense as an assertion or hypothesis,
but the latter ("a \in S") makes sense in any logical context.
(In particular, it makes sense to say "If a \notin S, then ..."
but not "If a: S is false, then ...".)

This can be extended to category theory as follows:
If "a" is a new variable and "C" is a term for a category,
then let "a: C" declare that a is an object of C.
If "f" is a new variable and "a" and "b" are terms for objects of C,
then let "f: a \to b" declare that f is a map in C from a to b,
which after all is already a very widely used notation.
(We can also write "f: a \to b: C" to declare everything at once.)
This generalises to higher categories, as long as you follow the plan
of naming sources and targets before you name whatever goes between them.
(If you follow the philosophy of avoiding "evil", then this is very natural.)
On the other hand, if D is a category and "C" is a term for a subcategory,
then "a \in C" has an unambiguous meaning as long as "a" is a term
for an element at some level (object or map) in D;
this also generalises to higher categories.


--Toby


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