\epsilon is often used informally to mean an object or an arrow is contained in a category C. I.e. a \in C is shorthand for a \in ob(C) or a \in ar(C), when its interpretation is obvious from context. Barr and Wells (Toposes, Triples and Theories, 1984) uses set inclusion notation in an interesting way, thinking of arrows rather as "generalised elements" of objections. So, they'll write something like f \in^A B instead of f : A -> B, which should be read "f is an A-element of B". When the category is sets and A = {*}, this is the usual notion of element. This notion gives the right kind of intuition when working and categories that are a lot like sets, especially toposes. The language also admits a particularly beautiful way to express the Yoneda lemma. "The (normal) elements of FA are the same as the hom(-,A)-elements of F." a On Wed, Feb 24, 2010 at 12:43 AM, <peasthope@shaw.ca> wrote:
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?
Thanks, ... Peter E.
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