Re: abstraction of notation from sets.

[Richard Garner <rhgg2@hermes.cam.ac.uk>]

> 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.

> 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.

I think it is more perspicuous to treat this, not as an
overloading of \in, but as an "implicit conversion"
associated to the notion of category; that is, we allow the
forgetful functor Cat->Set to be applied silently in contexts
which otherwise would not type-check. In fact the vast
majority of "abuses of notation" are of this character, when
applied to, for example, any forgetful functor from an
Eilenberg-Moore category; the discrete category functor
Set->Cat; the Yoneda embedding C -> [C^op, Set]; the
forgetful functor from universal cones to their vertex, etc,
etc. In principle this becomes problematic as soon as the
category generated by all such implicit conversions has
non-identity idempotents; in practice, this category is free
on a graph and we hope to identify a shortest path between
two vertices!

Richard


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