Am Mittwoch, 18. März 2009 05:36 schrieb wren ng thornton:
Wolfgang Jeltsch wrote:
Am Dienstag, 17. März 2009 10:54 schrieben Sie:
I'm reading the Barr/Wells slides at the moment, and they say the following:
"Thus a category can be regarded as a generalized monoid,
What is a “generalized monoid”? According to the grammatical construction (adjective plus noun), it should be a special kind of monoid, like a commutative monoid is a special kind of monoid. But then, monoids would be the more general concept and categories the special case, quite the opposite of how it really is.
Usually in math texts "a Y is a generalized X" means exactly "Ys are a generalization of Xs", and thus Y is the larger class of objects got by relaxing some law in X. It's a description, not a name. E.g. Hilbert space is a generalized Euclidean space, Heyting algebras are generalized Boolean algebras, modules are generalized vector spaces, etc.
I know these phrases but I always considered them as something, mathematicians use when they talk to each other informally, not what they would write in a book. Best wishes, Wolfgang