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. The compounding adjective+name=name scheme used for "commutative X" and such doesn't apply when the adjective happens to be "generalized". That scheme isn't a general rule of English anyways (only a common rule of mathematics), as with Dan Piponi's "fake gun". -- Live well, ~wren