[Haskell-cafe] Re: categories and monoids

[Wolfgang Jeltsch]

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