[Haskell-cafe] Re: categories and monoids
wren ng thornton
wren at freegeek.org
Wed Mar 18 00:36:09 EDT 2009
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".
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