[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".

Live well,

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