categories and monoids (was: Re: [Haskell-cafe] Design Patterns by Gamma or equivalent)

Am Dienstag, 17. März 2009 10:54 schrieben Sie:

Wolfgang Jeltsch writes:

...

By the way, the documentation of Control.Category says that a category is a monoid (as far as I remember). This is wrong. Category laws correspond to monoid laws but monoid composition is total while category composition has the restriction that the domain of the first argument must match the codomain of the second.

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. A category is not a “generalized monoid” but categories (as a concept) are a generalization of monoids. Each category is a monoid, but not the other way round. A monoid is clearly defined as a pair of a set M and a (total) binary operation over M that is associative and has a neutral element. So, for example, the category of sets and functions is not a monoid. First, function composition is not total if you allow arbitrary functions as its arguments. Second, the collection of all sets is not itself a set (but a true class) which conflicts with the above definition which says that M has to be a set.

or a 'monoid with many objects'"

What is a monoid with many objects? Best wishes, Wolfgang