On Tue, 2009-03-17 at 13:06 +0100, Wolfgang Jeltsch wrote: > Am Dienstag, 17. März 2009 10:54 schrieben Sie: > > Wolfgang Jeltsch <g9ks1...@acme.softbase.org> 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.
You mean ``each monoid is a category, 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? A categorical definition of a monoid (that is, a plain old boring monoid in Set) is that it is a category with a single object. A category is thus a monoid with the restriction to a single object lifted :) jcc _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe