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. 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 _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe