Am Dienstag, 17. März 2009 16:32 schrieben Sie:
On Tue, 2009-03-17 at 13:06 +0100, Wolfgang Jeltsch wrote:
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''.
Exactly. :-)
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 :)
Okay. Well, a monoid with many objects isn’t a monoid anymore since a monoid has only one object. It’s the same as with: “A ring is a field whose multiplication has no inverse.” One usually knows what is meant with this but it’s actually wrong. Wrong for two reasons: First, because the multiplication of a field has an inverse. Second, because the multiplication of a ring is not forced to have no inverse but may have one. It reminds me of a definition of “constant” in programming languages which occured in some literature: “A constant is a variable whose value cannot be changed.” :-) Best wishes, Wolfgang