[Haskell-cafe] Re: categories and monoids

[Wolfgang Jeltsch]

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