[Haskell-cafe] Re: categories and monoids
Wolfgang Jeltsch
g9ks157k at acme.softbase.org
Wed Mar 18 06:12:33 EDT 2009
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
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