[Haskell-cafe] Re: categories and monoids

[Gregg Reynolds]

On Thu, Mar 19, 2009 at 5:43 AM, Wolfgang Jeltsch
<g9ks157k at acme.softbase.org> wrote:
> Am Mittwoch, 18. März 2009 15:17 schrieben Sie:
>> Wolfgang Jeltsch schrieb:
>> > 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.
>>
>> “A ring is like a field, but without a multiplicative inverse” is, in my
>> eyes, an acceptable formulation. We just have to agree that “without”
>> here refers to the definition, rather than to the definitum.
>
> Note that you said: “A ring is *like* a field.”, not “A ring is a field.”
> which was the formulation, I criticized above.
>

"Alternatively, the fundamental notion of category theory is that of a
monoid ... a category itself can be regarded as a sort of generalized
monoid."

   --  Saunders MacLane, "Categories for the Working Mathematician"  (preface)