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Topic: The Monoid Category in DC Proof
Replies: 7   Last Post: Nov 11, 2012 10:37 PM

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Rotwang

Posts: 1,678
From: Swansea
Registered: 7/26/06
Re: The Monoid Category in DC Proof
Posted: Nov 10, 2012 11:54 AM
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On 10/11/2012 06:17, Dan Christensen wrote:
> On Nov 9, 2:34 pm, Dan Christensen <Dan_Christen...@sympatico.ca>
>> [...]
>>

>>> but the category corresponding to a monoid is
>>> easier to define if they aren't. Firstly, the unique node doesn't have
>>> to be the underlying set m; it can be anything, even an urelement. More
>>> importantly, though, you've defined a category whose arrows are the
>>> right actions of elements of m on m, and whose composition is given by
>>> function composition. But you could just as well have defined the arrows
>>> to be the elements of m, and defined composition to be given by +. That is,

>>
>>> nodes(C) = {*} /* where * is anything at all */
>>> arrows(C) = m /* the underlying set of your monoid */
>>> id(m) = 0 /* the identity of your monoid */
>>> (a @ m & b @ m) => comp(a, b) = a + b
>>> /* wasn't that nice and simple? */

>>
>
> Indeed. But I needed a bit more detail*:
>
> 1. ALL(a):[a @ nodes <=> a=m]
>
> 2. ALL(f):[f @ arrows <=> f @ m]
>
> 3. id(m)=0
>
> 4.* ALL(f):[f @ arrows => source(f)=m]
>
> 5.* ALL(f):[f @ arrows => target(f)=m]


You may notice that the material you snipped from my earlier post
included the words "Upon defining source() and target() in the only way
possible [...]".


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