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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


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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