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Topic: The Monoid Category in DC Proof
Replies: 5   Last Post: Nov 9, 2012 4:03 PM

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Rotwang

Posts: 1,678
From: Swansea
Registered: 7/26/06
Re: The Monoid Category in DC Proof
Posted: Nov 9, 2012 4:03 PM
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On 09/11/2012 20:56, Jesse F. Hughes wrote:
> Rotwang <sg552@hotmail.co.uk> writes:
>> [...]
>>
>> Sure. In fact this is what I was getting at with my "this is not a
>> coincidence" comments - that the map that sends an element of a monoid
>> to its left action is part of a functor from the monoid (considered a
>> category) to Set. More generally, if C is any small category then the
>> Yoneda functor C -> Set^{C^{op}}, followed by the functor that takes F:
>> C^{op} -> Set to the coproduct of F(a) for a in ob(C), is a faithful
>> functor from C to Set, which shows that any small category is
>> concretizable. In the case where C has a single object this simplifies
>> to the construction you give above.

>
> Ugh.
>
> Don't mention Yoneda to me, dammit. I'm pretty sure that's a part of
> category theory which I've repressed.
>
> Though thanks for the result. If I've ever known it, I've forgotten it.


I can't take credit, I found it on Wikipedia.


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