Re: Are there exactly 11 categories with 3 arrows?

[Aleks Kissinger <aleks0@gmail.com>]

Probably a good exercise to work out the details yourself, but here's
a hint. The 1-object case is exactly the same as counting monoids.

On Fri, Mar 26, 2010 at 10:07 PM, Mark Spezzano
<mark.spezzano@chariot.net.au> wrote:
> Hi,
>
> This is a beginner's question. I have a textbook (Walters) that asks to show that there are exactly 11 categories with 3 arrows.
>
> Now, my logic tells me that I need to cover three possibilities:
>
> a) One object with three arrows. How many are there of these?
>
> b) Two objects with three arrows. How many are there of these?
>
> c) Three objects with three arrows. I think that the answer to this is the easiest. The answer is 1 categories because they are all endomorphisms, each object containing just the identity morphism.
>
> So the other 10 arrows must come from a) and b), but I keep getting different answers like 12 and 13 categories as the total.
>
> Can someone please explain to me the combinations of categories that need to be covered and why some are missed out during the calculation.
>
> Any help would be immensely appreciated.
>
> Thanks,
>
> Mark Spezzano
>


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