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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