Are there exactly 11 categories with 3 arrows?

Are there exactly 11 categories with 3 arrows?

[Mark Spezzano]

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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Re: Are there exactly 11 categories with 3 arrows?

[David Roberts]

Hi Mark,

Let us phrase the question as:

Given three sets, {a},{b,c},{d,e,f}, how many categories are there
with one of these sets as its set of objects and exactly three arrows?
(any other set of objects is disallowed, as you pointed out)

This is to remove questions of 'up to equivalence' because
technically, there are an infinite number of categories with three
arrows, and taking categories up to equivalence will, I think, give
too few for your liking (for example, in my list of four categories
with two objects, we would only get two categories).

Here's a start: there are two categories b -> c, c -> b (omitting
identity arrows) with no non-trivial isomorphisms, and two categories
Z/2 \sqcup {*} where * is one of b or c, and the other is the identity
element of Z/2. Given your unique category with three elements this
brings us to five categories. Then there is Z/3 (with identity element
a), giving us six.

What other categories with three arrows have you identified? They
will, I imagine, have two objects.

David



On 27 March 2010 00:37, 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
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Are there exactly 11 categories with 3 arrows?

[Aleks Kissinger]

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
>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Are there exactly 11 categories with 3 arrows?

[Vaughan Pratt]

On 3/26/2010 6:01 PM, Aleks Kissinger wrote:
 > 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.

With the exception of one category, so is the general case.  It suffices
to identify the odd category out and then put the remainder in bijection
with the monoids of order at most 3 (see Sloane A058129).

For positive integer n, 3 is the least value of n for which the
categories with n arrows cannot be put in bijection with the monoids of
order at most n.

Vaughan


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Re: Are there exactly 11 categories with 3 arrows?

[Fred E.J. Linton]

On Fri, 26 Mar 2010, at 08:09:06 PM EDT, Mark Spezzano
<mark.spezzano@chariot.net.au> asked: 

> Are there exactly 11 categories with 3 arrows?

Some gentle hints, without giving away the answers:

> ... I need to cover three possibilities:
> 
> a) One object with three arrows. How many are there of these?

That's the heart of the problem: how many monoid structures
can be imposed on the identity and the two non-identity arrows?
 
> b) Two objects with three arrows. How many are there of these?

Where can the single non-identity arrow be?
 
> c) Three objects with three arrows. ... the answer to this ... is 1 ...

Agreed. HTH. Cheers, -- Fred




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