Double category question

Double category question

[Mike Stay]

Consider this setup:

*→*→*→*
↓⇙↓⇙↓⇙↓
*→*→*→*
↓⇙↓⇙↓⇙↓
*→*→*→*
↓⇙↓⇙↓⇙↓
*→*→*→*

What kind of higher category models the case where never have a path
that goes right twice in a row?  There are four paths from the upper
left to the lower right satisfying that condition:
→↓→↓→↓
→↓→↓↓→
→↓↓→↓→
↓→↓→↓→
We can almost do this with a double category:  we take the product of
the points above with {0,1} and then say for horizontal neighboring
points x, x' we have a single morphism
    (x, 0) -R-> (x', 1)
and for vertical neighboring points y, y' we have two morphisms
    (y, 0) -D1-> (y', 0)
    (y, 1) -D2-> (y', 0).
This way it's impossible to form the composition of two arrows going right.

The squares would need to be of the form R;D2 => D1;R, but the types
don't match:  R;D2 goes from (s,0) to (t,0) while D1;R goes from (s,0)
to (t,1).

Has anyone seen work on something like this?
-- 
Mike Stay - metaweta@gmail.com
http://www.cs.auckland.ac.nz/~mike
http://reperiendi.wordpress.com


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

Re: Double category question

[Robert Dawson]

On 7/21/2014 6:41 PM, Mike Stay wrote:
> Consider this setup:
>
> *???*???*???*
> ?????????????????????
> *???*???*???*
> ?????????????????????
> *???*???*???*
> ?????????????????????
> *???*???*???*
>
> What kind of higher category models the case where never have a path
> that goes right twice in a row?  There are four paths from the upper
> left to the lower right satisfying that condition:
> ??????????????????
> ??????????????????
> ??????????????????
> ??????????????????
> We can almost do this with a double category:  we take the product of
> the points above with {0,1} and then say for horizontal neighboring
> points x, x' we have a single morphism
>      (x, 0) -R-> (x', 1)
> and for vertical neighboring points y, y' we have two morphisms
>      (y, 0) -D1-> (y', 0)
>      (y, 1) -D2-> (y', 0).
> This way it's impossible to form the composition of two arrows going right.
===================================================================

This structure does not feel very categorical... if you can't compose
horizontal arrows, it would seem that you have,in general, no horizontal
composition of cells.  So, if I've understood this correctly, the
cell-and-arrow notation doesn't mean what it usually would.

Robert Dawson




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