Double category question

[Mike Stay <metaweta@gmail.com>]

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


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