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/ ]