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/ ]