(simple?) question about 2-category-like structure
In a standard 2-category, every homset C(A,B) is a category. In C we have structures like: f A, B in C ---------------> f, g : A --> B || A || s B s : f ==> g \/ ---------------> g What I need is a structure which resembles a 2-category, but in which the _union_ of all homsets is a category. In such a category we have structures like: f A, B, A', B' in C A ---------------> B f : A --> B || g : A' --> B' || s s : f ==> g \/ A' ---------------> B' g and we have horizontal and vertical composition which satisfy the interchange law (s . s') o (t . t') = (s o t) . (s' o t'), 2-functors etc. I would like to know if such a structure has been explored, if it has a name, and references to papers describing it. -- Nico Verwer | verwer@cs.ruu.nl Dept. of Computer Science, University of Utrecht | +31 30 533921 p.o. box 80.089 3508 TB Utrecht The Netherlands
What I need is a structure which resembles a 2-category, but in which the _union_ of all homsets is a category. The apparent localization of 1-categories to constant (single) homsets comes from looking *only* at the vertical compositions, which have a stationary or static character with respect to endpoints. Use the horizontal compositions as well as the vertical when you need to move the endpoints of homsets around. Lax comma categories (if I recall correctly these are in John Gray's 1974 or so LNM book on 2-categories) give a further generalization of this when you need to restrict or otherwise circumscribe the domain of variation of the endpoints in order to make your category. But this is just icing on top of the essential machinery provided by 2-categories. Vaughan Pratt
participants (2)
-
Nico Verwer -
pratt＠cs.stanford.edu