Categories: Between Cubes and Globes. Sketch I
For a finite partially ordered set I, we define an abstract polytope PI which is a cube or a globe in the cases of discrete or linear poset, respectively. For a poset P, we have built a small category ♦P with finite lower subsets in P as objects. This category ♦P = ♦P+♦P- is factorized into a product of two wide subcategories ♦P+ of faces and ♦P- of degenerations. One can imagine a degeneration from I to J ⊂ I as a projection of an abstract polytope PI to the subspace spanned by J. Morphisms in ♦P+ with fixed target I are identified with faces of PI . The composition in ♦P admits the natural geometric interpretation. On the category ♦I of presheaves on ♦I , we construct a monad of free category in two steps: for a terminal presheaf, the free category is obtained via a generalized nerve construction; in the general case, the cells of a nerve are colored by elements of the initial presheaf. Strict P-fold categories are defined as algebras over this monad. All constructions are functorial in P. The usual theory of globular and cubical higher categories can be translated in a natural way into our general context.
M.A. Batanin. Monoidal globular categories as natural environment for the theory of weak n-categories. Advances in Math. 136, 39 (1998). https://doi.org/10.1006/aima.1998.1724
M.A. Batanin. The Eckmann-Hilton argument, higher operads and En-spaces. Advances in Math. 217, 334 (2008). https://doi.org/10.1016/j.aim.2007.06.014
R. Brown, P.J. Higgins, R. Sivera. Nonabelian Algebraic Topology. Filtered spaces, Crossed Complexes, Cubical Homotopy Groupoids. EMS Tracts in Math. 15, EMS (2011). https://doi.org/10.4171/083
I. Gleason, I. Hubard. Products of abstract polytopes. J. Combinatorial Theory. Series A 157, 287 (2018). https://doi.org/10.1016/j.jcta.2018.02.002
T. Leinster. Higher Operads, Higher Categories (Cambridge University Press, 2003). https://doi.org/10.1017/CBO9780511525896
P. McMullen, E. Schulte. Abstract Regular Polytopes Encyclopedia of Mathematics and its Applications (Cambridge University Press, 2002), vol. 92. https://doi.org/10.1017/CBO9780511546686