One of the axioms is a globularity axiom, which says that the source of a source (that is, the union of sources of all -faces in the source of a -face) equals the source of the target, and similarly that the target of a source equals the target of the target. Thus, orientals mediate between the simplicial and the globular world of infinity-categories.
The first few orientals look as follows:
The construction of orientals is designed to be compatible with face and degeneracy maps. Therefore the orientals arrange themselves into a cosimplicialω-category, i.e., a functor
Regarding the standard -simplex as a filtered space with the standard filtering, and denoting for a filtered space by the fundamental filter-respecting -groupoid of , we obtain a cosimplicial omega-groupoid
It should be true that is the free -groupoid on . Is that right?
Relation to cyclic polytopes
There is a convex-geometric perspective on orientals outlined in Kapranov-Voevodsky We will try to explain the punch-line first, then go into details.
In , denote the source of the -morphism corresponding to the -face in by , and denote its target by . Since these are source and target of the same morphism, they both have the same source, and both have the same target. Write for this source, and for this target. Inductively define and for all .
The promised punchline is that the -morphisms from to are in natural correspondence to the triangulations of the -dimensional cyclic polytope with vertices.
For example, when , we have is the 1-morphism , and is the 1-morphism . The 2-morphisms from to consist of triangulations of the -gon. See the diagrams above for examples.
We are going to embed the simplex inside , so we can think (convex-)geometrically about it.
Let is get the main convex-geometric definitions out of the way at the beginning.
The moment curve in is defined parametrically by .
A cyclic polytope of dimension with vertices is the convex hull of points on the moment curve in . For many purposes, the choice of the points is irrelevant; for example, the face lattice of the polytope does not depend on which points were chosen.
A triangulation of a polytope is a subdivision of the polytope into maximal-dimensional simplices, which overlap only on their boundaries, and all of whose vertices are vertices of the original polytope.
To fix an embedding of into , choose real numbers , and embed inside so that its -th vertex is at .
Now, as already described above, we have that and are compositions of the morphisms corresponding to some -faces, so that the morphism coming from the unique -face of goes from to . Which faces belong to which -morphism?
The answer is very simple: is the composition of the lower faces, while is the composition of the upper faces, where lower and upper are always taken with respect to the final co-ordinate.
Now, project into by forgetting the last co-ordinate. The image of inside is the convex hull of points on the moment curve in , that is to say, it is a cyclic polytope with vertics in dimension .
consists of the lower faces of this cyclic polytope (where, as always, we take upper/lower with respect to the final co-ordinate, though note that in this case, the final co-ordinate is the -st co-ordinate, since we have already forgotten the -th co-ordinate). Similarly, consists of the upper faces of this cyclic polytope.
The projections of and , meanwhile, are triangulations of this cyclic polytope. The cyclic polytope of dimension with vertices has only two triangulations, so these are all of them.
In general, consists of the lower faces of the image of the projection of into by forgetting the last coordinates, while consists of the upper faces of the image. And then , correspond to particular triangulations of this image and, indeed, the -morphisms from to correspond bijectively to triangulations of the image (which is a cyclic polytope).
The -nerve of an omega-category is a simplicial set which generalizes the nerve of an ordinary category: the collection of -simplices in is the collection of images of the -th oriental in , i.e.
This naturally extends to a functor
The nerve functor is faithful. This means that omega categories can be regarded as simplicial sets equipped with extra structure. The precise nature of this structure was identified by Dominic Verity in terms of complicial sets in his work on the Doplicher-Roberts conjecture.
Free -Category on a simplicial set
The -nerve has a left adjoint, the free category on a simplicial set
Ross Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283-335; MR89a:18019 (pdf).
The link to cyclic polytopes is discussed in
Mikhail Kapranov and Vladimir Voevodsky, Combinatorial-geometric aspects of polycategory theory: pasting schemes and higher Bruhat orders (list of results). International Category Theory Meeting (Bangor, 1989 and Cambridge, 1990). Cahiers Topologie Géom. Différentielle Catég. 32 (1991), no. 1, 11–27. (pdf).
The -groupoids are discussed in detail in
Ronnie Brown, Philip J. Higgins and Rafael Sivera, Nonabelian algebraic topology (pdf)
An introductory survey of the role of orientals in Steet’s definition of an -category is given in section 6 of