Orientals are “oriented simplices”: the -th oriental is the simplicial -simplex equipped with source and target relations, assigning to each -face a set of -faces called its source and a set of -faces called its target, subject to some natural axioms. Each oriental freely generates (see below) a structure of a strict omega-category , such that -morphisms in are pasting diagrams of -faces in .
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:
Here’s the same figure, in SVG. I think it looks better. Opinions may vary, however. – Jacques
In principle I think the SVG is great, always better than embedded images. This particular one is rendering a bit weird though on my screen; the letters are covered up.
Weird! Renders rather differently in MovableType versus in Instiki. So much for portability! Anyway, this version should look better. – Jacques
Mmm I think it might be a bit better but it’s still not rendering quite right. The last few pixels of each equation as well as some of the numbers seem to be getting cut off. – Bruce
Need a diagrams package for Instiki. I had an idea based on modifying itexml to translateterms eg. in an \xymatrix formula into MathML, but leave the & symbols alone. Now you have an interim XHTML page with MathML on it, with the commutative diagrams rendered in math but not displayed in a matrix format. Then you get javascript to query the HTML elements, ask them “what is your size?”. It uses this information to lay out a diagram, resulting in the final HTML page with the diagram layed out. – Bruce
The construction of orientals is designed to be compatible with face and degeneracy maps. Therefore the orientals arrange themselves into a cosimplicial omega-category, i.e., a functor
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.
(N(C))_k := Hom_{\omega Cat}(O([k]), C)
\,.
This naturally extends to a functor
N : \omega Cat \stackrel{Hom_{\omega Cat}(O([-_2]),-_1)}{\to}
SimplicialSets
\,.
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
(Here one uses that is naturally tensored over : the notation refers to a coproduct of an -indexed collection of copies of . See also enriched category theory.)
Because the nerve functor is faithful, the counit of the adjunction ,
\varepsilon_C: F N C \to C,
is an epimorphism for omega categories .
Is faithful? It seems to be… If not, how does it fail to be faithful?
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
\Pi_\omega(\Delta^{-})
:
\Delta \to \omega Grpd
It should be true that is the free -groupoid on . Is that right?
Literature
The orientals were introduced in
Ross Street, The algebra of oriented simplexes, J. Pure Appl. Algebra 49 (1987) 283-335; MR89a:18019 (pdf).
The -groupoids are discussed in detail in
Ronald Brown, Philip J. Higgins and Rafael Sivera, Nonabelian algebraic topology (pdf)
Further resources
We had related blog discussion in the following -Café-entries: