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I have added some discussion to the page on orientals (in the sense of Ross Street), regarding the link to the convex geometry of cyclic polytopes (as discussed by Kapranov and Voevodsky).
My selfish motive for doing so is that I am curious if my recent work with Steffen Oppermann which includes a new description of the triangulations of (even-dimensional) cyclic polytopes, has any relevance to the study of orientals, or higher category theory more broadly. (In particular, if there are explicit questions about the internal structure of orientals which are of interest, I would like to hear about them.)
A particularly speculative version of my question, would be whether there is a natural connection between orientals and the representation theory which we are studying in that paper (which necessitated a detour into convex geometry). We biject triangulations of an even-dimensional cyclic polytope to (a nice class of) tilting objects for a certain algebra. The simplest version of this (which was already known) is that triangulations of an $n$-gon correspond to tilting objects for the path algebra of the quiver consisting of a directed path with $n-2$ vertices. (These tilting objects then give derived equivalences between the derived category of this path algebra, and the derived category of the endomorphism ring of this tilting object.)
Questions, speculations, or suggestions would be very welcome.
Hugh
Thanks!
i have now reorganized the entry slightly, putting your cyclic polytopes together with the $\omega$-groupoids in a section “Relation to other concept”. Check out if you can live with it.
Concerning your paper and your speculations: sounds interesting, but I’d need to think about it. It’s been a while since i actively thought about derived categories over quivers and functors coming from tilting modules. What’s the intuition here for why these ought to be related to triangulations?
Maybe we need entries tilting module etc…
I’m very happy with the reorganization. Thanks for making the other changes to the entry as well.
I’m afraid I don’t have much of an intuition for the link from the representation theory to triangulations – it just works. Well, okay, I guess I have something. For triangulations of a polygon, there is an operation called “diagonal flip” which removes a diagonal from a polygon and replaces it by the other possible diagonal. (The graph whose vertices are triangulations and whose edges connect triangulations related by a flip, is then the 1-skeleton of the associahedron.)
In the hereditary case, there is a parallel notion of “mutation” of tilting modules: you can remove an indecomposable summand and replace it in exactly one way by a different indecomposable summand. In the present situation, the two notions happen to line up perfectly (indecomposable summands correspond to diagonals; diagonals which fit together to make a triangulation correspond to a tilting object). My paper with Steffen is then a higher dimensional generalization of this.
(I am lying slightly in the previous paragraph: under some circumstances, a summand of a tilting module cannot be replaced by a different summand. In the situation of the path algebra of a directed path, this applies to exactly one summand, which is in all tilting modules. I am tacitly ignoring it. It doesn’t correspond to a diagonal.)
I agree that an entry on tilting modules would be good. I will try to write something.
I agree that an entry on tilting modules would be good. I will try to write something.
That would be much appreciated!
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