The $(k,n)$-hypersimplex is a convex polytope$\Delta(k, n)$ (or $\Delta_{k,n}$) in $\mathbb{R}^n$ which is the convex hull of the $\left(\array{n \\ k}\right)$ points $e_{i_1}+ e_{i_2}+ \cdots + e_{i_k}$, $1\leq i_1\lt i_2\lt \ldots \lt i_k\leq n$, where the $e_{j}$ are the standard basis vectors. $\Delta(k,n)$ may be seen as a convex hull of the barycenters of the $(k-1)$-dimensional faces of $a (n-1)$-dimensional simplex. Special cases, $\Delta(1,n)$ and $\Delta(n-1,n)$ are themselves simplices of dimension $(n-1)$.

The combinatorics of hypersimplices extends the combinatorics of distinguished triangles and octahedra in the standard triangulated categories; in fact they are postulated in Maltsiniotis‘s strong version of a triangulated category. The octahedron is a $(2,4)$-hypersimplex. Higher hypersimplices were indeed obtained from $A_\infty$-enrichments by Volodymyr Lyubashenko, so one could expect that they can also be obtained from the homotopy category of a stable (∞,1)-category. There are also connections between distinguished hypersimplices and Postnikov towers in triangulated categories.

References

I.M.Gelfand, R.D.MacPherson, Geometry in Grassmannians and a generalization of the dilogarithm, Adv. Math., 44 A982), 279–312; Coll. pap. of I.M. Gelfand v. 3, Springer 1989, p. 492–525.

I. M. Gelfand, M. Kapranov, A. Zelevinsky, Discriminants, resultants, and multidimensional determinants, 1994.

Sergei Gelfand, Yuri Manin, Methods of homological algebra, transl. from the 1988 Russian (Nauka Publ.) original. Springer 1996. xviii+372 pp.; 2nd corrected ed. 2002.

Fred J. Rispoli, The graph of the hypersimplex, arxiv/0811.2981

Yu. Bespalov, Volodymyr Lyubashenko, O. Manzyuk, Pretriangulated $A_\infty$-categories, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 (ps.gz) (chapter 13)

A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Astérisque 100 (1982). MR 86g:32015 (diagrams in Rem. 1.1.14 are hypersimplices)