Let $\mathcal C$ be a pre-triangulated dg-category (or a stable $\infty$-category, if you wish).


* An object $X$ in $\mathcal C$ gives a "point":
$$X$$
* A morphism $X\xrightarrow f Y$ in $\mathcal C$ gives a "triangle":
$$\begin{matrix}X&&\to&&Y\cr&\nwarrow&&\swarrow\cr&&\operatorname{cone}(f)\end{matrix}$$
* A chain of two morphisms $X\xrightarrow fY\xrightarrow gZ$ in $\mathcal C$ gives an "octahedron":
$$\begin{matrix}\operatorname{cone}(g)&&\leftarrow&&Z\cr&\searrow&&\nearrow\cr\downarrow&&Y&&\uparrow\cr&\swarrow&&\nwarrow\cr\operatorname{cone}(f)&&\to&&X\end{matrix}\qquad\qquad\begin{matrix}\operatorname{cone}(g)&&\leftarrow&&Z\cr&\nwarrow&&\swarrow\cr\downarrow&&\operatorname{cone}(gf)&&\uparrow\cr&\nearrow&&\searrow\cr\operatorname{cone}(f)&&\to&&X\end{matrix}$$
(copied from [wikipedia](https://en.wikipedia.org/wiki/Triangulated_category#TR_4:_The_octahedral_axiom), which also has a more three-dimensional rendition of the octahedron).


Given a longer chain $X\_0\xrightarrow{f\_1}\cdots\xrightarrow{f\_p}X\_p$ in $\mathcal C$, is there a canonical $(p+1)$-dimensional convex polytope $V\_p$ organizing together the natural morphisms between $X\_0,\ldots,X\_p$ and all possible cones $\operatorname{cone}(f\_j\circ f\_{j-1}\circ\cdots\circ f\_i)$? Note that it is natural to expect that:


* One of the facets of $V\_p$ is the $p$-simplex $\Delta^p$ of morphisms $X\_0\xrightarrow{f\_1}\cdots\xrightarrow{f\_p}X\_p$.
* The polytopes associated to the chains obtained by "forgetting" one of the $X\_i$ also occur as facets of $V\_p$.



Complete the following sequence: point, triangle, octahedron, . . . in a dg-category
I believe these are called 'hypersimplices'. See


1) Gelfand, Manin "Methods of homological algebra", Ex. IV.2 1(c), p. 260.


2) Belinson, Bernstein, Deligne "Faisceaux pervers", Remarque 1.1.14, p. 26.


3) The diagrams for $n$ up to 4: <http://students.mimuw.edu.pl/~pa235886/pdf/hypersimplices.pdf>


