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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, 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 $V_p$.

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    $\begingroup$ the picture at the bottom of p.24 here: math.harvard.edu/~lurie/papers/HA.pdf has an evident generalization to any n-tuple of morphisms. If you write it down you'll join the group of m people who have independently discovered the notion of an "n-angulated category". The shape you seek comes from collapsing that unrolled version in some probably not-so-trivial way. $\endgroup$ – Dylan Wilson Apr 18 '16 at 18:55
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    $\begingroup$ (It is not known whether any triangulated thing is n-angulated, but any stable $\infty$-category is $n$-angulated for any $n$, and the proof is the same as in loc. cit.) $\endgroup$ – Dylan Wilson Apr 18 '16 at 18:57
  • $\begingroup$ @DylanWilson Are you sure it is on p.24? I see there (TR4) and it is certainly not evident for me how to generalize this. $\endgroup$ – მამუკა ჯიბლაძე Apr 18 '16 at 19:26
  • $\begingroup$ The diagram is produced algorithmically. If you want to do it for n composable morphisms, draw a them in a line. Now extend the line to the right with a zero, and down on the left with a zero. Now fill in this rectangle with pushout diagrams (so you're looking at an [n+1] \times [1] diagram.) Now on the far lower right you can extend by a map to 0, and on the lower left offset by one you can extend by zero... rinse wash and repeat. You get a staircase lookin' thing. $\endgroup$ – Dylan Wilson Apr 18 '16 at 19:30
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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

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  • $\begingroup$ The hypersimplex is normally considered a two parameter family of polytopes: en.wikipedia.org/wiki/Hypersimplex. Can you say which hypersimplices we get? $\endgroup$ – Sam Hopkins Apr 19 '16 at 12:07
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    $\begingroup$ The one which is the convex hull of the midpoints of the edges of the $(n+1)$-simplex (cf. Gelfand–Manin, reference 1 above). I guess that should be $\Delta_{n+1, 2}$? $\endgroup$ – Piotr Achinger Apr 19 '16 at 18:32
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As pointed out in Piotr Achinger's answer, what follows is just an attempt to solve Exercise IV.2 1(c) (p. 260) in Gelfand & Manin.

Trying to use the above comments by Dylan Wilson, seems like $V_p$ is the convex hull of edge midpoints of $\Delta^{p+1}$.

Its facets are $p+2$ copies of $V_{p-1}$ (convex hulls of edge midpoints of each of the $p+2$ facets of $\Delta^{p+1}$) and $p+2$ copies of $\Delta^p$ (obtained at each of the $p+2$ vertices of $\Delta^{p+1}$ as a result of truncation (of length up to edge midpoints, which is the harshest possible truncation until truncating hyperplanes begin to intersect inside the polyhedron)).

In addition to the facets described in two bullets at the end of the question, there is one more copy of $V_{p-1}$ corresponding to the composable $(p-1)$-tuple$$\operatorname{cone}(f_1)\to\operatorname{cone}(f_2f_1)\to...\to\operatorname{cone}(f_p\cdots f_1),$$and $p+1$ more copies of $\Delta^p$ corresponding to composable $p$-tuples \begin{align*} &\operatorname{cone}(f_1)\to\operatorname{cone}(f_2f_1)\to...\qquad \to\operatorname{cone}(f_{p-1}\cdots f_1)\to\operatorname{cone}(f_p\cdots f_1)\xrightarrow*X_0,\\ &\operatorname{cone}(f_2)\to\operatorname{cone}(f_3f_2)\to...\qquad\qquad\qquad \to\operatorname{cone}(f_p\cdots f_2)\xrightarrow*X_1\to\operatorname{cone}(f_1),\\ &\operatorname{cone}(f_3)\to\operatorname{cone}(f_4f_3)\to...\qquad\qquad\qquad\ \quad \xrightarrow*X_2\to\operatorname{cone}(f_2f_1)\to\operatorname{cone}(f_2),\\ &\vdots\\ &\operatorname{cone}(f_i)\to...\to\operatorname{cone}(f_p\cdots f_i)\xrightarrow*X_{i-1}\to\operatorname{cone}(f_{i-1}\cdots f_1)\to...\to\operatorname{cone}(f_{i-1}),\\ &\vdots\\ &\operatorname{cone}(f_p)\xrightarrow*X_{p-1}\to...\ \to\operatorname{cone}(f_{p-1}f_{p-2}f_{p-3})\to\operatorname{cone}(f_{p-1}f_{p-2})\to\operatorname{cone}(f_{p-1}),\\ &X_p\to\operatorname{cone}(f_p\cdots f_1)\to...\ \quad\to\operatorname{cone}(f_pf_{p-1}f_{p-2})\to\operatorname{cone}(f_pf_{p-1})\to\operatorname{cone}(f_p), \end{align*} where maps with stars denote maps with degree shift by one; that is, $A\xrightarrow*B$ means a map of the form $A\to\Sigma B$ (or else $A\to B[1]$, or $\Omega A\to B$, or...).

Hope the maps and the patterns are clear from the above. If not, tell me, I'll try to formulate more details.

Here is a projection of $V_3$ to the 3-space ($C_{ij}$ stands for $\operatorname{cone}(f_j\cdots f_i)$; red means degree 1)

enter image description here

One can see it inside out on Wikipedia under the name of rectified 5-cell

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