Tamari's diagram

Scans of Fig. 1 in Tamari's 1951 thesis MR 51833 can be found here:

• Tamari memorial Festschrift 2012 MR 3235205:
  • Loday 2012 MR 3221534 (page 74)
  • Stasheff 2012 MR 3221533 (page 46) [see image]
• Stasheff 2019 MR 3939050 (page 94)
  • arXiv:1809.02526v2 [math.QA] (page 4)

Stasheff, Jim. How I ‘met’ Dov Tamari. Associahedra, Tamari lattices and related structures, 45-63. Progr. Math., 299, 2012. MR 3221533.

Here is a TeX/TikZ emulation:

[high resolution image]

The following colored variant has reversed arrows and perspective. It illustrates the strong deformation retractions

ℳ₄ → ℳ₃ → ℳ₂ → ℳ₁

The embeddings

ℳ₁ → ℳ₂ → ℳ₃ → ℳ₄

are given by prepending "1" to the labels. The green/red/blue arrows indicate the homotopies. In terms of the cubical view, these retract the cuboid linearly to a side.

[high resolution image]

The next variant shows the duals. The dual (of the boundary) of the associahedron is a simplicial complex build from non-crossing polygon diagonals. It is a triangulation of the sphere. See

• Lee, Carl W. The associahedron and triangulations of the n-gon. European J. Combin. 10 (1989), no. 6, 551--560. MR 1022776.

[high resolution image]

Source code

• TeX/TikZ code of the first diagram
• See also: Section Tikz Source of File:Associahedron K5 back.svg (Wikimedia Commons)

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