Tamari's diagram

A TeX/TikZ diagram after Fig. 1 in Tamari's 1951 thesis MR 51833:

[higher 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.

[higher resolution image]

The TeX/TikZ code of these diagrams is partially based on section Tikz Source of File:Associahedron K5 front.svg (Wikimedia Commons).

Scans of the original diagram can be found here:

• Tamari memorial Festschrift 2012 MR 3235205:
  • Loday 2012 MR 3221534 (page 74)
  • Stasheff 2012 MR 3221533 (page 46) [see below]
• 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.

Go to: Notes on the associator · Publications and Preprints · Markus Rost's Web Page