Séance Séminaire

Séminaire Algèbre Géométrie Algébrique Topologie Algébrique

General coherence theorems on CW-complexes and polyhedral complexes

We formulate and prove a coherence theorem on regular CW-complexes: 1-cells determine cellular paths, and the theorem states that any two such parallel paths (i.e. with the same end 0-cells) are provably equivalent by repetitive discrete transformations along a 2-cell if and only if each path component of the complex is simply connected. A number of coherence theorems of the literature follow as a corollary ((permuto)associahedra for (symmetric) monoidal categories, etc.). The proof is very different from Mac Lane's original proof which uses rewriting, even if the vocabulary of rewriting theory was not available then. It substantiates Kapranov's claim of an «instant one-step proof of MacLane’s theorem».We then give a second strictly less general proof of coherence, applying to polyhedral complexes satisfying a certain condition (which is in particular satisfied by all polytopes), that relies on an orientation given by some generic vector, and that retains most of the features of Mac Lane's original proof. Finally, we shall present a condition on polytopes of a particular kind, the hypergraph polytopes of Došen and Petrić, that allows to retain all the Ingredients of Mac Lane's original proof. (Joint work with Guillaume Laplante-Anfossi)