Mathematical Colloquium með Pierre-Louis Curien

Mathematical Colloquium
Fyrirlesari: Pierre-Louis Curien (INRIA)
Dags.: Fimmtudagurinn 20. mars, kl. 13:20
Staðsetning: 158 in VR-II.

Titill: General coherence theorems on CW-complexes and polyhedral complexes

Ágrip: We formulate and prove a coherence theorem on regular CW-complexes: 1-cells determine so-called cellular paths, and the theorem states that,  if each path component of the complex is simply connected,  any two such parallel paths (i.e. with the same end 0-cells) are provably equivalent by repeated transformations along  2-cell (this is in fact an « if and only if »). in other words, continuous homotopy agrees with a discrete  and cellular version of homotopy.

A number of coherence theorems of the literature of category theory follow as a corollary, via a geometrical reading of the relevant data. For example, for monoidal categories (introduced by Mac Lane), the associated polytopes are called associahedra.

But most of those theorems were originally proved in a quite different way, using techniques now well-established in computer science under the name of rewriting systems.

We give a second strictly less general proof of geometrical 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, if we further restrict our attention to a class of polytopes called nestohedra,  that have a nice combinatorial description, and to a certain subclass of those, we show that we get even closer to Mac Lane’s original proof.

(Joint work with Guillaume Laplante-Anfossi)