L'incontournabilité des teisi

(The unavoidability of teisi)

Sjoerd Crans, Séminaire Itinérant de Catégories, Amiens, 10 November 2001

(This is joint work with Clemens Berger)
(This is work in progress)

Everything starts from the two monoidal structures on Cat: the cartesian structure and the funny structure, which are both closed, the first one giving rise to natural transformations and the second one to not necessarily natural transformations. Enriching, one obtains 2-categories and sesqui-categories respectively. On the category of 2-categories there is still the cartesian monoidal structure and on the category of sesqui-categories there is still the funny structure, etc., then in higher dimensions there are omega

-categories and NI-categories (``no interchange'' categories).

Let Glob be the category of globular sets, and let N and M be the monads on Glob such that their algebras are NI-categories and omega

- categories respectively. There is a surjection of monads N - >> M (because each omega

- category is a NI-category and each horizontal composition can be written as a vertical composition, perhaps in several ways).

N1 -> M1 is a surjection of M-collections, and N1 is an M-operad. The idea is to construct F as ``the smallest contractible M-collection containing N1'' (after the talk I realized that an additional condition on F will be needed) and G as ``the greatest quotient of F which is an M-operad''.

N1 -->> M1
v       ^
|       |
|       |
v       |
F --->> G

The G-algebras should be the teisi.

Definition An n-stage tree is m-sprout if

for all i m there is exactly one vertex at height i;
for all i > m there is at most one input edge for every vertex at height i.

Definition Let T be an n-stage tree which is m-sprout. The tas-dimension of T is given by

td (T) = ( Sigma _{v input vertex of T} (ht (v) - m - 1) ) + m + 1 .

The 0-sprout trees in A of dimension 0 form a globular set, the 0-sheets of dimension 0 in A.
given the globular set S^A_{k, m-1} of m-sheets of dimension k' for (m', k') in the region
```
 (m,k) 
   |
   v
    -----
   .|   |
----    |
\       |
 \      |
  \     |
   \    |
    \   |
     \  |   k
      \ |  ^
       \|  |
     <-
    m
```
an m-sheet of dimension k in A is a diagram of trees over S^A_{k, m-1} where the base tree T_m is m-sprout of tas-dimension k and on the input vertices of T_m one has (m-1)-sheets.
The m'-sheets of dimension k' for k' < k or k' = k and m' m form a globular set.

(FA)_n = { m-sheets of dimension n in A }.

One has FA -> M...MA -> MA so F1 is an M-collection.

Conjecture F1 est contractible.

Conjecture F1 is the smallest contractible M-collection containing N1.

To define a multiplication on F it suffices to consider m-sprouts of m'-sprouts for m'

m, but this multiplication is not associative. So, one defines G as the greatest quotient of F such that the multiplication becomes associative.

Conjecture G est contractible.

Now there are two possibilities:

either every weak n-category is equivalent to a G-algebra, and teisi are obviously unavoidable,
or G is too small and one has only that every weak n-category is equivalent to a G'-algebra for G' an M-operad different from G, hence it will be necessary to compare G' and G, and teisi are still unavoidable.

A more extensive abstract of this talk is available as a ps-file en français

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