Camell Kachour - Globular perspective for Grothendieck ∞-topos and Grothendieck (∞,n)-topos
Channel: Institut des Hautes Études Scientifiques (IHÉS)
Date: 2016-09-15

In this short talk we first briefly recall [4] how to build, for each integers n0, monads Tn on the category Glob of globular sets
which algebras are globular models of (1; n)-categories, which
have the virtue to be weak 1-categories of Penon and thus also to
be weak 1-categories of Batanin (see [2,6]). On the other hand
we are also briefly explain how the difficult problem to prove the
existence of the weak higher category of the weak higher categories on the globular setting can be replaced by a very precise technical problem, on the level of globular operads in Batanin’s sense (see [1,5]). In the conclusion of the thesis [3] we give some general pictures of how to define right and left weak higher adjunctions for weak higher functors, and also weak higher (co)limits for weak 1 higher functors, by using globular operads. According to an easy characterisation of Grothendieck topos (by using presheaves on a small category), we finish our talk by sketching mains tools which permit to build globular models of Grothendieck 1-topos and Grothendieck (1; n)-topos (by using weak higher prestacks on a small category).

References
[1] Michael Batanin, Monoidal globular categories as a natural environment for the theory of weak-n-categories, Advances in Mathematics (1998), Volume 136, Pages 39–103 .

[2] Michael Batanin, On the Penon method of weakening algebraic
structures, Journal of Pure and Applied Algebra (2002), Volume
172 (1), Pages 1–23 .

[3] Camell Kachour, Aspects of Globular Higher Category Theory
,Ph.D Thesis, Macquarie University (2013). 

[4] Camell Kachour, Algebraic Definition of weak (1; n)-Categories,
Published in Theory and Applications of Categories (2015), Volume
30, No. 22, pages 775–807.

[5] Camell Kachour, Steps toward the Weak !-category of the Weak!-categories in the globular setting, Published in Categories and General Algebraic Structures with Applications (2015).

[6] Jacques Penon, Approche polygraphique des 1-catégories non-strictes, Cahiers de Topologie et de Géométrie Différentielle Catégorique (1999), Pages 31–80.