2 new papers on higher category theory

[Mark.Weber@pps.jussieu.fr]

Greetings.

The following 2 papers

Algebras of higher operads as enriched categories II (joint with Michael
Batanin and Denis-Charles Cisinski)
http://arxiv1.library.cornell.edu/abs/0909.4715

Abstract:  One of the open problems in higher category theory is the
systematic construction of the higher dimensional analogues of the Gray
tensor product. In this paper we continue the work of [2] to adapt the
machinery of globular operads [1] to this task. The resulting theory
includes the Gray tensor product of 2-categories and the Crans tensor
product [4] of Gray categories. Moreover much of the previous work on the
globular approach to higher category theory is simplified by our new
foundations, and we illustrate this by giving an expedited account of many
aspects of Cheng's analysis [3] of Trimble's definition of weak
n-category. By way of application we obtain an ``Ekmann-Hilton'' result
for braided monoidal 2-categories, and give the construction of a tensor
product of A-infinity algebras.

Free Products of Higher Operad Algebras
http://arxiv1.library.cornell.edu/abs/0909.4722

Abstract:  In this paper we continue the developments of [2] and [5] by
understanding the natural generalisations of Gray's little brother, the
funny tensor product of categories. In fact we exhibit for any higher
categorical structure definable by an n-operad in the sense of Batanin
[1], an analogous tensor product which forms a symmetric monoidal closed
structure on the category of algebras of the operad.

[1] M. Batanin. Monoidal globular categories as a natural environment for
the theory of weak n-categories. Advances in Mathematics, 136:39–103,
1998.
[2] M. Batanin and M. Weber. Algebras of higher operads as enriched
categories. Applied Categorical Structures, 38 pages, 2008.
[3] E. Cheng. Comparing operadic theories of n-category, preprint, 2008.
[4] S. Crans. A tensor product for Gray categories. Theory and
applications of categories, 5:12–69, 1999.
[5] M. Batanin, D-C. Cisinski, and M. Weber. Algebras of higher operads as
enriched categories II, preprint 57 pages, 2009.

have recently been completed and are available at the above URL's or my
new home-page

http://sites.google.com/site/markwebersmaths/

Mark Weber



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