re: Semi-strict n-categories

[Mark Weber <mark.weber.math@googlemail.com>]

Dear Jamie,

An inductive definition of semi-strict n-category in which semi-strict
(n+1)-categories are categories enriched in a category of semi-strict
n-categories equipped with an appropriate monoidal structure is too much to
hope for by a heuristic argument given by Crans in [1]. More recently,
Bourke and Gurski described a more precise obstruction to this naive scheme
in [2]. Thus present work on semi strict n-categories is directed at
setting up some variation of this naive scheme. For instance, instead of
insisting on a tensor product of semi strict n-categories which forms part
of an honest monoidal structure, one can widen the search to include lax
monoidal structures. One could perhaps also put the tensor product on a
category whose objects are semi strict n-categories, but the morphisms are
weaker.

In the setting of the higher operads of Batanin, the problem of defining
semi strict n-categories is that of describing for each n the n-operad
"S_n" whose algebras are those structures. There is a useful interplay
between the theory of n-operads and that of lax monoidal structures which
is described in [3] from which any inductive definition scheme of the sort
that we wish to identify for semi-strict n-categories ought to arise. Using
that interplay
one can give such an inductive definition scheme for weak n-categories with
strict units as explained in [4]. The n-operads that describe n-dimensional
structures with strict units, called "reduced n-operads", are better
behaved algebraically than general n-operads. Since all the weakness of a
semi strict n-category will be
concentrated in the interchangers, S_n will be reduced, and so a part of
the search for S_n is to develop further the theory of reduced n-operads.

Taking the Gray tensor product of 2-categories and "ignoring everything
going on in dimension 2" gives the "funny tensor product" on Cat. As
explained in [5] one has an analogous tensor product for the algebras of
any nice enough n-operad. In particular reduced n-operads are nice enough,
and moreover in this case one has comparison maps from the funny tensor
product to the
cartesian product. In the case of 2-categories, the Gray tensor product can
be recovered by factoring these canonical comparisons.

As I'm sure you know, the closed structure on 2-Cat corresponding to the
Gray tensor product involves *pseudo* natural transformations -- i.e.
"Hom(A,B)" is the 2-category of 2-functors, pseudo nats and modifications
between A and B. Thus a true understanding of semi-strict n-categories
should include their relationship with weak higher morphisms. Garner showed
us how to define weak morphisms of higher categories in a general way in
[6] and some first steps in defining and organising weak higher
transformations
operadically were taken by Kachour in [7] and [8]. The thinking on the homs
to be associated with higher Gray tensor products is inspired very much by
work on the Deligne conjecture. See Tamarkin [9] and Batanin and Markl [10]
and especially example 82 of this last article.

[1] S. Crans. A tensor product for Gray categories. TAC 5:12–69 1999.
[2] J. Bourke and N. Gurski. A cocategorical obstruction to tensor products
of gray-categories. ArXiv:1412.1320.
[3] M. Weber, Multitensors as monads on categories of enriched graphs. TAC
28:857-932 2013.
[4] M. Batanin, D-C. Cisinski and M. Weber, Multitensor lifting and
strictly unital higher category theory. TAC 28:804-856 2013.
[5] M. Weber, Free products of higher operad algebras. TAC 28:24-65 2013.
[6] R. Garner, Homomorphisms of higher categories. Advances in Mathematics
224(6):2269–
2311, 2010.
[7] C. Kachour. Operadic definition of non-strict cells. ArXiv:1007.1077
and ArXiv:1211.2314.
[8] C. Kachour. ω-Operads of coendomorphisms for higher structures.
ArXiv:1211.2310.
[9] D.E. Tamarkin. What to dg-categories form? Compositio 143:1335–1358,
2007.
[10] M. Batanin and M. Markl. Centres and homotopy centres in enriched
monoidal categories. Advances in Mathematics 230:1811–1858, 2012.


With best regards,

Mark Weber


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