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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]