## Some geometry of 2-categorical k-tensor calculus

### Sjoerd Crans – 11 February 1998

I outlined a generalization of Joyal and Street's ``The geometry of tensor calculus, I''. Recall definition of semistrict braided monoidal 2-category, with strict functoriality. Undecided about whether braiding is equivalence or iso. A few words about generalized 2-graphs. Progressiveness will mean we will only have planes, edges and nodes. Progressive polarized boxed anchored 2-graphs: embed in 4D box, with outer edges and nodes on appropriate faces. Free braided monoidal 2-categories on what? Not on 2-computads, as that cannot be represented for computational purposes, but on ``2-tensor 2-schemes'', which involve words of words (Ross suggested ``braided rewrite systems''). Objects of free are deformation classes of points on line, where one can deform as long as no singularities are introduced. Arrows are deformation classes of string diagrams without singularities, where a singularity is, e.g., a node on top of a crossing, or a triple crossing. 2-arrows are deformation classes of progressive polarized boxed anchored 2-graphs, which I interpreted as movies of string diagrams where some (i.e., a finite number) of the frames may have one singularity. Deformations of these may have at most two singularities, or one singularity of degree 2, e.g., a quadruple crossing. More in a future talk.

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