This paper aims to develop the theory of duals for Gray categories. The principal tool is the authors’ diagrammatic calculus, which can be viewed as a higher-categorical three-dimensional analogue of the diagrams used for computations in pivotal categories. Many of the algebraic results on Gray categories with duals can be understood in terms of the geometry of the corresponding diagrams.
The synopsis of the paper goes as follows.

§ 2

introduces diagrams for Gray categories without duals, which are a generalization of the diagrammic calculus for braided monoidal categories [A. Joyal andR. Street, Adv. Math. 88, No. 1, 55–112 (1991;Zbl 0738.18005)]. A braided monoidal category is to be viewed as a Gray category with a single object and 1-morphism. The evaluation of diagrams for braided monoidal categories is generalized to the evaluation of Gray category diagrams. It is shown (Theorem 2.32) that
Theorem. Let \(D\), \(D^{\prime}\)be generic Gray category diagrams that are isotopic by a one-parameter family of isomorphisms of progressive diagrams. Then the evaluations of \(D\)and \(D^{\prime}\)are equal.

§ 3

introduces Gray categories with duals, using the definition ofJ. C. Baez andL. Langford [Adv. Math. 180, No. 2, 705–764 (2003;Zbl 1039.57016)], but with some minor modifications. The Gray categories possess two types of duals, \(\ast\)and \(\#\), which correspond to 180 degree rotations around two distinct coordinate axes. The \(\ast\)-duals are familiar from pivotal or ribbon categories.

§ 4

is concerned with the algebraic structure of the duality operations, establishing the first main result (Theorem 4.3, Lemma 4.4 and Theorem 4.5).
Theorem. The duals extend in a canonical way to (partially contravariant) functors of 2-strict tricategories\[\ast,\#:\mathcal{G}\rightarrow\mathcal{G}\]with \(\ast\ast=1\), defining natural isomorphisms\begin{align*}\Gamma & :\ast\#\ast\#\rightarrow1\\\Theta & :\#\#\rightarrow1\end{align*}

§ 5

establishes the second main result (Theorems 5.2 and 5.3), which is a strictification theorem for the duals.
Theorem. Every spatial Gray category with duals can be strictified to a Gray category whose duals\[\ast,\#:\mathcal{G}\rightarrow\mathcal{G}\]abide by\begin{align*}\ast\ast & =1\\\#\# & =1\\\ast\#\ast\# & =1\end{align*}

§ 6

explores in more depth the relation between Gray categories with duals and their diagrams. The main theorem (Theorem 6.9) claims that the evaluations of standard surface diagrams are invariant under a set of moves that are the PL counterparts of the moves induced by projecting an isotropy in the smooth setting. Under the conjecture (Conjecture 6.8) that these are also all the moves arising from projecting PL isotropies, it implies that oriented isomorphisms of standard surface diagrams leave their evaluations invariant.

Appendix A

defines functors of strict tricategories and their natural transformations and modifications by specializing the standard definitions for functors of (strict) 2-categories [G. M. Kelly andR. Street, Lect. Notes Math. 420, 75–103 (1974;Zbl 0334.18016);T. Leinster, “Basic bicategories”, Preprint,arXiv:math/9810017].

Reviewer: Hirokazu Nishimura (Tsukuba)

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