ScalingStacks

The main tool in constructing the 4-manifold invariants $\mathcal{S}^{N}_{0}$ is a family of 4-categories with sufficiently well-behaved duals. This is in analogy with the case of quantum invariants of 3-manifolds, which—in one way or another—all depend on a suitable 3-category, such as the ribbon category $\Rep(U_{q}(\mathfrak{gl}_{N}))$ of finite-dimensional representations of quantum $\mathfrak{gl}_{N}$.

In fact, the 4-categories we construct should be thought of as categorified representation categories¹¹ 1 These are related, but not identical, to categories of higher representations of categorified quantum $\mathfrak{gl}_{N}$. of quantum $\mathfrak{gl}_{N}$. They are defined to have unique 0- and 1-morphisms and

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  2-morphisms are indexed by finite sets of points in a disk,

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  3-morphisms are indexed by tangles in a ball,

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  4-morphisms between two tangles $T_{1}$ and $T_{2}$ are elements of the Khovanov–Rozansky homology $\mathrm{KhR}_{N}(T_{1}\sqcup\overline{T_{2}})$ of the link obtained by reflecting $T_{2}$ and gluing it with $T_{1}$ along their corresponding endpoints.

The various ways of composing $k$-morphisms are purely geometric for $k\leq 3$ and use certain cobordism maps between Khovanov–Rozansky homologies to define composition of $4$-morphisms. We give two constructions of such 4-categories, following the axioms of a disklike 4-category in §5 and of a braided monoidal 2-category in §6.

We invite the reader to use Khovanov–Rozansky link homology to build interesting examples of 4-categories following different axiomatizations, and to explore the appropriate incarnations of the sweep-around property in these settings.