Infinity Links L, infinity-4-Manifolds M_L and Kirby Categories

Abstract

We construct what we call a Kirby category, a monoidal category whose morphisms are smooth 4-manifolds, projecting down to another monoidal category whose morphisms are orientable 3-manifolds, the projection being induced by the boundary map on manifolds. We construct a higher categorical generalization of such concepts and introduce the notion of ribbon $\infty$-categories, a generalization of braided monoidal $\infty$-categories (\cite{Lu1}), which gives rise to the concepts of $\infty$-links, $\infty$-4-manifolds as well as the more general notion of walled $\infty$-4-manifolds if one focuses attention on $\infty$-4-manifolds built from gluing thickened sheets on ribbons. These fall into a larger class of constrained $\infty$-4-manifolds whose classical 4-dimensional counterparts are constrained 4-manifolds on which we consider physical theories. We regard pairs of constrained 4-manifolds and Lagrangians densities depicting physical theories defined on such spaces as morphism objects in an enhanced Kirby category, whose objects are regarded as events. We define a universal category $\Lambda$ of all events that we relate to the $\infty$-category of ribbon $\infty$-categories and conclude in part that Lagrangian field theories can be superseded by using $\infty$-categories.