Lander Hermans: Virtual double categories as coloured box operads

24th of October, 2024 ——— Virtual double categories are a 2-categorification of multicategories and compare to double categories as multicategories compare to monoidal categories. In algebraic topology, multicategories are also known as coloured operads and are extensively used to encode algebraic operations, thus generalizing operads. By viewing virtual double categories as coloured versions of box operads, we shift our point of view: from objects of study to algebraic gadgets encoding higher operations. This is exemplified by our main application: we present a box operad Lax encoding lax functors from U to Cat(k) into the category of k-linear categories. They appear in algebraic geometry as prestacks generalizing structure sheaves and (noncommutative) deformations thereof. In the second part of the talk, I will sketch key components of our main result: a Koszul duality for box operads. For example, to every box operad we can associate a canonical L_\infty-algebra. A salient feature is that these results can be explained purely in terms of (virtual) double categorical diagrams, or in our terms, stackings of boxes. If time permits, I will explain how it applies to Lax in order to tackle their deformation and homotopy theory.