Categorical Interactions in Algebra, Geometry and Physics: Cubical Structures and Truncations There are several interactions between algebra and geometry coming from polytopic complexes as for instance demonstrated by several versions of Deligne's conjecture. These are related through blow-ups or truncations. The polytopes and their truncations also appear naturally as regions of integration for products, which is an area of active study. Two fundamental polytopes are cubes and simplices. The importance of cubes as a basic appears naturally in various situations on which we will concentrate. In particular, we will discuss cubical Feynman categories, which afford a W-construction that is a cubical complex. These relate combinatorics to geometry. Furthermore using categorical notions of push-forwards, we show how to naturally construction Moduli Spaces of curves and several of their compactifications. The combinatorial ingredients are graphs and there is a universal way of decorating them to study different types. This makes the theory applicable to several different geometries appearing in Moduli Spaces and Outer space. With respect to physics, there is an additional relationship coming through Hopf algebras which in turn also are related to multiple zeta values. We will discuss these constructions and relations on concrete examples.