The general goal is the computation of diagrams and the analysis of multi linearity of morphisms between algebraic structures. Specifically, an “algebraic structure” is a category equipped with one or two operations making it into the analog of a group, abelian group, or a ring. Morphisms between algebraic structures are functors between their underlying categories respecting the operations. There are well-known machines turning these structures into diagrams of spaces, or combinatorial versions thereof, which are therefore amenable to be investigated using tools from homotopy theory. A specific topic would be to use these classifying diagrams to study the algebraic K-theory of “rig” categories and their 2-categories of modules.