Classifying diagrams of double categories with shared isomorphisms

Abstract: The classifying diagram construction, originally due to Rezk [4], provides a way to produce a complete Segal space from an ordinary category, in a way that refines the usual nerve construction. In particular, two categories are equivalent if and only if their classifying diagrams are levelwise equivalent as simplicial spaces. The key feature of the classifying diagram, compared to the ordinary nerve, is that it provides a way to distinguish isomorphisms from ordinary morphisms in the category.

To make sense of an analogous classifying diagram for a double category, it turns out that we need the horizontal and vertical categories to have the same isomorphisms. Making the necessary conditions precise, we arrive at the definition of double categories with shared isomorphisms, which have interesting features in their own right. In joint work with Shapiro and Zakharevich, we identify when a simplicial space is the classifying diagram of a category, and when a bisimplicial space is the classifying diagram of a double category with shared isomorphisms, in terms of various lifting conditions [2].

Our motivation for defining classifying diagrams for double categories was when we surprisingly found them to provide a useful characterization of which pointed stable double categories in the sense of [1] correspond to CGW categories in the sense of [3], thus providing the relationship between two general inputs for algebraic K-theory constructions.