Mathematics

Certain classical cohomology theories, such as group or (variants of) Hochschild cohomology, classify homotopy types in low degrees. Furthermore, morphisms between homotopy types in low degrees can be efficiently computed by way of special diagrams called Butterflies, owing to their shape. In a geometric context, instead of homotopy types we consider stacks equipped with monoidal or, in more recent developments, bimonoidal structures. I will survey the main points and some of the applications of the theory in the former case first, and then to discuss the latter case of stacks which are categorical rings (ring-like, for want of a better name). Ring-like stacks are related to truncated cotangent complexes, and connections to Shukla, MacLane cohomology of rings, and more generally functor cohomology, can be found. These cohomology objects provide one of the main motivations for looking into this subject and they will be the starting point for our discussion.