Have you ever watched an Olivetti talk and immediately gone, “huh, what if I formally dualized all of that?” Well, it’s your lucky day! In this talk, we will discuss cogroups and their role in homotopy theory. We will ask:

• “if cohomology comes from groups, does homology come from cogroups?” (no)
• “is the notion of an infinite suspension space as interesting as the notion of infinite loop space?” (definitely not)
• “is there some more convenient setting for homotopy theory - perhaps related to the notion of a chain complex - which does satisfy a strong notion of duality, making it more amenable to computational techniques and offering what appears to be the most promising route towards progress on longstanding open problems?” (actually, yes!)