This is an on-going project and the story goes like that. We know several different approaches to introduce co-categories of a space, which could not apparently be the same. These notions are closely related to the desuspension problem of a suspended space. In this talk, we show how we are trying to use the co-$A_\infty$ structure of a suspension space to desuspend. At this moment, we are facing a problem in a process to connect to the Mike Hopkins co-category, which is caused by the existence of a homotopy unit, and so we are led to use a version of Muro-Tonks unital associahedra.