During the feudal period all work was confined within the home (farm) and entertainment, if it occurred at all, was strictly within certain buildings outside of the home. And then there was a transition during and after industrialization which led to the separation of work from home. Work occurred outside of the home and entertainment gradually became confined to the living room. We can see here an inverse operation: if, at one time, entertainment was extrinsic to the outside area of the home (e.g., it was outside the home but still exclusive to a particular building and largely hidden away), then, at another time, entertainment was intrinsic to the home and was placed at the center (e.g., typically around the television set). We have today an entirely new situation: everything today, work and entertainment, is brought together in the potency of the pure outside. Thus, we have moved from a world where set theory made sense (e.g., the focus on intrinsic functions and objects: the home had within it an excessive part which we named the living room, and so on) to a world where category theory makes sense (e.g., the focus on extrinsic relationships where objects are nothing more than identity relations: the home is a machine-assemblage which is connected absolutely to everything else, including, for example, the cafe.) To visualize this latest development you can simply point to social media: I can check the cafe hours and who has ‘checked in’ from any location.
What the feudal and industrial capitalist logic required was a theory of the empty set (“there exists a set such that no set is a member of it”). Thus, Zermelo set theory was decades ahead of its time and posed a serious challenge to capitalist society, the consequences of which are still to be understood. What contemporary society requires however is not category theory which simply provides us with the syntax of neoliberalism but rather a syntax of our fidelity to the Zermelo axiom of the empty set. I’m not sure what this means precisely but I know that it means that we can not return simply to set theory to solve all of the problems of category theory, or the recent push toward the outside. We need something different if we are going to understand why cloud computing, airbnb, uber, social media, and so on, are today a part of the basic paradigm of communicative capitalism.