We construct a Goodwillie tower of categories which interpolates between the category pointed spaces and spectra. This refines identity functor in precise sense. More generally, we such for large class ∞ <...

We construct an equivalence between the categories of vertex operator algebras and vertex operator coalgebras. We then investigate to what degree weak modules, generalized modules and ordinary modules carry corresponding comodule structures, as well as when various comodules carry module structure.

We define a notion of weak cubical category, abstracted from the structure of n-cubical cospans x : ∧ → X in a category X, where ∧ is the ‘formal cospan’ category. These diagrams form a cubical set with compositions x +i y in all directions, which are computed using pushouts and behave ‘categorically’ in a weak sense, up to suitable comparisons. Actually, we work with a ‘symmetric cubical struc...

We define a notion of weak cubical category, abstracted from the structure of n-cubical cospans x : ∧ → X in a category X, where ∧ is the ‘formal cospan’ category. These diagrams form a cubical set with compositions x +i y in all directions, which are computed using pushouts and behave ‘categorically’ in a weak sense, up to suitable comparisons. Actually, we work with a ‘symmetric cubical struc...

We investigate the categories of weak maps associated to an algebraic weak factorisation system (awfs) in the sense of Grandis–Tholen [14]. For any awfs on a category with an initial object, cofibrant replacement forms a comonad, and the category of (left) weak maps associated to the awfs is by definition the Kleisli category of this comonad. We exhibit categories of weak maps as a kind of “hom...