In higher algebra, we study algebraic objects endowed with a multiplication that is associative only up to (coherent) homotopy, or commutative up to (coherent) homotopy. In this Brave new algebra, we study algebras and modules that includes the classical theory of algebra. The ground ring is not the ring of integer anymore, it is the sphere spectrum. Rigidification results (or sometimes called rectification) state that some of these highly coherent algebras over some rings can have their multiplication rigidified into a strictly associative multiplication. This has been used in many instances using the tool of model categories. In fact, in the 90s, many symmetric monoidal model categories of spectra were introduced such that strictly associative associative algebras were representing A_\infty-algebras, and similarly E_\infty-algebras.
Thursday, September 24, 2020 - 4:30pm
In this talk, we will explore the dual algebraic objects of coalgebras and comodules in higher algebra. Instead of a multiplication, we have a comultiplication\coaction that we now require to be co-associative up to higher homotopy. I will show that higher algebras are enriched over higher coalgebras and thus, coalgebras provide insight on the structure for algebras. However, we will see that these objects are much more mysterious than algebras. I will show that none of the current monoidal model categories of spectra represent well the higher coalgebras in spectra. This is will hint that the correct language to study higher coalgebra is infinity-categories. I will also show that it is challenging but possible to rigidify coaction of comodules when using connective spectra over a field. This result allows to define a derived cotensor product of comodules which has not been possible before.