My general research is in algebraic topology. I am particulary interested in stable homotopy theory, in which I study spectra with additional structures. Homotopy theorists have generalized the notion of rings with multiplication that are associative and/or commutative up to higher homotopies. These objects are called E_1 or E_∞ ring spectra. There is a natural dual notion of coalgebras, which are spectra or modules X, in which, instead of an associative unital multiplication X ⊗ X → X, we require a co-unital comultiplication X → X ⊗ X, which is coassociative up to higher homotopies. These are called E_1-coalgebras. If we require cocommutativity to be up to higher homotopies, we call them E_∞-coalgebras. There is also a notion of comodules over a coalgebra, dual to modules over an algebra. In my research, I use both model categories and quasicategories (i.e. ∞-categories) to describe the homotopy theory of coalgebras and comodules.
Publications & Preprints
The Coalgebraic Enrichment of Algebras in Higher Categories, (to appear in JPAA).
Rigidification of Connective Comodules, (submitted).
Highly Structured Coalgebras and Comodules, PhD thesis (May 2020).
Coalgebras in symmetric monoidal categories of spectra, with Brooke Shipley, Homology, Homotopy and Applications 21 (2019), no.1, 1-18.