Mateusz Stroiński: Module categories in the absence of adjunctions

Abstract

This thesis consists of an introduction and three research articles in the field of module categories. In Paper I, we we establish a biequivalence between the bicategory of cyclic module categories, Tambara modules and their morphisms for a fixed monoidal category, and the bicategory of monoids, bimodules and bimodule morphisms in the category of Tambara modules on the same monoidal category. This yields additional functoriality properties to the reconstruction theory of module categories for tensor categories, and gives a weak generalization thereof to the setting of general, in particular non-rigid, monoidal categories. Further, we prove an action-via-enrichment result for Tambara modules, extending the results of Wood into the non-closed setting.

In Paper II, we define a notion of dual objects in semigroup categories. We show that if a semigroup category is rigid, then this semigroup category is promonoidal. We also solve a problem of lifting module categories for semigroup categories, and characterize finite tensor categories in terms of their semigroup categories of projective objects.

In Paper III, we develop a reconstruction theory for abelian module categories over abelian monoidal categories which is very close to the tensor-categorical theory. Rather than (co)monoids in the base monoidal category , we obtain lax module (co)monads on it, generalizing (co)monoids, and we give counter-examples to the existence of a reconstructing monoid. For the monoidal category of comodules over a bialgebra, we show that such comonads are given by Hopf trimodule algebras. This gives categorical proofs of the theorem of Hopf trimodules of Hausser and Nill and the Hopf-monadic theorem of Hopf modules of Bruguières, Lack and Virelizier. Towards these results, we show an Eilenberg-Watts theorem for lax module monads, and extend the formalism of multicategories of Linton coequalizers of Aguiar, Haim and López Franco to the multiactegorical setting.