2019/03/05 by Krantz, Thomas H. M.
#Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.02021
We explain two related constructions on the data of two monoidal symmetric closed categories \mathscrA and \mathscrE and monoidal functors F: \mathscrE→ \mathscrA and G: \mathscrA→ \mathscrE. In a first part, we recall and partly extend work of A. Kock: In case F is left-adjoint to G, and this adjunction is monoidal, we can equip the Eilenberg-Moore category \mathscrET for T being the canonical monad associated to the adjunction, with the structure of symmetric monoidal closed category, provided \mathscrE has equalizers and \mathscrET co-equalizers. In a second part, inspired by the Chu-construction, we build a category \mathscrRG, which is symmetric monoidal closed as well, under the condition that \mathscrE has pullbacks. Similarly we build a category \mathscrLF which is symmetric monoidal closed under the condition that \mathscrA has what we call F-pushouts and F-pullbacks. In case F \dashv G is a monoidal adjunction, we show that \mathscrLF and \mathscrRG are isomorphic as symmetric monoidal closed categories. We show also how \mathscrET is related to both.