2021/03/14 by Di Liberti, Ivan, González, Julia Ramos
#14A22 #14F06 #18B25 #18C35 #18E10 #18F10 #18M05 #Algebraic Geometry (math.AG) #Category Theory (math.CT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2103.07876
We introduce and describe the 2-category Grt\flat of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories \boxtimes restricts nicely to Grt\flat. Then, we characterize exponentiable objects with respect to \boxtimes: these are continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme X, the category of quasi-coherent sheaves Qcoh(X) is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials.