Sheaves on semicartesian monoidal categories and applications in the quantalic case - Tenorio, Ana Luiza da Conceição (2023)
In this doctoral thesis, we introduce the notion of Grothendieck prelopologies, which is a notion of covering for semicartesian monoidal categories that generalizes Grothendieck pretopologies. Moreover, this generalization encompasses a certain notion of sheaves in semicartesian quantales Q, introduced in this thesis, which is more general than the usual definition of sheaves on locales L. We observe that the respective sheaf categories, Sh(Q) and Sh(L), share certain properties; however, Sh(Q) does not always form a Grothendieck topos. The analysis of the lattice of subobjects of the terminal sheaf in Sh(Q) suggests that the notion of sheaves for Grothendieck prelopologies has a linear internal logic rather than an intuitionistic one. Furthermore, we develop a Cech cohomology in which the coefficients are sheaves on a quantale, and we find a morphism between the locale of open sets of a topological space X and the quantale of ideals of the ring C(X) of continuous functions on that allows us to relate the Cech cohomology of X and the (expanded) Cech cohomology of C(X).