Enriched coverages and sheaves under change of base
Ariel E. Rosenfield
TL;DR
This work analyzes how changing the enriching base along a faithful, conservative right adjoint $G$ interacts with enriched coverages and sheaves on a small $\mathcal{V}$-category. It develops a base-change framework using $G_*$ and $\widetilde{G}$, proving injective correspondences on subobjects and on enriched coverages, and showing commutativity of the enriched associated-sheaf construction under fullness of $G$. The authors establish that $\mathcal{W}$-coverages form a complete lattice and that base change preserves/refines coverages, with detailed examples including Gabriel topologies and Lawvere metric-space settings. They also reveal that faithfulness is essential for injectivity in base-change phenomena, via graded Gabriel-topology counterexamples, and connect these results to localizations of presheaf categories and enriched sheaf theory. Overall, the paper provides tools for transferring enriched topological structures across base-change functors and clarifies the role of faithfulness in preserving finer enrichment data.
Abstract
We investigate how change of enriching base category via a faithful, conservative right adjoint functor interacts with enriched coverages and sheaves on a given enriched category. We prove that change of base via such a functor gives rise both to an injective mapping on subobjects in enriched presheaf categories, and to an injective mapping on enriched coverages. In case the base change functor is also full, the enriched associated sheaf construction on a presheaf category commutes with base change.