We study a number of categorical quasi-uniform structures induced by functors. We depart from a category C with a proper (E,M)-factorization system, then define the continuity of a C-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) on C and use it to describe the quasi-uniformities induced by pointed and copointed endofunctors of C. In particular, we demonstrate that every quasi-uniformity on a reflective subcategory of C can be lifted to a coarsest quasi-uniformity on C for which every reflection morphism is continuous.
Thinking of categories supplied with quasi-uniformities as large "spaces", we generalize the continuity of C-morphisms (with respect to a quasi-uniformity) to functors. We prove that for an M-fibration or a functor that has a right adjoint, we can obtain a concrete construction of the coarsest quasi-uniformity for which the functor is continuous. The results proved are shown to yield those obtained for categorical closure operators. Various examples considered at the end of the paper illustrate our results.
Keywords: Closure operator, Syntopogenous structure, Quasi-uniform structure, (co)pointed endofunctor and Adjoint functor
2020 MSC: 18A05, 18F60, 54A15, 54B30
Theory and Applications of Categories, Vol. 39, 2023, No. 17, pp 519-534.