Quasi-uniform structures and functors

We study a number of categorical quasi-uniform structures induced by functors. We depart from a category $\mathcal{C}$ with a proper $(\mathcal{E}, \mathcal{M})$-factorization system, then define the continuity of a $\mathcal{C}$-morphism with respect to two syntopogenous structures (in particular with respect to two quasi-uniformities) on $\mathcal{C}$ and use it to describe the quasi-uniformities induced by pointed and copointed endofunctors of $\mathcal{C}$. In particular, we demonstrate that every quasi-uniformity on a reflective subcategory of $\mathcal{C}$ can be lifted to a coarsest quasi-uniformity on $\mathcal{C}$ for which every reflection morphism is continuous. Thinking of categories supplied with quasi-uniformities as large ``spaces'', we generalize the continuity of $\mathcal{C}$-morphisms (with respect to a quasi-uniformity) to functors. We prove that for an $\mathcal{M}$-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.