Topological functors and right adjoints

Abstract

Let T:$\text{A}$ → $\text{L}$ be an ($\text{L}$, $\text{M}$)-topological functor and S:$\text{B}$ → $\text{Y}$ a faithful functor. Let F:$\text{L}$ → $\text{Y}$ and L:$\text{A}$ → $\text{B}$ be functors with a:FT → SL an epi natural transformation. We are concerned with the question of when L has a right adjoint given that F has a right adjoint. We give two characterizations of the existence of a right adjoint to L. One involves just the “topological data” and the other is an application of Freyd's adjoint functor theorem. As a consequence, we characterize when a category which is monoidal and ($\text{L}$, $\text{M}$)-topological over a monoidal closed category is also closed.