Return to Question

Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call geometric morphisms open by analogy to an open map. Similarly is the case for closed geometric morphisms, $T_0$ geometric embeddings etc. Considering the lack of adjoints, is there a way to make this process of generalization precise? Maybe a relative adjoint to $Sh \circ Op$, that would cover this process for large class of geometric morphism properties?

Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call geometric morphisms open by analogy to an open map. Similar are the cases for closed geometric morphisms, $T_0$ geometric embeddings etc. Considering the lack of adjoints, is there a way to make this process of generalization precise? Maybe a relative adjoint to $Sh \circ Op$, that would cover this process for large class of geometric morphism properties?

Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call geometric morphisms open by analogy to an open map, and that geometric morphisms $Sh(Op(X)) \rightarrow Sh(Op(Y))$ coming fro are proper precisely when the map $X \rightarrow Y$ is proper. Similarly is the case for closed geometric morphisms, $T_0$ geometric embeddings etc. Considering the lack of adjoints, is there a way to make this process of generalization precise? Maybe a relative adjoint to $Sh \circ Op$, that would cover this process for large class of geometric morphisms properties?

Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call geometric morphisms open by analogy to an open map. Similarly is the case for closed geometric morphisms, $T_0$ geometric embeddings etc. Considering the lack of adjoints, is there a way to make this process of generalization precise? Maybe a relative adjoint to $Sh \circ Op$, that would cover this process for large class of geometric morphism properties?

Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$ does not have any adjoints as discussed here on MO before. Yet we call geometric morphisms open by analogy to an open map, and that geometric morphisms $Sh(Op(X)) \rightarrow Sh(Op(Y))$ coming fro are proper precisely when the map $X \rightarrow Y$ is proper. Similarly is the case for closed geometric morphisms, $T\_0$ geometric embeddings etc. Considering the lack of adjoints, is there a way to make this process of generalization precise? Maybe a relative adjoint to $Sh \circ Op$, that would cover this process for large class of geometric morphisms properties?

Let's note that the bifunctor $Sh \circ Op : Top \rightarrow Topos$, taking Topos as Grothendieck topoi + geometric morphisms, does not have any adjoints as discussed here on MO before. Yet we call geometric morphisms open by analogy to an open map, and that geometric morphisms $Sh(Op(X)) \rightarrow Sh(Op(Y))$ coming fro are proper precisely when the map $X \rightarrow Y$ is proper. Similarly is the case for closed geometric morphisms, $T_0$ geometric embeddings etc. Considering the lack of adjoints, is there a way to make this process of generalization precise? Maybe a relative adjoint to $Sh \circ Op$, that would cover this process for large class of geometric morphisms properties?