We prove that the opposite of the category of coalgebras for the Vietoris endofunctor on the category of compact Hausdorff spaces is monadic over $\mathsf {Set}$. We deliver an analogous result for the upper, lower, and convex Vietoris endofunctors acting on the category of stably compact spaces. We provide axiomatizations of the associated (infinitary) varieties. This can be seen as a version of Jónsson–Tarski duality for modal algebras beyond the zero-dimensional setting.

C-systems were defined by Cartmell as models of generalized algebraic theories. B-systems were defined by Voevodsky in his quest to formulate and prove an initiality conjecture for type theories. They play a crucial role in Voevodsky’s construction of a syntactic C-system from a term monad. In this work, we construct an equivalence between the category of C-systems and the category of B-systems, thus proving a conjecture by Voevodsky.

Using Butz and Moerdijk's topological groupoid representation of a topos with enough points, a ‘syntax-semantics’ duality for geometric theories is constructed. The emphasis is on a logical presentation, starting with a description of the semantic topological groupoid of models and isomorphisms of a theory. It is then shown how to extract a theory from equivariant sheaves on a topological groupoid in such a way that the result is a contravariant adjunction between theories and groupoids, the restriction of which is a (...) duality between theories with enough models and semantic groupoids. Technically a variant of the syntax-semantics duality constructed in 1 for first-order logic, the construction here works for arbitrary geometric theories and uses a slice construction on the side of groupoids—reflecting the use of ‘indexed’ models in the representation theorem—which in several respects simplifies the construction and the characterization of semantic groupoids. (shrink)