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.

Balanced pairs appear naturally in the realm of relative homological algebra associated with the balance of right-derived functors of the Hom functor. Cotorsion triplets are a natural source of such pairs. In this paper, we study the connection between balanced pairs and cotorsion triplets by using recent quiver representation techniques. In doing so, we find a new characterization of abelian categories that have enough projectives and injectives in terms of the existence of complete hereditary cotorsion triplets. We also provide a short proof of the lack of balance for derived functors of Hom computed using flat resolutions, which extends the one given by Enochs in the commutative case.

To a Lie groupoid over a compact base $M$, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing $M$). Moreover, it gives rise to an adjunction between the category of Lie groupoids over $M$ and the category of Lie groups acting on $M$. In the last section we then show how to promote this adjunction to almost an equivalence of categories.

Recently we have introduced an enriched cohomology theory for categories that are tripleable (algebraic) over a category of modules. The cohomology admits a circle product, related to the obstruction problem for algebraic deformations, making the total complex a graded ring. We here offer similar constructions in two other situations - coalgebraic and bialgebraic categories. Examples include categories of bialgebras, sheaves of modules, and sheaves of algebras over a sheaf of rings.

Let be the category of all homomorphisms (i.e. functions preserving satisfaction of atomic formulas) between models of a set of sentences T in a finitary first-order language L. Functors between two such categories are said to be canonical if they commute with the forgetful functors. The following properties are characterized syntactically and also in terms of closure of for some algebraic constructions (involving products, equalizers, factorizations and kernel pairs): There is a canonical isomorphism from to a variety (resp. quasivariety) in a finitary expansion of L which assigns to a model its (unique) expansion. This solves a problem of H. Volger.

In the case of a purely algebraic language, the properties are equivalent to:“ is canonically isomorphic to a finitary variety (resp. quasivariety)” and, for the variety case, to “the forgetful functor of is monadic (tripleable)”.