It is well-known that the construction of Zariski spectra (commutative) rings yields a dual adjunction between category and locally ringed spaces. There are many constructions algebras in various contexts giving such adjunctions. Michel Coste unified them language categorical logic by showing that, for an appropriate triple $ (T_0,T,\Lambda) (which we call spatial context), each T_0 $-model can...

We study lax epimorphisms in 2-categories, with special attention to Cat and V-Cat. show that any 2-category convenient colimits has an orthogonal LaxEpi-factorization system, we give a concrete description of this factorization Cat.

We give a sharper version of a theorem of Rosický, Trnková and Adámek [12], and a new proof of a theorem of Rosický [13], both about colimits in categories of structures. Unlike the original proofs, which use category-theoretic methods, we use set-theoretic arguments involving elementary embeddings given by large cardinals such as α-strongly compact and C(n)extendible cardinals.

The paper is in essence a survey of categories having φ-weighted colimits for all the weights φ in some class Φ. We introduce the class Φ of Φ-flat weights which are those ψ for which ψ-colimits commute in the base V with limits having weights in Φ; and the class Φ− of Φ-atomic weights, which are those ψ for which ψ-limits commute in the base V with colimits having weights in Φ. We show that bo...

This paper studies colimits of sequences of finite Chu spaces and their ramifications. Besides generic Chu spaces, we consider extensional and biextensional variants. In the corresponding categories we first characterize the monics and then the existence (or the lack thereof) of the desired colimits. In each case, we provide a characterization of the finite objects in terms of monomorphisms/inj...

For deterministic systems, expressed as coalgebras over polynomial functors, every tree t (an element of the final coalgebra) turns out to represent a new coalgebra At. The universal property of these coalgebras, resembling freeness, is that for every state s of every system S there exists a unique coalgebra homomorphism from a unique At which takes the root of t to s. Moreover, the tree coalge...