Elementary equivalences and accessible functors

Elementary Equivalences and Accessible Functors

We introduce the notion of λ-equivalence and λ-embeddings of objects in suitable categories. This notion specializes to L∞λ-equivalence and L∞λ-elementary embedding for categories of structures in a language of arity less than λ, and interacts well with functors and λ-directed colimits. We recover and extend results of Feferman and Eklof on “local functors” without fixing a language in advance....

Dualities and Equivalences Induced by Adjoint Functors

Monoidal Company for Accessible Functors

Distributive laws between functors are a fundamental tool in the theory of coalgebras. In the context of coinduction in complete lattices, they correspond to the so-called compatible functions, which enable enhancements of the coinductive proof technique. Amongst these, the greatest compatible function, called the companion, has recently been shown to satisfy many good properties. Categorically...

Coequational logic for accessible functors

Covarieties of coalgebras are those classes of coalgebras for an endofunctor H on the category of sets that are closed under coproducts, subcoalgebras and quotients. Equivalently, covarieties are classes of H-coalgebras that can be presented by coequations. Adámek introduced a logic of coequations and proved soundness and completeness for all polynomial functors on the category of sets. Here th...

On Stable Equivalences Induced by Exact Functors

Let A and B be two Artin algebras with no semisimple summands. Suppose that there is a stable equivalence α between A and B such that α is induced by exact functors. We present a nice correspondence between indecomposable modules over A and B. As a consequence, we have the following: (1) If A is a self-injective algebra, then so is B; (2) If A and B are finite dimensional algebras over an algeb...