We study the notion of Morita equivalence in various categories. We start with Morita equivalence and Morita duality in pure algebra. Then we consider strong Morita equivalence for C-algebras and Morita equivalence for W-algebras. Finally, we look at the corresponding notions for groupoids (with structure) and Poisson manifolds.

let $h$ be a hopf algebra and $a$ an $h$-bimodule algebra‎. ‎in this paper‎, ‎we investigate gorenstein global dimensions for hopf‎ ‎algebras and twisted smash product algebras $astar h$‎. ‎results from‎ ‎the literature are generalized‎.

We investigate when an exact functor F ∼= − ⊗Λ MΓ : mod-Λ → mod-Γ which induces a stable equivalence is part of a stable equivalence of Morita type. If Λ and Γ are finite dimensional algebras over a field k whose semisimple quotients are separable, we give a necessary and sufficient condition for this to be the case. This generalizes a result of Rickard’s for self-injective algebras. As a corol...

Morita equivalence is a widely used tool for rings with identity. (Two rings are said to be Morita equivalent if the categories of unitary modules over them are equivalent.) For monoids, this notion is not really useful: in most cases it reduces to isomorphism. As the theory of Morita equivalence could be developed for the more general case of rings with local units, and then for idempotent rin...

For a fixed inverse semigroup S, there are two natural categories of left actions of S: the category Fact of unitary actions of S on sets X meaning actions where SX = X, and the category Étale of étale actions meaning those unitary actions equipped with a function p : X → E(S), to the set of idempotents of S, such that p(x)x = x and p(sx) = ses∗, where s∗ denotes the inverse of s. The category ...

In the representation theory of finite groups, the stable equivalence of Morita type plays an important role. For general finite-dimensional algebras, this notion is still of particular interest. However, except for the class of self-injective algebras, one does not know much on the existence of such equivalences between two finite-dimensional algebras; in fact, even a nontrivial example is not...

We characterize abstract Morita contexts in several bicategories. In particular, we use heteromorphisms for the bicategory Prof of categories and profunctors and coreflective subcategories for Cat (categories and functors). In addition, we prove general statements concerning strict Morita contexts, and we give new equivalent forms to the standard notions of adjointness, category equivalence and...

We introduce a natural notion of strong Morita equivalence of twisted actions of a locally compact group on C*-algebras, and then show that the corresponding twisted crossed products are strongly Morita equivalent. This result is a generalization of the result of Curto, Muhly and Williams concerning strong Morita equivalence of crossed products by actions.