It is well-known that a countably injective module is Σ-injective. In Proc. Amer. Math. Soc. 316, 10 (2008), 3461-3466, Beidar, Jain and Srivastava extended it and showed that an injective module M is Σ-injective if and only if each essential extension of M(א0) is a direct sum of injective modules. This paper extends and simplifies this result further and shows that an injective module M is Σ-i...

I discuss the general formalism of two-dimensional topological field theories defined on open-closed oriented Riemann surfaces, starting from an extension of Segal’s geometric axioms. Exploiting the topological sewing constraints allows for the identification of the algebraic structure governing such systems. I give a careful treatment of bulk-boundary and boundary-bulk correspondences, which a...

1.1. Injective resolutions. Let C be an abelian category. An object I ∈ C is injective if the functor Hom(−, I) is exact. An injective resolution of an object A ∈ C is an exact sequence 0→ A→ I → I → . . . where I• are injective. We say C has enough injectives if every object has an injective resolution. It is easy to see that this is equivalent to saying every object can be embedded in an inje...

It is no exaggeration to say that the theory of separably injective spaces is quite different from that of injective spaces. In this chapter we will explain why. Indeed, we will enter now in the main topic of the monograph, namely, separably injective spaces and their “universal” version. After giving the main definitions and taking a look at the first natural examples one encounters, we presen...

In this paper, we first present a new type of the concept of open sets by expressing some properties of arbitrary mappings on a power set. With the generalization of the closure spaces in categorical topology, we introduce the generalized topological spaces and the concept of generalized continuity and become familiar with weak and strong structures for generalized topological spaces. Then, int...

Abstract Assume that k is an algebraically closed field and A a finite-dimensional wild -algebra. Recently, L. Gregory M. Prest proved in this case the width of lattice all pointed -modules undefined. Hence result Ziegler implies there exists super-decomposable pure-injective -module, if base countable. Here we give different straightforward proof fact. Namely, show special family -modules, cal...

This paper is a continuation of the paper [4]. In [4] it was shown that there exists a duality Ψa between the category DSkeLC (introduced there) and the category SkeLC of locally compact Hausdorff spaces and continuous skeletal maps. We describe here the subcategories of the category DSkeLC which are dually equivalent to the following eight categories: all of them have as objects the locally co...

It is no exaggeration to say that the theory of separably injective spaces is quite different from that of injective spaces. In this chapter we will explain why. Indeed, we will enter now in the main topic of the monograph, namely, separably injective spaces and their “universal” version. After giving the main definitions and taking a look at the first natural examples one encounters, we presen...

Abstract We show that the image of a subshift X under various injective morphisms symbolic algebraic varieties over monoid universes with variety alphabets is finite type, respectively sofic subshift, if and only so . Similarly, let G be countable A , B Artinian modules ring. prove for every closed submodule $\Sigma \subset A^G$ -equivariant uniformly continuous module homomorphism $\tau \colon...