For a self-small abelian group A of torsion-free rank 1, we characterize A-reflexive abelian groups which are induced by the contravariant functor Hom(−, A) in two cases: the range of Hom(−, A) is the category of all abelian groups, respectively the range of Hom(−, A) is the category of all left E-modules, where E is the endomorphism ring of A.

In this paper the notion of Rees short exact sequence for S-posets is introduced, and we investigate the conditions for which these sequences are left or right split. Unlike the case for S-acts, being right split does not imply left split. Furthermore, we present equivalent conditions of a right S-poset P for the functor Hom(P;-) to be exact.

5 Mapping supermanifolds and the functor of points formalism 31 5.1 Internal hom objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 5.2 The functor of points formalism . . . . . . . . . . . . . . . . . . . . . . . . . 34 5.3 The supermanifold of endomorphisms of R0|1 . . . . . . . . . . . . . . . . . . 36 5.4 Vector bundles on supermanifolds . . . . . . . . . . . . . . . ....

Let H be a Hopf algebra over a field k, and A an Hcomodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives.

Modules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract framework to explore the theory of structures. In this book we generalize and redevelop the basic notions and results of category theory using this framework of modules. Topics Chapter 1 introduces modules and cells among them. A moduleM ∶ X⇢A from a...

Let $${\mathcal {C}}$$ be an n-angulated category. We prove that its idempotent completion $$\widetilde{{\mathcal {C}}}$$ admits a unique structure such the canonical functor $$\iota : {\mathcal {C}}\rightarrow \widetilde{{\mathcal is n-angulated. Moreover, $$ induces equivalence $$Hom _{n-ang }(\widetilde{{\mathcal {C}}},{\mathcal {D}})\cong Hom }({\mathcal {C}},{\mathcal {D}})$$ for any c...

First of all, we find some further properties of the characterization of fiber product preserving bundle functors on the category of all fibered manifolds in terms of an infinite sequence A of Weil algebras and a double sequenceH of their homomorphisms from [5]. Then we introduce the concept of Weilian prolongation WA HS of a smooth category S over N and of its action D. We deduce that the func...

We prove existence of equalizers in certain categories of cocomplete cocategories. This allows us to complete the proof of the fact that A∞-functor categories arise as internal Hom-objects in the category of differential graded cocomplete augmented cocategories.