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...

The notion of commutative monad was denned by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the frontand end-adjunctions are closed tran...

Let $G$ be a $p$-adic reductive group and $\mathfrak{g}$ its Lie algebra. We construct functor from the extension closure of Bernstein-Gelfand-Gelfand category $\mathcal{O}$ associated to into locally analytic representations $G$, thereby expanding on an earlier construction Orlik-Strauch. A key role in this new is played by logarithms tori. This shown exact with image subcategory admissible se...

In [6, Theorem 2.2] Doi gave a Hopf-algebraic proof of a generalization of Oberst’s theorem on affine quotients of affine schemes. He considered a commutative Hopf algebra H over a field, coacting on a commutative H-comodule algebra A. If AcoH denotes the subalgebra of coinvariant elements of A and β : A ⊗AcoH A −→ A ⊗H the canonical map, he proved that the following are equivalent: (a) AcoH ⊂ ...

We show that the quotient of a Hom-finite triangulated category C by the kernel of the functor HomC(T, −), where T is a rigid object, is preabelian. We further show that the class of regular morphisms in the quotient admit a calculus of left and right fractions. It follows that the Gabriel-Zisman localisation of the quotient at the class of regular morphisms is abelian. We show that it is equiv...

We provide explicit constructions for various ingredients of right exact monoidal structures on the category finitely presented functors. As our main tool, we prove a multilinear version universal property so-called Freyd categories, which in turn is used proof correctness constructions. Furthermore, compare construction with Day convolution arbitrary additive always yields closed structure all...

We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic information underlying classical moduli space. The main tool in our construction is representability theorem geometry, which has been established previous work. provides us with sufficient and necessary conditions for an functor to possess structure stack. In order verify ...

such that C and D are compact objects in S (an object X in S is compact if the representable functor Hom(X,−) preserves arbitrary coproducts). The concept of a coherent functor has been introduced explicitly for abelian categories by Auslander [1], but it is also implicit in the work of Freyd on stable homotopy [9]. In this paper we characterize coherent functors in a number of ways and use the...

Abstract Let $\textsf{T}$ be a triangulated category with shift functor $\Sigma \colon \textsf{T} \to \textsf{T}$ . Suppose $(\textsf{A},\textsf{B})$ is co-t-structure coheart $\textsf{S} = \Sigma \textsf{A} \cap \textsf{B}$ and extended $\textsf{C} \Sigma^2 \textsf{B} \textsf{S}* \textsf{S}$ , which an extriangulated category. We show that there bijection between co-t-structures $(\textsf{A}^{...

The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom-Lie algebras, we describe the notion and some properties of Homalgebras and provide examples. We introduce Hom-coalgebr...