A functorial approach to monomorphism categories for species I

We introduce a very general extension of the monomorphism category as studied by Ringel and Schmidmeier which in particular covers generalized species over locally bounded quivers. We prove that analogues of the kernel and cokernel functor send almost split sequences over the path algebra and the preprojective algebra to split or almost split sequences in the monomorphism category. We derive this from a general result on preservation of almost split morphisms under adjoint functors whose counit is a monomorphism. Despite of its generality, our monomorphism categories still allow for explicit computations as in the case of Ringel and Schmidmeier.

Why does Necker Cube appear as the three-dimensional shape?

Visual perception, receiving a two-dimensional (2D) visual input, often constructs the three-dimensional (3D) perceptual image. Although there are generally multiple structures in the external world that give an equivalent two-dimensional retinal image, the perceptual process naturally and easily infer only one 3D structure as the solution. However, the following problems are not obvious at all: what kind of structure can be obtained as a 3D perceptual image from certain 2D information, and why do we get a three-dimensional perceptual image instead of a two-dimensional one. In the present study, we investigate this problem by untangling the Necker cube phenomenon, and propose a novel theory of three-dimensional visual perception from the viewpoint of the efficiency of information coding. Among the possible structures that can yield the 2D retinal image of the Necker cube, the structure of the typical three-dimensional perceptual image of the Necker cube maximizes the symmetry (in group theory). This maximization of symmetry is characterized by the pairs of adjoint functors (in category theory). Therefore, according to this proposed theory, "the Necker cube" in the three-dimensional space is perceived as the most efficient encoding of the two-dimensional retinal image.

Localization, Isomorphisms and Adjoint Isomorphism in the Category Comp(A - Mod)

A and B are considered to be non necessarily commutative rings and X a complex of (A - B) bimodules. The aim of this paper is to show that: The functors \overline{EXT}^n_{Comp(A-Mod)}(X,-): Comp(A-Mod) \longrightarrow Comp(B-Mod) and Tor_n^{Comp(B-Mod)}(X,-): Comp(B-Mod) \longrightarrow Comp(A-Mod) are adjoint functors. The  functor S_C^{-1}() commute with  the functors X\bigotimes - , Hom^{\bullet}(X,-) and their corresponding derived functors  \overline{EXT}^n_{Comp(A-Mod)}(X,-) and  Tor_n^{Comp(B-Mod)}(X,-).

Internal coalgebras in cocomplete categories: Generalizing the Eilenberg–Watts theorem

The category of internal coalgebras in a cocomplete category [Formula: see text] with respect to a variety [Formula: see text] is equivalent to the category of left adjoint functors from [Formula: see text] to [Formula: see text]. This can be seen best when considering such coalgebras as finite coproduct preserving functors from [Formula: see text], the dual of the Lawvere theory of [Formula: see text], into [Formula: see text]: coalgebras are restrictions of left adjoints and any such left adjoint is the left Kan extension of a coalgebra along the embedding of [Formula: see text] into [Formula: see text]. Since [Formula: see text]-coalgebras in the variety [Formula: see text] for rings [Formula: see text] and [Formula: see text] are nothing but left [Formula: see text]-, right [Formula: see text]-bimodules, the equivalence above generalizes the Eilenberg–Watts theorem and all its previous generalizations. By generalizing and strengthening Bergman’s completeness result for categories of internal coalgebras in varieties, we also prove that the category of coalgebras in a locally presentable category [Formula: see text] is locally presentable and comonadic over [Formula: see text] and, hence, complete in particular. We show, moreover, that Freyd’s canonical constructions of internal coalgebras in a variety define left adjoint functors. Special instances of the respective right adjoints appear in various algebraic contexts and, in the case where [Formula: see text] is a commutative variety, are coreflectors from the category [Formula: see text] into [Formula: see text].

Cotorsion pairs and adjoint functors in the homotopy category of N-complexes

In this paper, we first construct some complete cotorsion pairs on the category [Formula: see text] of unbounded [Formula: see text]-complexes of Grothendieck category [Formula: see text], from two given cotorsion pairs in [Formula: see text]. Next, as an application, we focus on particular homotopy categories and the existence of adjoint functors between them. These are an [Formula: see text]-complex version of the results that were shown by Neeman in the category of ordinary complexes.

Homological dimensions of skew group algebras

Let [Formula: see text] be a finite dimensional algebra over a field [Formula: see text] and [Formula: see text] a subgroup of a finite group [Formula: see text]. In this paper, we consider the Gorenstein global dimensions and representation dimensions of the skew group algebras [Formula: see text] and [Formula: see text]. Under the assumption that [Formula: see text] is a separable extension over [Formula: see text], we show that [Formula: see text] and [Formula: see text] share the same Gorenstein global dimensions and representation dimensions. As an application, we give an affirmative answer for a conjecture raised in [Adjoint functors and representation dimensions, Acta Math. Sinica 22(2) (2006) 625–640]. Several known results are then obtained as corollaries.

A LOGICAL AND ALGEBRAIC CHARACTERIZATION OF ADJUNCTIONS BETWEEN GENERALIZED QUASI-VARIETIES

We present a logical and algebraic description of right adjoint functors between generalized quasi-varieties, inspired by the work of McKenzie on category equivalence. This result is achieved by developing a correspondence between the concept of adjunction and a new notion of translation between relative equational consequences.