Let $mathcal{C}$ be a triangulated category. We first introduce the notion of balanced pairs in $mathcal{C}$, and then establish the bijective correspondence between balanced pairs and proper classes $xi$ with enough $xi$-projectives and enough $xi$-
injectives. Assume that $xi:=xi_{mathcal{X}}=xi^{mathcal{Y}}$ is the proper class induced by a balanced pair $(mathcal{X},mathcal{Y})$. We prove that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is an extriangulated category. Moreover, it is proved that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is a triangulated category if and only if $mathcal{X}=mathcal{Y}=0$; and that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is an exact category if and only if $mathcal{X}=mathcal{Y}=mathcal{C}$. As an application, we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.
Herschend-Liu-Nakaoka introduced the notion of $n$-exangulated categories. It is not only a higher dimensional analogue of extriangulated categories defined by Nakaoka-Palu, but also gives a simultaneous generalization of $n$-exact categories and $(n
+2)$-angulated categories. In this article, we give an $n$-exangulated version of Auslanders defect and Auslander-Reiten duality formula. Moreover, we also give a classification of substructures (=closed subbifunctors) of a given skeletally small $n$-exangulated category by using the category of defects.
In this paper, we first provide an explicit procedure to glue complete hereditary cotorsion pairs along the recollement $(mathcal{A},mathcal{C},mathcal{B})$ of abelian categories with enough projective and injective objects. As a consequence, we inve
stigate how to establish recollements of triangulated categories from recollements of abelian categories by using the theory of exact model structures. Finally, we give applications to contraderived categories, projective stable derived categories and stable categories of Gorenstein injective modules over an upper triangular matrix ring.
It is proved that a finite intersection of special preenveloping ideals in an exact category $({mathcal A}; {mathcal E})$ is a special preenveloping ideal. Dually, a finite intersection of special precovering ideals is a special precovering ideal. A
counterexample of Happel and Unger shows that the analogous statement about special preenveloping subcategories does not hold in classical approximation theory. If the exact category has exact coproducts, resp., exact products, these results extend to intersections of infinite families of special peenveloping, resp., special precovering, ideals. These techniques yield the Bongartz-Eklof-Trlifaj Lemma: if $a colon A to B$ is a morphism in ${mathcal A},$ then the ideal $a^{perp}$ is special preenveloping. This is an ideal version of the Eklof-Trlifaj Lemma, but the proof is based on that of Bongartz Lemma. The main consequence is that the ideal cotorsion pair generated by a small ideal is complete.
Let $T$ be a right exact functor from an abelian category $mathscr{B}$ into another abelian category $mathscr{A}$. Then there exists a functor ${bf p}$ from the product category $mathscr{A}timesmathscr{B}$ to the comma category $(Tdownarrowmathscr{A}
)$. In this paper, we study the property of the extension closure of some classes of objects in $(Tdownarrowmathscr{A})$, the exactness of the functor ${bf p}$ and the detail description of orthogonal classes of a given class ${bf p}(mathcal{X},mathcal{Y})$ in $(Tdownarrowmathscr{A})$. Moreover, we characterize when special precovering classes in abelian categories $mathscr{A}$ and $mathscr{B}$ can induce special precovering classes in $(Tdownarrowmathscr{A})$. As an application, we prove that under suitable cases, the class of Gorenstein projective left $Lambda$-modules over a triangular matrix ring $Lambda=left(begin{smallmatrix}R & M O & S end{smallmatrix} right)$ is special precovering if and only if both the classes of Gorenstein projective left $R$-modules and left $S$-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them.
Let $T=left( begin{array}{cc} R & M 0 & S end{array} right) $ be a triangular matrix ring with $R$ and $S$ rings and $_RM_S$ an $R$-$S$-bimodule. We describe Gorenstein projective modules over $T$. In particular, we refine a result of Enoch
s, Cort{e}s-Izurdiaga and Torrecillas [Gorenstein conditions over triangular matrix rings, J. Pure Appl. Algebra 218 (2014), no. 8, 1544-1554]. Also, we consider when the recollement of $mathbb{D}^b(T{text-} Mod)$ restricts to a recollement of its subcategory $mathbb{D}^b(T{text-} Mod)_{fgp}$ consisting of complexes with finite Gorenstein projective dimension. As applications, we obtain recollements of the stable category $underline{T{text-} GProj}$ and recollements of the Gorenstein defect category $mathbb{D}_{def}(T{text-} Mod)$.
