Related papers: Relatively divisible and relatively flat objects i…

We develop a general theory of partial morphisms in additive exact categories which extends the model theoretic notion introduced by Ziegler in the particular case of pure-exact sequences in the category of modules over a ring. We relate…

In an abelian category $\mathscr{A}$ with small ${\rm Ext}$ groups, we show that there exists a one-to-one correspondence between any two of the following: balanced pairs, subfunctors $\mathcal{F}$ of ${\rm Ext}^{1}_{\mathscr{A}}(-,-)$ such…

In this paper, by using functor rings and functor categories, we study finiteness and purity of subcategories of the module categories. We give a characterisation of contravariantly finite resolving subcategories of the module category of…

We identify two categories of locally compact objects on an exact category A. They correspond to the well-known constructions of the Beilinson category lim A and the Kato category k(A). We study their mutual relations and compare the two…

Drinfeld recently suggested to replace projective modules by the flat Mittag--Leffler ones in the definition of an infinite dimensional vector bundle on a scheme $X$. Two questions arise: (1) What is the structure of the class $\mathcal D$…

We develop techniques for constructing model structures on chain complexes valued in accessible exact categories, and apply this to show that for a closed symmetric monoidal, locally presentable exact category $\mathpzc{E}$ with exact…

We give a complete list of indecomposable exact module categories over the finite tensor category $\mathrm{Rep}(u_q(\mathfrak{sl}_2))$ of representations of the small quantum group $u_q(\mathfrak{sl}_2)$, where $q$ is a root of unity of odd…

If $X$ is a quasi-compact and quasi-separated scheme, the category $Qcoh(X)$ of quasi-coherent sheaves on $X$ is locally finitely presented. Therefore categorical flat quasi-coherent sheaves naturally arise. But there is also the standard…

We extend all known results about transferred model structures on algebraically cofibrant and fibrant objects by working with weak model categories. We show that for an accessible weak model category there are always Quillen equivalent…

We start with observing that the only connected finite dimensional algebras with finitely many isomorphism classes of indecomposable bimodules are the quotients of the path algebras of uniformly oriented $A_n$-quivers modulo the radical…

An important example of a model category is the category of unbounded chain complexes of R-modules, which has as its homotopy category the derived category of the ring R. This example shows that traditional homological algebra is…

We study properties of relative modular categories and derive sufficient conditions for their existence. In particular, we derive sufficient conditions for relative pre-modular categories to be non-degenerate and relative modular, and for…

A locally coherent exact category is a finitely accessible additive category endowed with an exact structure in which the admissible short exact sequences are the directed colimits of admissible short exact sequences of finitely presentable…

We introduce the notion of exact tilting objects, which are partial tilting objects $T$ inducing an equivalence between the abelian category generated by $T$ and the category of modules over the endomorphism algebra of $T$. Given a chain of…

We give a construction of triangulated categories as quotients of exact categories where the subclass of objects sent to zero is defined by a triple of functors. This includes the cases of homotopy and stable module categories. These…

In all $\kappa$-accessible additive categories, $\kappa$-pure monomorphisms and $\kappa$-pure epimorphisms are well-behaved, as shown in our previous paper arXiv:2311.02418. This is known to be not always true in $\kappa$-accessible…

We use the category of linear complexes of tilting modules for the BGG category O, associated with a semi-simple complex finite-dimensional Lie algebra g, to reprove in purely algebraic way several known results about O obtained earlier by…

This is an exposition of homotopical results on the geometric realization of semi-simplicial spaces. We then use these to derive basic foundational results about classifying spaces of topological categories, possibly without units. The…

We give a characterization of the sets of objects of the derived category of a block of a finite group algebra (or other symmetric algebra) that occur as the set of images of simple modules under an equivalence of derived categories. We…

We introduce the new concept of cartesian module over a pseudofunctor $R$ from a small category to the category of small preadditive categories. Already the case when $R$ is a (strict) functor taking values in the category of commutative…