A functorial approach to monomorphism categories for species I

Abstract

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.

Multi-layer quivers and higher slice algebras

“…For the case of n-properly-graded algebra, such sequences can either be regarded as pairs (M • , g) [0,n+1] of certain complex M • of length n + 2 concentrated in the interval [0, n + 1] in the category of complexes C(Λ) of Λ-modules with a map g its end term, or as pairs of representation ( M , g) of the the quiver A n+1 of type A n+1 with linear order over the algebra Λ with a Λ-map from the last one to the first one. The representation theory of Λ and Γ are related to the morphism categories Mor n (Λ) and Mor n (Γ) (see [33,10]).…”

“…We show that their representation theory are related using the representation theory of diagram of quivers. The diagrams of a small categories are used recently in representation theory and related researches [7,8,9,10,30]. Using the multilayer quiver, we can describe the (n + 1)-slice algebra Γ as a triangulate matrix algebra with diagonal entries in Γ (Theorem 6.4), and as a tensor algebra of certain bimodules over Γ n+1 (direct sum of n + 1 copies of Γ) (Theorem 6.5).…”

Representations over diagrams of categories and abelian model structures

“…2.4 Example. Gao, Külshammer, Kvamme and Psaroudakis introduced in [28] the notion of a phylum on a quiver Q = (Q 0 , Q 1 ) as an extension of Gabriel's notion of species [27]. Recall from [28,Definition 4.1] that a phylum U on Q consists of the following data:…”

“…We notice that D-Rep unifies many special cases such as comma categories, module categories of Morita context rings, categories of additive functors from I to an abelian category (which are called representations of I by representation theorists), categories of representations of (generalised) species and phyla studied in e.g. [15,16,27,28,29,46,47]; see Examples 2.2 and 2.4. Another example comes from a recent work by Estrada and Virili [26].…”

A functorial approach to monomorphism categories for species I

“…In a second paper in preparation [GKKP19] we will provide an analogue of the explicit description of the Auslander-Reiten translation obtained in [RS08] for the monomorphism category S(A) to the subcategory G P proj(C T (X) ) using an explicit description of the (minimal) right approximation with respect to the subcategory G P proj(C T (X) ).…”