Endomorphism category of an abelian category

ABSTRACT

Let 𝒞 be an additive category. Denote by End(𝒞) the endomorphism category of 𝒞, i.e., the objects in End(𝒞) are pairs (C,c) with C∈𝒞,c∈End_𝒞(C), and a morphism f:(C,c)→(D,d) is a morphism f∈Hom_𝒞(C,D) satisfying fc = df. This paper is devoted to an approach of the general theory of the endomorphism category of an arbitrary additive category. It is proved that the endomorphism category of an abelian category is again abelian with an induced structure without nontrivial projective or injective objects. Furthermore, the endomorphism category of any nontrivial abelian category is nonsemisimple and of infinite representation type. As an application, we show that two unital rings are Morita equivalent if and only if the endomorphism categories of their module categories are equivalent.

KEYWORDS:

• Abelian category
• additive category
• endomorphism category
• Morita equivalence
• triangulated category

2010 Mathematics Subject Classification:

• Primary: 16E65
• Secondary: 16G10
• 16G50

Acknowledgment

The third author is very grateful to Professor Steffen König for his hospitality and many helpful suggestions during the visit of University of Stuttgart.