The Structure of Endomorphism Monoids in Conjugate Categories

Document Type

Article

Journal Title

International Journal of Algebra

Publication Date

2009

Abstract

We invoke structure theorems from the theory of semigroups to analyze the anatomy of categories arising from a type of conjugation by a subcategory. The equivalence classes of Green’s H-relation admit group structures when they contain idempotents, and we use this to study the endomorphism monoids of the objects in a conjugate category. We prove that in any such category, the H-group associated to an idempotent endomorphism is always realizable as the automorphism group of its “rank object,” which serves as a generalized image. As all H-groups are automorphism groups of lower rank, we establish a resulting homogeneity of these H-groups, giving stringent necessary conditions for the existence of conjugate categories.

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