On the equivalence between the existence of $n$-kernels and $n$-cokernels

TL;DR

This paper proves that in certain categories, the existence of n-kernels is equivalent to n-cokernels, simplifying the understanding of their structure and implications for module categories.

Contribution

It provides an elementary proof of the equivalence between n-kernels and n-cokernels in idempotent complete preadditive categories with weak kernels and cokernels.

Findings

Equivalence of n-kernels and n-cokernels under specified conditions

Elementary proofs of global dimension equalities for module categories

Abstract

We give an elementary proof of the statement that if an idempotent complete preadditive category has weak kernels and weak cokernels, then it has $n$-kernels if and only if it has $n$-cokernels, where $n$ is a nonnegative integer. As a consequence, elementary proofs of two results concerning the equality between the global dimensions of certain right and left module categories are obtained.