Exactness of limits and colimits in abelian categories revisited

Abstract

Let $\Sigma$ be a small category and $\mathcal{A}$ be a $\Sigma$-co-complete (resp. $\Sigma$-complete) abelian category. It is a well-known fact that the category $\operatorname{Fun}(\Sigma,\mathcal{A})$ of functors of $\Sigma$ in $\mathcal{A}$ is an abelian category, and that the functor $\mathsf{colim}_\Sigma(-):\operatorname{Fun}(\Sigma,\mathcal{A})\rightarrow\mathcal{A}$ (resp. $\mathsf{lim}_{\Sigma}(-):\operatorname{Fun}(\Sigma,\mathcal{A})\rightarrow\mathcal{A}$) is left (resp. right) adjoint to $\kappa^{\Sigma}:\mathcal{A}\rightarrow\operatorname{Fun}(\Sigma,\mathcal{A})$, where $\kappa^{\Sigma}$ is the associated constant diagram functor. In this paper we will show that the functor $\mathsf{colim}_\Sigma(-)$ (resp. $\mathsf{lim}_{\Sigma}(-)$) is exact if and only if the pair of functors $\left(\mathsf{colim}_\Sigma(-),\kappa^{\Sigma}\right)$ (resp. $\left(\kappa^{\Sigma},\mathsf{lim}_{\Sigma}(-)\right)$) is Ext-adjoint. As an application of our findings, we will give new proofs of known results on the exactness of limits and colimits in abelian categories.