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 $\mathrm{Fun}(\Sigma ,\mathcal{A})$ of functors of $\Sigma$ in $\mathcal{A}$ is an abelian category, and that the functor ${\mathsf{colim}}_{\Sigma }(-):\mathrm{Fun}(\Sigma ,\mathcal{A})\to \mathcal{A}$ (resp. ${\mathsf{lim}}_{\Sigma }(-):\mathrm{Fun}(\Sigma ,\mathcal{A})\to \mathcal{A}$) is left (resp. right) adjoint to ${\kappa }^{\Sigma }:\mathcal{A}\to \mathrm{Fun}(\Sigma ,\mathcal{A})$, 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.

arXiv comment16 pages