论文标题
重新审视阿贝尔类别中极限和界限的精确性
Exactness of limits and colimits in abelian categories revisited
论文作者
论文摘要
令$σ$为小型类别，$ \ Mathcal {a} $为$σ$ -co-complete（resp。$σ$ -COMPLETE）ABELIAN类别。众所周知的事实是，$ \ operatorname {fun}（σ，\ nathcal {a}）$ $ \ mathcal {a} $ in $σ$的函数是ABELIAN类别，并且是函数$ \ MATHSF {COLIM}_σ（ - ）：\ operatotorName {fun}（σ，\ Mathcal {a}）\ rightarrow \ Mathcal \ Mathcal {a} $（resp。 $ \ MATHSF {lim}_σ（ - ）：\ operatorName {fun}（σ，\ Mathcal {a}）\ rightArrow \ rightarrow \ Mathcal {a} $）留下（resp。右） $κ^σ：\ MATHCAL {a} \ rightArrow \ propatatorName {fun}（σ，\ Mathcal {a}）$，其中$κ^σ$是关联的常数函数。在本文中，我们将证明函数$ \ mathsf {colim}_σ（ - ）$（resp。$ \ \ \ \ \ \ \ \ \ \ \}_σ（ - ）$）是确切的，并且仅当且仅当一对fuctionors $ \ left（\ mathsf {colim}_σ（colim}_σ（ - ），κ^σ\ fightort $ \ left（\ mathsf {colim}），κ^σ\ first）。 $ \ left（κ^σ，\ mathsf {lim}_σ（ - ）\ right）$）是Ext-Adjoint。为了应用我们的发现，我们将提供有关阿伯利亚类别中极限和限制的确切性结果的新证据。
Let $Σ$ be a small category and $\mathcal{A}$ be a $Σ$-co-complete (resp. $Σ$-complete) abelian category. It is a well-known fact that the category $\operatorname{Fun}(Σ,\mathcal{A})$ of functors of $Σ$ in $\mathcal{A}$ is an abelian category, and that the functor $\mathsf{colim}_Σ(-):\operatorname{Fun}(Σ,\mathcal{A})\rightarrow\mathcal{A}$ (resp. $\mathsf{lim}_Σ(-):\operatorname{Fun}(Σ,\mathcal{A})\rightarrow\mathcal{A}$) is left (resp. right) adjoint to $κ^Σ:\mathcal{A}\rightarrow\operatorname{Fun}(Σ,\mathcal{A})$, where $κ^Σ$ is the associated constant diagram functor. In this paper we will show that the functor $\mathsf{colim}_Σ(-)$ (resp. $\mathsf{lim}_Σ(-)$) is exact if and only if the pair of functors $\left(\mathsf{colim}_Σ(-),κ^Σ\right)$ (resp. $\left(κ^Σ,\mathsf{lim}_Σ(-)\right)$) 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.
