内部关系IV的封闭性：用减法表示范畴的可加性
Closedness properties of internal relations IV: Expressing additivity of a category via subtractivity
摘要：
作者最近引入的减范畴的概念是泛代数的（点）减泛代数簇概念的“范畴版本”，这是由于A.乌尔西尼我们证明了一个减法簇$\C$，其理论包含一个唯一常数，是阿贝尔的（即$\C$是一个固定环上的模簇），当且仅当$\C$的对偶范畴$\C^\mathrm{op}$是减法的。更一般地说，我们证明了$\C$是加性的当且仅当$\C$和$\C^\mathrm{op}$都是减性的，其中$\C$是一个任意的带二进制和的完全点范畴，并且使得$\C$中的每个态射$f$可以表示为复合$f=me$，其中$m$是单态射，$e$是满态射。
英文摘要：
The notion of a subtractive category, recently introduced by the author, is a ``categorical version'' of the notion of a (pointed) subtractive variety of universal algebras, due to A. Ursini. We show that a subtractive variety $\C$, whose theory contains a unique constant, is abelian (i.e. $\C$ is the variety of modules over a fixed ring), if and only if the dual category $\C^\mathrm{op}$ of $\C$, is subtractive. More generally, we show that $\C$ is additive if and only if both $\C$ and $\C^\mathrm{op}$ are subtractive, where $\C$ is an arbitrary finitely complete pointed category, with binary sums, and such that each morphism $f$ in $\C$ can be presented as a composite $f=me$, where $m$ is a monomorphism and $e$ is an epimorphism.
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