Outdated Archival Version

These pages are not updated anymore. They reflect the state of 6 June 2010. For the current production of this journal, please refer to http://www.springer.com/mathematics/geometry/journal/40062.

Closedness properties of internal relations IV: Expressing additivity of a category via subtractivity

Zurab Janelidze

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.

Journal of Homotopy and Related Structures, Vol. 1(2006), No. 1, pp. 219-227

http://jhrs.rmi.acnet.ge/volumes/2006/n1a10/