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

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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/