The notion of a *subtractive category* recently introduced by the
author, is a pointed categorical counterpart of the notion of a
*subtractive variety* of universal algebras in the sense of
A.~Ursini (recall that a variety is subtractive if its theory contains a
constant 0 and a binary term s satisfying s(x,x)=0 and s(x,0)=x).
Let us call a pointed regular category $\mathbb{C}$ *normal* if
every regular epimorphism in $\mathbb{C}$ is a normal epimorphism. It is
well known that any *homological category* in the sense of
F. Borceux and D. Bourn is both normal and subtractive. We prove that in
any subtractive normal category, the upper and lower $3\times 3$ lemmas
hold true, which generalizes a similar result for homological categories
due to D. Bourn (note that the middle $3\times 3$ lemma holds true if and
only if the category is homological). The technique of proof is new: the
pointed subobject functor
$\mathcal{S}=\mathrm{Sub}(-):\mathbb{C}\rightarrow\mathbf{Set}_*$ turns
out to have suitable preservation/reflection properties which allow us to
reduce the proofs of these two diagram lemmas to the standard
diagram-chasing arguments in $\mathbf{Set}_*$ (alternatively, we could use
the more advanced embedding theorem for regular categories due to
M.~Barr). The key property of $\mathcal{S}$, which allows to obtain these
diagram lemmas, is the preservation of *subtractive spans*.
Subtractivity of a span provides a weaker version of the *rule of
subtraction* --- one of the *elementary rules for chasing
diagrams*
in abelian categories, in the sense of S. Mac Lane. A pointed regular
category is subtractive if and only if every span in it is subtractive,
and moreover, the functor $\mathcal{S}$ not only preserves but also
reflects subtractive spans. Thus, subtractivity seems to be exactly what
we need in order to prove the upper/lower $3\times 3$ lemmas in a normal
category. Indeed, we show that a normal category is subtractive if and
only if these $3\times 3$ lemmas hold true in it. Moreover, we show that
for any pointed regular category $\mathbb{C}$ (not necessarily a normal
one), we have: $\mathbb{C}$ is subtractive if and only if the lower
$3\times 3$ lemma holds true in $\mathbb{C}$.

Keywords: subtractive category; normal category; homological category; homological diagram lemmas; diagram chasing

2000 MSC: 18G50, 18C99

*Theory and Applications of Categories,*
Vol. 23, 2010,
No. 11, pp 221-242.

http://www.tac.mta.ca/tac/volumes/23/11/23-11.dvi

http://www.tac.mta.ca/tac/volumes/23/11/23-11.ps

http://www.tac.mta.ca/tac/volumes/23/11/23-11.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/23/11/23-11.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/23/11/23-11.ps