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 $\mathbb{C}$.
