The pointed subobject functor, 3 x 3 lemmas, and subtractivity of spans

Zurab Janelidze

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.

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