We show that, in any Mal'tsev (and a fortiori protomodular) category
**E**, not only the fibre *Grd*_X **E** of internal groupoids
above the object *X* is a naturally Mal'tsev category, but moreover
it shares with the category *Ab* of abelian groups the property
following which the domain of any split epimorphism is isomorphic with the
direct sum of its codomain with its kernel. This allows us to point at a
new class of ``non-pointed additive'' categories which is necessarily
protomodular. Actually this even gives rise to a larger classification
table of non-pointed additive categories which gradually take place
between the class of naturally Mal'tsev categories and the one of
essentially affine categories. As an application, when furthermore the
ground category **E** is efficiently regular, we get a new way to
produce Baer sums in the fibres *Grd*_X **E** and, more generally,
in the fibres *n-Grd*_X **E**.

Keywords: Mal'tsev, protomodular, naturally Mal'tsev categories; internal group; Baer sum; long cohomology sequence

2000 MSC: 18E05,18E10, 18G60, 18C99, 08B05

*Theory and Applications of Categories,*
Vol. 20, 2008,
No. 4, pp 48-73.

http://www.tac.mta.ca/tac/volumes/20/4/20-04.dvi

http://www.tac.mta.ca/tac/volumes/20/4/20-04.ps

http://www.tac.mta.ca/tac/volumes/20/4/20-04.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/4/20-04.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/20/4/20-04.ps