Star-multiplicative graphs in pointed protomodular categories
Protomodularity, in the pointed case, is equivalent to the Split
Short Five Lemma. In this work we combine this and two other conditions on a
category B, in order to study and caracterize the relations between several
internal categorical structures in B, such as: groupoids, categories,
multiplicative graphs, star-multiplicative graphs and reflexive graphs.
As it is well known, Split Short Five Lemma implies that every internal
category is in fact an internal groupoid. Also the known condition that
every split epi is jointly epic (i.e., assuming pointedness and kernels of
split epis, every pair (k,s), with k a kernel and s a section of a split
epi, is jointly epic), implies that a multiplicative graph is in fact an
The novelty of this work is a condition, called the Kernel Reflected
Admissibility Property, which implies that every star-multiplicative graph
is in fact a multiplicative graph, as it is in the case, for example, in
Groups and Rings.
When combined, these three conditions provide a simple description for the
category of internal groupoids in B; this description is very close to the
categorical notion of crossed module.
Nelson Martins-Ferreira (CDRSP/IPLeiria)
February 10, 2009