#
Star-multiplicative graphs in pointed protomodular categories

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N. Martins-Ferreira

Protomodularity, in the pointed case, is equivalent to the Split Short
Five Lemma. It is also well known that this condition implies that every
internal category is in fact an internal groupoid. In this work, this is
condition (II) and we introduce two other conditions denoted (I) and
(III). Under condition (I), every multiplicative graph is an internal
category. Under condition (III), every star-multiplicative graph can be
extended (uniquely) to a multiplicative graph, a problem raised by G.
Janelidze in the semiabelian context.

When the three conditions hold, internal groupoids have a simple
description, that, in the semiabelian context, correspond to the notion of
internal crossed module, in the sense of Janelidze.

Keywords:
Internal category, internal groupoid, reflexive graph,
multiplicative graph, star-multiplicative graph, jointly epic pair,
admissible pair, jointly epic split extension, split short five lemma,
pointed protomodular

2000 MSC:
18D35

*Theory and Applications of Categories,*
Vol. 23, 2010,
No. 9, pp 170-198.

http://www.tac.mta.ca/tac/volumes/23/9/23-09.dvi

http://www.tac.mta.ca/tac/volumes/23/9/23-09.ps

http://www.tac.mta.ca/tac/volumes/23/9/23-09.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/23/9/23-09.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/23/9/23-09.ps

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