Algebra, Logic and Topology

 

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Title
Star-multiplicative graphs in pointed protomodular categories

Abstract
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 internal category. 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.

Speaker(s)
Nelson Martins-Ferreira (CDRSP/IPLeiria)

Date
February 10, 2009

Time
15.00

Room
Sala 2.5

 
     
 

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