Qu’est-ce l’admissibilité de Nelson Martins-Ferreira ?

In his Ph.D.thesis [3], Nelson Martins-Ferreira introduced a technical condition (for a certain type of diagram in a category) which he called admissibility. His first aim was to efficiently describe internal categorical structures, but the flexibility of the condition allowed him to use it for expressing commutativity conditions as well. Admissibility allowed us to conveniently describe the so-called Smith is Huq condition [5, 2] and its close relationship with weighted commutativity [1, 6]. We were, however, not entirely happy to be using a pure technical definition which at first sight does not seem to have a conceptual meaning. Clearly it should model commuting in some sense, but for which kind of objects? The aim of my talk is to explain that admissibility is indeed a commutativity condition, namely for the same objects that answer the question raised in [4]. [1] M. Gran, G. Janelidze, and A. Ursini, Weighted commutators in semi-abelian categories, J.Algebra 397 (2014), 643–665. [2] M. Hartl and T. Van der Linden, The ternary commutator obstruction for internal crossed modules, Adv.Math. 232 (2013), no. 1, 571–607. [3] N. Martins-Ferreira, Low-dimensional internal categorical structures in weakly Mal’cev sesquicategories, Ph.D. thesis, University of Cape Town, 2008. [4] N. Martins-Ferreira, A. Montoli, A. Ursini, and T. Van der Linden, What is an ideal a zero-class of?, Pré-Publicações DMUC 15-05 (2015), 1–14. [5] N. Martins-Ferreira and T. Van der Linden, A note on the “Smith is Huq” condition, Appl. Categ. Structures 20 (2012), no. 2, 175–187. [6] N. Martins-Ferreira and T. Van der Linden, A decomposition formula for the weighted commutator, Appl. Categ. Structures 22 (2014),899–906