Coverings, factorization systems and closure operators in the category of quandles

by Valérian Even

Institution: Université Catholique de Louvain
Year: 2016
Posted: 02/05/2017
Record ID: 2120215
Full text PDF: http://hdl.handle.net/2078.1/171914


Quandles are mathematical structures that have been mostly studied in knot theory, where they determine a knot invariant that is complete up to orientation. The aim of this thesis is to capture some categorical properties of the variety of quandles. More specifically, we study two adjunctions in the variety of quandles: the first one with its subvariety of trivial quandles and the second one with its subvariety of abelian symmetric quandles. We show that both of them are admissible in the sense of the categorical Galois theory developed by G. Janelidze, and we characterize the corresponding coverings. In particular, we show that the coverings arising from the adjunction with the subvariety of trivial quandles correspond to the quandle coverings introduced and studied by M. Eisermann. We prove that the category of quandle cove- rings is a reflective subcategory of the category of surjective quandle homomorphisms, and we give an explicit description of this reflection. We also investigate some factorization systems for surjective quandle homomor- phisms as well as closure operators of subobjects in the variety of quandles. (SC - Sciences)  – UCL, 2016 Advisors/Committee Members: UCL - SST/IRMP - Institut de recherche en mathématique et physique, UCL - Faculté des Sciences, Gran, Marino, Everaert, Tomas, Clementino, Maria Manuel, Vercruysse, Joost, Vitale, Enrico, Willem, Michel.