Normal Subobjects and Abelian Objects in Protomodular Categories

By Dominique Bourn. 2000

68 citations recorded by the source corpus.

w x Recent works about Mal’cev categories 6, 7, 9]11 , a notion arising from considerations about Mal’cev varieties, led the way for an alternative approach to problems that confronted universal algebra. This approach focuses on semantical aspects of the theories rather than the syntactical ones, and singles out some new points of views. Actually the notion of a Mal’cev category is a link in a chain of notions which are organized, the existence of finite limits being assumed, by the w x following implications 3

Subfields: Computational Theory and Mathematics, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/304d5948ab89b6896ec3fb69c69fb433fff56c05

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Citations in the corpus

Normalities and commutators (2010, background): “phism. Notice that kis a monomorphism, so that kernels are indeed sub-objects. 3 Normal subobjects. The categorical notion of normality has been introduced for a ﬁnitely complete category by Bourn in [Bou00]. Let (R,r1,r2) be an equivalence relation on an object A. We call a map k: K→ Anormal to Rwh”
A semi-abelian extension of a theorem by Takeuchi (2018, background): “Indeed, it follows from Proposition 9 in [7] that the abelian objects in any semi-abelian category C are precisely the objects X with the property that the diagonal ∆: X → X×X is a normal monomorphism ( = a kernel of some morphism in C).”
On the second cohomology group in semi-abelian categories (2005, background): “In the pointed protomodular case, this construction determines a bijection between the equivalence relations on X and the normal subobjects of X [8].”
Centrality and connectors in Maltsev categories (2002, background): “By assumption the arrow (p0 ,r 0 ): (C, R) → (A × A, A )d etermines a discrete fibration of internal equivalence relations, so that by Corollary 2 in [ 4 ], the square (1) is a pullback; then R is isomorphic to Q × A .B y Proposition 4. 2t he fact that the equivalences R[r0 ]a ndR[r1 ]a re connected”
Baer invariants in semi-abelian categories II: Homology (2004, unknown): “However, it is quite clear from the construction of the connecting morphism that the sequence is, moreover, natural.”
Central extensions in semi-abelian categories (2002, background): “By Proposition 12 in [4], we can assert that the kernel pair R[f] of f is given by (K[d0]× A; A; J d1) as in the diagram K[d0]× A J d1 −−−−−→ A”
Centrality and normality in protomodular categories (2001, unknown): “In this paper we investigate the relationship between the classical notion of centrality of equivalence relations [22] and the abstract notion of normal monomorphism introduced in [4].”
Characterization of protomodular varieties of universal algebras (2003, unknown): “Moreover any protomodular category being Maltsev [3], we have immediately the second part of the following string of inclusions, whose first part will be a consequence of our main theorem: K 1 ⊂ K 2 ⊂ K 3 , where: 1.”
Topological semi-abelian algebras (2005, background): “The category V is protomodular (see [9]) when it admits pullbacks and all the inverse image functors v∗ between the categories of points reflect isomorphisms.”