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New exactness conditions involving split cubes in protomodular categories. (English) Zbl 1401.18002

From the introduction: The main purpose of the paper is to introduce and compare several new exactness conditions involving what we call split cubes. Some of these conditions can be thought of as non-pointed analogues of known categorical conditions. In particular, our conditions include non-pointed analogues of: (i) the condition introduced and studied by F. Borceux et al. [Theory Appl. Categ. 14, 244–286 (2005; Zbl 1103.18006)] which they briefly called the axiom of normality of unions; and (ii) the condition required that an internal graph is multiplicative as soon as it is star multiplicative, introduced and studied by G. Janelidze [Georgian Math. J. 10, No. 1, 99–114 (2003; Zbl 1069.18009)]. In addition to these conditions, we introduce a condition which is new even in the semi-abelian context where it has the following consequences: (i) Huq commutativity is reflected by the change of base functors of the fibration of points that is, \(\mathbb{C}\) satisfies (SSH)in the sense of T. Van der Linden and the second author in [Appl. Categ. Struct. 23, No. 4, 527–541 (2015; Zbl 1327.18016)] (Proposition 4.13); (ii) Huq commutators distribute over binary joins in each fiber of the fibration of points (Corollary 3.7).
The notion of semi-abelian category was introduced by G. Janelidze et al. [J. Pure Appl. Algebra 168, No. 2–3, 367–386 (2002; Zbl 0993.18008)], and it plays a similar role for the categories of groups, algebras, and other related structures as abelian categories play for abelian groups and modules. A category is semi-abelian if it is pointed, has binary coproducts, is protomodular [D. Bourn, Lect. Notes Math. 1488, 43–62 (1991; Zbl 0756.18007)] and is exact in the sense of M. Barr [Lect. Notes Math. 236, 1–120 (1971; Zbl 0223.18010)].
A split cube is a cubic diagram with the following morphisms: \(h_j:X\rightarrow D_j, f_j:I\rightarrow E_j,X\overset{\chi}{\longrightarrow} I\overset{\eta}{\longrightarrow} X, D_j\overset{\delta_j}{\longrightarrow} E_j\overset{\epsilon_j}{\longrightarrow} D_j, g_j:E_j\rightarrow B , i_j:D_j\rightarrow A, j=1,2, A\overset{\alpha}{\longrightarrow }B\overset{\beta}{\longrightarrow }A\), with \(\alpha\beta=1_B, \delta_j\epsilon_j=1_{E_j}\), and \(\chi\eta=1_I\), such that the diagrams obtained by removing all upward and all downward directed arrows, respectively, commute.
Then the authors say that the above split cube is:
(i) of type LE1 if the two back faces are split pullbacks (i.e., \((X,h_1,\chi)\) and \((X,h_2,\chi)\) are pullbacks of \(\delta_1\) and \(f_1\), and \(\delta_2\) and \(f_2\) , respectively );
(ii) of type LE2 if it is of type LE1 and the two front faces are split pullbacks (i.e., it is of type LE1, and \((D_j,i_j,\delta_j)\), are pullbacks of \(\alpha\) and \(g_j, j=1,2\).
(iii) of type RE if \(A\) together with the morphisms \(i_1, i_2\) and \(\beta\) is a colimit of the complementary diagram in the split cube.
For a category \(C\) the authors compare the conditions:
1.8. Condition. For each split cube of type LE1 such that \(g_1\) and \(g_2\) form a join (i.e., are a pair of jointly strongly epimorphic monomorphisms), if \(i_1\) and \(i_2\) form a join, then the split cube is of the type LE2.
1.9. Condition. For each split cube of type LE1 such that \(g_1\) and \(g_2\) form a join, if \(i_1\) and \(i_2\) are monomorphisms and the split cube is of type RE, then it is of type LE2.
1.10. Condition. For each split cube of type LE1 such that \(g_1\) and \(g_2\) form a join, if it is of type LE2, then it is of type RE.
Proposition 1.12. The category of commutative unital rings satisfies Condition 1.8. Corollary 3.7. Let \(\mathbb{C}\) be an exact protomodular category with finite colimits. If \(C\) satisfies Condition 1.10, then for each object \(B\) in \(\mathbb{C}\) , Huq commutators in the category \(\mathbf{Pt}(B)\) distribute over binary joins.
(For the category \(\mathbf{Pt}(B)\) see [Bourn, loc. cit.] and for Huq commutators see [S. A. Huq, Q. J. Math., Oxf. II. Ser. 19, 363–389 (1968; Zbl 0165.03301)]).
Theorem 4.13. Let \(\mathbb{C}\) be a regular protomodular category such that each object has global support. If \(\mathbb{C}\) satisfies Condition 1.10, then (SSH) holds in \(\mathbb{C}\) (i.e., change of base functors of the fibration of points reflect Huq commutativity).
(All the notions that appear here are defined in the Introduction ).
A series of pointed and non-pointed examples are given in the end of the paper.
Reviewer: Ioan Pop (Iaşi)

MSC:

18A10 Graphs, diagram schemes, precategories
18A25 Functor categories, comma categories
18A30 Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.)
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References:

[1] M. Barr. Exact categories. in: Lecture Notes in Mathematics, 236:1–120, 1971. · Zbl 0223.18010
[2] F. Borceux and D. Bourn.Mal’cev, protomodular, homological and semi-abelian categories. Kluwer Academic Publishers, 2004. · Zbl 1061.18001
[3] F. Borceux, G. Janelidze, and G. M. Kelly. On the representability of actions in a semi-abelian category. Theory and Applications of Categories, 14(11):244–286, 2005. · Zbl 1103.18006
[4] D. Bourn.Normalization equivalence, kernel equivalence and affine categories. Springer Lecture Notes Math., 1488:43–62, 1991. · Zbl 0756.18007
[5] D. Bourn and J. R. A. Gray. Aspects of algebraic exponentiation. Bulletin of the Belgian Mathematical Society, 19(5):821–844, 2012. · Zbl 1264.18003
[6] R. Brown and G. Janelidze. Van kampen theorems for categories of covering morphisms in lextensive categories. Journal of Pure and Applied Algebra, 119(3):255–263, 1997. · Zbl 0882.18005
[7] J. R. A. Gray. Algebraic exponentiation in general categories. Applied Categorical Structures, 20(6):543–567, 2012. · Zbl 1276.18002
[8] J. R. A. Gray. A note on the distributivity of the Huq commutator over finite joins. Applied Categorical Structures, 22(2):305–310, 2014. · Zbl 1303.18015
[9] S. A. Huq. Commutator, nilpotency and solvability in categories. Quarterly Journal of Mathematics, 19(1):363–389, 1968. · Zbl 0165.03301
[10] G. Janelidze. Internal crossed modules. Georgian Mathematical Journal, 10(1):99– 114, 2003. · Zbl 1069.18009
[11] G. Janelidze, L. M´arki, and W. Tholen. Semi-abelian categories. Journal of Pure and Applied Algebra, 168:367–386, 2002. 1058J. R. A. GRAY AND N. MARTINS-FERREIRA · Zbl 0993.18008
[12] G. Janelidze and W. Tholen. Facets of descent. I. Applied Categorical Structures. A Journal Devoted to applications of categorical methods in algebra, analysis, order, topology and computer science, 2(3):245–281, 1994. · Zbl 0805.18005
[13] G. Janelidze and W. Tholen. Facets of descent. II. Applied Categorical Structures, 5(3):229–248, 1997. · Zbl 0880.18007
[14] S. Mac Lane. Duality for groups. Bulletin of the American Mathematical Society, 56(6):485–516, 1950.
[15] N. Martins-Ferreira. Weakly Mal’cev categories. Theory and Applications of Categories, 21(6):91–117, 2008. · Zbl 1166.18005
[16] N. Martins-Ferreira and T. Van der Linden. A note on the “Smith is Huq” condition. Applied Categorical Structures, 20(2):175–187, 2012. · Zbl 1255.18008
[17] N. Martins-Ferreira and T. Van der Linden. Further remarks on the “Smith is Huq” condition. Applied Categorical Structures, 23(4):527–541, 2015. · Zbl 1327.18016
[18] B. Mesablishvili. Descent in categories of (co)algebras. Homology, Homotopy and Applications, 7(1):1–8, 2005. · Zbl 1081.18001
[19] G. Orzech. Obstruction theory in algebraic categories I, II. Journal of Pure and Applied Algebra, 2(4):287–314 and 315–340, 1972. · Zbl 0251.18016
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