Categorical, homological, and homotopical properties of algebraic objects (Q2404934)

Chapter 1 of this monograph is devoted to the study of internal categories and categories of groups with operations. A characterization of equivalences between two internal categories is given. In the next chapter, the cohomology of an internal category in the category of groups with operations is defined and computed. In Chapter 3, categories of interest as defined by \textit{G. Orzech} [J. Pure Appl. Algebra 2, 287--314 (1972; Zbl 0251.18016)] are considered. Homologically and cohomologically equivalences of internal categories are defined and it is shown that equivalent internal categories are homologically and cohomologically equivalent. Finally, internal categories \textbf{C} in the category of groups with operations satisfying \(H^0({\mathbf C},-)=0\) and \(H^1({\mathbf C},-)=0\) are characterized. Now, Kan extensions of internal functors in the category of groups are introduced. Necessary and sufficient conditions are given for the existence of such extensions. In Chapter 5, actor objects in categories of interest are studied. The existence and construction of such objects are obtained. This problem is considered for groups, modules over a ring, Lie, Leibniz, associative, associative commutative algebras, crossed modules and precrossed modules in groups. The study of noncommutative Leibniz-Poisson algebras is the content of Chapter 6. These algebras can be seen as algebra with bracket, as introduced by \textit{J. M. Casas} and \textit{T. Pirashvili} [Manuscr. Math. 119, No. 1, 1--15 (2006; Zbl 1100.18005)], such that the braket satisfies the Leibniz identity. The cohomology theory of this kind of algebras is studied. The author defines a functor from the category of groups with action on itself to the category of Lie-Leibniz algebras, this extends the classical functor from groups to Lie algebras constructed by \textit{E. Witt} [J. Reine Angew. Math. 177, 152--160 (1937; Zbl 0016.24401)] and gives also a functor to Leibniz answering a question of \textit{J.-L. Loday} [Enseign. Math. (2) 39, No. 3--4, 269--293 (1993; Zbl 0806.55009)]. A new functor is constructed sending free objects to free Leibniz algebras. The last chapter deals with the homotopy theory of chain functors, introduced by \textit{F. W. Bauer} [Homology Homotopy Appl. 3, No. 1, 55--85 (2001; Zbl 0978.55009)]. The presented material is closely related to the papers by the \textit{F. W. Bauer} and the author [Topology Appl. 131, No. 2, 101--128 (2003; Zbl 1027.55018); J. Algebra Appl. 5, No. 4, 379--401 (2006; Zbl 1116.55003)].