Browsing by Author "Janelidze, G"

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    Algebraic exponentiation and internal homology in general categories
    (2010) Gray, James Richard Andrew; Janelidze, G
    We study two categorical-algebraic concepts of exponentiation:(i) Representing objects for the so-called split extension functors in semi-abelian and more general categories, whose familiar examples are automorphism groups of groups and derivation algebras of Lie algebras. We prove that such objects exist in categories of generalized Lie algebras defined with respect to an internal commutative monoid in symmetric monoidal closed abelian category. (ii) Right adjoints for the pullback functors between D. Bourns categories of points. We introduce and study them in the situations where the ordinary pullback functors between bundles do not admit right adjoints in particular for semi-abelian, protomodular, (weakly) Maltsev, (weakly) unital, and more general categories. We present a number of examples and counterexamples for the existence of such right adjoints. We use the left and right adjoints of the pullback functors between categories of points to introduce internal homology and cohomology of objects in abstract categories.
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    Foundation of relative non-abelian homological algebra
    (2009) Janelidze, Tamar; Künzi, Hans-Peter A; Tholen, WP; Janelidze, G