Algebraic exponentiation and internal homology in general categories

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.

Grantor dc:publisher.institution

Department of Mathematics and Applied Mathematics

Year dc:date.issued

2010

Author dc:creator

• Gray, James Richard Andrew

Advisor dc:contributor.advisor

• Janelidze, G

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/11427/10519

OAI identifier oai:identifier

oai:open.uct.ac.za:11427/10519