Kan Extension

Abstract

Under a set of conditions, the Kan extension of an additive homology theory over smaller admissible subcategory is again a homology theory. A homology theory over the category of simply connected based topological spaces and continuous maps arising through Kan extension process, from an additive homology theory over a smaller subcategory always admits global Adams cocompletion. A cohomology theory on the category of spaces whose homotopy groups are in a Serre class of abelian groups, which is moreover an acyclic ideal of abelian groups, extends to a cohomology theory on the category of 1-connected based topological spaces having the homotopy type of a CW-complex; also under more restrictive conditions a cohomology on the latter category and the Kan extension of its restriction to the former category are suitably related.