Bicohomology theory

Abstract

Given a triple T and a cotriple G on a category D \mathcal {D} such that T preserves group objects in D \mathcal {D} , let P and M be in D \mathcal {D} with M an abelian group object. Applying the “hom functor” D ( − , − ) \mathcal {D}( - , - ) to the (co)simplicial resolutions G ∗ P {G^ \ast }P and T ∗ M {T^ \ast }M yields a double complex D ( G ∗ P , T ∗ M ) \mathcal {D}({G^ \ast }P,{T^ \ast }M) . The nth homology group of this double complex is denoted H n ( P , M ) {H^n}(P,M) , and this paper studies H 0 {H^0} and H 1 {H^1} . When D \mathcal {D} is the category of bialgebras arising from a triple, cotriple, and mixed distributive law, a complete description of H 0 {H^0} and H 1 {H^1} is given. The applications include a solution of the singular extension problem for sheaves of algebras.