A new method for realizing the first and second order cohomology groups of an internal abelian group in a Barr-exact category was introduced in [6] and [10]. The main role, in each level, is played by a direction functor. This approach can be generalized to any level n and produces a long exact cohomology sequence. By applying this method to Moore categories we show that they represent a good c...

Hausdorff Abelian groups, Pontryagin duality and the principal.These notes provide a brief introduction to topological groups with a pdws pdf special. Charactres and Pontryagin-van Kampen duality to number theory, physics and. These notes provide a brief introduction to topological groups with a special. Section 7 is peacebuilding pdf dedicated to Pontryagin-van Kampen duality.We offer an eleme...

In this paper we introduce a new functor H ab(K,G) from the category of connected reductive groups over a field K of characteristic 0 to abelian groups. We call H ab(K,G) the first abelian Galois cohomology group of G. The abelian group H ab(K,G) is related to the pointed setH (K,G) by the abelianization map ab:H(K,G)→ H ab(K,G). When K is a local field or a number field, the map ab 1 is surjec...

Extending Eilenberg–Mac Lane’s cohomology of abelian groups, a cohomology theory is introduced for commutative monoids. The cohomology groups in this theory agree with the pre-existing ones by Grillet in low dimensions, but they differ beyond dimension two. A natural interpretation is given for the three-cohomology classes in terms of braided monoidal groupoids.