# Representations of topological affine group scheme in the fibered category of profinite topological modules

The purpose of this talk is to give a framework of representations of topological affine group schemes represented by topological Hopf algebras over a field. We first review fundamental facts on the category of complete topological modules briefly and show that the fibered category of complete topological modules is a “cartesian closed fibered category”. So we have thee equivalent definitions of a representation of a topological affine group scheme on profinite topological modules; that is, the first one is the original one, the second one is given by using the “products” of the fibered category and the third one is given by using the existence of “exponents” of the fibered category. This enables us to consider the obits, the fixed points and left & right induced representations of given representations. We also remark that the Milner coaction relates the second and the third definitions of the representations.