Towards Riemannian diffeology

We introduce a framework for Riemannian diffeology. To this end, we use the tangent functor in the sense of Blohmann and one of the options of a metric on a diffeological space in the sense of Iglesias-Zemmour. As a consequence, the category consisting of weak Riemannian diffeological spaces and isometries is established. With a technical condition for a definite weak Riemannian metric, we show that the pseudodistance induced by the metric is indeed a distance. As examples of weak Riemannian diffeological spaces, an adjunction space of manifolds, a space of smooth maps and the mixed one are considered.