Super, quantum and non commutative species

We introduce an approach to the categorification of rings, via the notion of distributive
categories with negative objects, and use it to lay down categorical foundations for the study
of super, quantum and non-commutative combinatorics. Via the usual duality between algebra and geometry, these constructions provide categorifications for various types of affine
spaces, thus our works may be regarded as a starting point towards the construction of a
categorical geometry.