inductive graded rings, hyperfields and quadratic forms

In [6] we developed a k-theory for the category of hyperbolic hyperfields (a category that contains a copy of the category of (pre)special groups): this construction extends, simultaneously, milnor’s k-theory ([20]) and dickmann-miraglia’s k-theory ([13]). an abstract environment that encapsulate all them, and of course, provide an axiomatic approach to guide new extensions of the concept of k-theory in the context of the algebraic and abstract theories of quadratic forms is given by the concept of inductive graded rings a concept introduced in [9] in order to provide a solution of marshall’s signature conjecture in realm the algebraic theory of quadratic forms for pythagorean fields. the goal of this work is twofold: (i) to provide a detailed analysis of some categories of inductive graded ring - a concept introduced in [9] in order to provide a solution of marshall’s signature conjecture in the algebraic theory of quadratic forms; (ii) apply this analysis to deepen the connections between the category of special hyperfields ([6]) - equivalent to the category of special groups ([10]) and the categories of inductive graded rings.