Shahid Beheshti University Categories and General Algebraic Structures with Applications 2345-5853 23 1 2025 07 01 Inductive graded rings, hyperfields and quadratic forms 104696 10.48308/cgasa.2024.232583.1417 EN Kaique Matias De Andrade Roberto Institute of Mathematics and Statistics, University of São Paulo, Brazil. 0000-0001-7136-8951 Hugo Luiz Mariano Institute of Mathematics and Statistics, University of São Paulo, Brazil. 0000-0002-9745-2411 Journal Article 2023 08 04 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. https://cgasa.sbu.ac.ir/article_104696_d7ea43f526970c0dcc1908cd9d77679e.pdf