Abstract We construct a calculus of functors in the spirit orthogonal calculus, which is designed to study ‘functors with reality’ such as Real classifying space functor, . The produces Taylor tower, n -th layer classified by spectrum an action further give model categorical considerations, producing zigzag Quillen equivalences between spectra and structure on category input captures homotopy t...

We prove that St(n,A) is a crossed module over GL(n,A) under local stable rank condition on an algebra A commutative ring. Our proof uses only elementary localization techniques in terms of pro-groups and stability results for K1 K2. also similar result the Steinberg group associated with any sufficiently isotropic general linear constructed by quasi-finite algebra.