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.

1.1. Fundamental groups. Let FET/X be the category of schemes finite étale over X. Choose x̄ a geometric point of X. The fiber functor Fx̄ : FET → SET is defined as Fx̄(Y ) = HomX(x̄, Y ). Then the étale fundamental group is defined as π1(X, x̄) = AutFx̄ π1(X, x̄) is a profinite group. If f : X → Y is a morphism, then it induces f∗ : π1(X, x̄) → π1(Y, f(x̄)). In the following, we will often omit the ”ba...

The branching (resp. merging) space functor of a flow is a left Quillen functor. The associated derived functor allows to define the branching (resp. merging) homology of a flow. It is then proved that this homology theory is a dihomotopy invariant and that higher dimensional branchings (resp. mergings) satisfy a long exact sequence.

For any finite groupG, we define a bivariant functor from the Dress category of finite G-sets to the conjugation biset category, whose objects are subgroups of G, and whose morphisms are generated by certain bifree bisets. Any additive functor from the conjugation biset category to abelian groups yields a Mackey functor by composition. We characterize the Mackey functors which arise in this way.

The main purpose of this paper is to introduce a concept of$L$-fuzzifying topological groups (here $L$ is a completelydistributive lattice) and discuss some of their basic properties andthe structures. We prove that its corresponding $L$-fuzzifyingneighborhood structure is translation invariant. A characterizationof such topological groups in terms of the corresponding$L$-fuzzifying neighborhoo...