Abstract Free algebras are always free as modules over the base ring in classical algebra. In equivariant algebra, incomplete Tambara functors play role of and Mackey modules. Surprisingly, often fail to be functors. this paper, we determine for all finite groups conditions under which a functor is functor. For solvable groups, show that flat precisely when these hold. Our results imply almost ...

For a differential graded k-quiver Q we define the free A∞-category FQ generated by Q. The main result is that the restriction A∞-functor A∞(FQ,A) → A1(Q,A) is an equivalence, where objects of the last A∞-category are morphisms of differential graded k-quivers Q → A. A∞-categories defined by Fukaya [Fuk93] and Kontsevich [Kon95] are generalizations of differential graded categories for which th...

Smooth K-functors are introduced and the smooth K-theory of locally convex algebras is developed. It is proved that the algebraic and smooth K-functors are isomorphic on the category of quasi ⊗̂-stable real (or complex) Fréchet algebras.

Let G be a finite group. In [5], Hambleton, Taylor and Williams have considered the question of comparing Mackey functors for G and biset functors defined on subgroups of G and bifree bisets as morphisms. This paper proposes a different approach to this problem, from the point of view of various categories of G-sets. In particular, the category G-set of fused G-sets is introduced, as well as th...

Tannaka duality describes the relationship between algebraic objects in a given category and functors into that category; an important case is that of Hopf algebras and their categories of representations; these have strong monoidal forgetful “fibre functors” to the category of vector spaces. We simultaneously generalize the theory of Tannaka duality in two ways: first, we replace Hopf algebras...

We study Translation functors and Wall-Crossing functors on infinite dimensional representations of a complex semisimple Lie algebra using D -modules. This functorial machinery is then used to prove the Endomorphism-theorem and the Structure-theorem, two important results established earlier by W. Soergel in a totally different way. Other applications to the category O of Bernstein-GelfandGelfa...

Kato has constructed reflection functors for KLR algebras which categorify the braid group action on a quantum group by algebra automorphisms. We prove that these reflection functors are monoidal.