We describe an isomorphism of categories conjectured by Kontsevich. If M and M̃ are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on M̃. We prove this equivalence when M is an elliptic curve and M̃ is its dual curve, exhibiting the dictionary in detail. † email: apolish@m...

Various frameworks that generalise the notion of contextuality in theories of physics have been proposed; one is the sheaf-theoretic approach by Abramsky and Brandenburger; an other is the equivalence-based approach by Spekkens. We show that these frameworks are equivalent for scenarios with preparations and measurements, whenever factorizability is justified. This connection gives rise to a ca...

We consider presentations that were derived in [3] for the interval groups associated with proper quasi-Coxeter elements of Coxeter group W(Dn). use combinatorial methods to derive alternative groups, and these new show a element W(Dn) cannot be isomorphic Artin type Dn. While specific problems we solve arise from study their solution provides an illustration how techniques indicated by computa...

Remark 4.2. For those familiar with category theory, one can define inverse limits for any category. In most cases the result will be another object of the same category (unique up to isomorphism), in which case the projection maps are then morphisms in that category. We will restrict our attention to the familiar categories of sets, groups, and rings. One can also generalize the index set {n} ...

We develop the theory of Hopf bimodules for a finite rigid tensor category C. Then we use this theory to define a distinguished invertible object D of C and an isomorphism of tensor functors δ : V ∗∗ → D⊗∗∗V ⊗D. This provides a categorical generalization of Radford’s S formula for finite dimensional Hopf algebras [R1], which was proved in [N] for weak Hopf algebras, in [HN] for quasi-Hopf algeb...

We introduce the compactness locus of a geometric functor between rigidly-compactly generated tensor-triangulated categories, and describe it for several examples arising in equivariant homotopy theory and algebraic geometry. It is a subset of the tensor-triangular spectrum of the target category which, crudely speaking, measures the failure of the functor to satisfy Grothendieck-Neeman duality...

Dade [D1, Theorem 7.4] obtained an important result on the equivalence of categories, extending the classical stable Clifford theory. He used the theory of strongly graded rings. Recently, this work has been generalized to arbitrary graded rings, see E. Dade [D2], [D3], J.L. Gómez Pardo and C. Nǎstǎsescu [GN ], C. Nǎstǎsescu and F. Van Oystaeyen [NVO2]. In the classical case the stable Clifford...

This paper is a collection of remarks on control categories, including answers to some frequently asked questions. The paper is not self-contained and must be read in conjunction with [3]. We clarify the definition of response categories, and show that most of the conditions can be dropped. In particular, the requirement of having finite sums can be dropped, leading to an interesting new CPS tr...

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolu...

In the theory of operads we consider generalized symmetric power functors defined by sums of coinvariant modules. One observes classically that the symmetric functor construction provides an isomorphism from the category of symmetric modules to a split subcategory of the category of functors on dgmodules (if dg-modules form our ground category). The purpose of this article is to obtain a simila...