We categorify the R-matrix isomorphism between tensor products of minuscule representations of Uq(sln) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of Uq(sl2) which are related to representations of Uq(sln) by quant...

Let (L;C ) be the (up to isomorphism unique) countable homogeneous structure carrying a binary branching C-relation. We study the reducts of (L;C ), i.e., the structures with domain L that are first-order definable in (L;C ). We show that up to existential interdefinability, there are finitely many such reducts. This implies that there are finitely many reducts up to first-order interdefinabili...

A sovereign monoidal category is an autonomous monoidal category endowed with the choice of an autonomous structure and an isomorphism of monoidal functors between the associated left and right duality functors. In this paper we define and study the algebraic counterpart of sovereign monoidal categories: cosovereign Hopf algebras. In this framework we find a categorical characterization of invo...

It is shown that all the assumptions for symmetric monoidal categories flow out of a unifying principle involving natural isomorphisms of the type (A ∧B) ∧ (C ∧D) → (A ∧ C) ∧ (B ∧D), called medial commutativity. Medial commutativity in the presence of the unit object enables us to define associativity and commutativity natural isomorphisms. In particular, Mac Lane’s pentagonal and hexagonal coh...

It is known that Plotkin’s reduction theorem is very important for his theory of universal algebraic geometry [1, 2]. It turns out that this theorem can be generalized to arbitrary categories containing two special objects and in this case its proof becomes considerable more simple. This new proof and applications are the subject of the present paper. INTRODUCTION An automorphism φ of a categor...

Categories with algebraic structure—the most prominent example being monoidal categories—satisfy equational axioms only up-to coherent isomorphisms. Therefore they are pseudo algebras. We extend Lawvere’s functorial semantics to such pseudo structure: in contrast to standard strict algebras which are identified with productpreserving functors, pseudo algebras are product-preserving pseudo funct...

Rooth (1992) and Fiengo and May (1994) argue that elliptical structures are subject to both a semantic and a syntactic parallelism requirement. Pseudosluicing, where an elliptical cleft takes a non-cleft as its antecedent, pose a problem for the idea that there is a syntactic parallelism requirement. In this talk, we will look at Spanish data (building on the analysis of Rodrigues et al. 2008) ...

1.1 Adjoint functors Let C and D be categories and let f∗ : C → D and f∗ : D → C be functors. Then f∗ is a right adjoint to f∗ and f∗ is a left adjoint to f∗ if, for each D ∈ D and C ∈ C there is a natural vector space isomorphism HomC(fD,C) Φ −→HomD(D, f∗C). For C ∈ C and D ∈ D define τC = Φ(idf∗C) ∈ HomC(ff∗C,C) and φD = Φ(idf∗D) ∈ HomD(D, f∗fD). In general τC and φD are neither injective nor...

There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is natur...