This paper concerns contravariant functors from the category of rings to the category of sets whose restriction to the full subcategory of commutative rings is isomorphic to the prime spectrum functor Spec. The main result reveals a common characteristic of these functors: every such functor assigns the empty set to Mn(C) for n ≥ 3. The proof relies, in part, on the Kochen-Specker Theorem of qu...

For each standard affine data satisfying some condition on the coroots, we define a group functor on the category of all rings which is a subgroup of some group functor R 7→ G(R[t, t−1])o T (R), where G is a Chevalley scheme and T is a torus. This functor is then shown to satisfy the conditions of the five axioms required by [Ti4] to be a ‘minimal’ Kac-Moody group.

HomX denotes the sheaf-Hom functor of OX -complexes: HomX(E,F )(U) := Hom • U (E|U , F |U ) (U ⊂ X open), the restriction map for U ′ ⊂ U being the obvious one. This “dynamic” sheafified version of Hom• has a derived functor RHom•, defined as usual via q-injective resolutions (which always exist!). Similarly, we have a sheaf-theoretic version of ⊗, and its left-derived functor ⊗ = , defined via...

Building on work done by Myra VanInwegen and Elsa Gunter, we encode the dynamic semantics of the Module system of SML in HOL. We extend this to a possible dynamic semantics for a Module system with higher order functors. We relate these two semantics by embeddings and projections and outline how we are using these to state and prove that the new evaluation relations conservatively extend the ol...

We introduce and develop the notion of displayed categories. A displayed category over a category C is equivalent to ‘a category D and functor F : D → C’, but instead of having a single collection of ‘objects of D’ with a map to the objects of C, the objects are given as a family indexed by objects of C, and similarly for the morphisms. This encapsulates a common way of building categories in p...

Covarieties of coalgebras are those classes of coalgebras for an endofunctor H on the category of sets that are closed under coproducts, subcoalgebras and quotients. Equivalently, covarieties are classes of H-coalgebras that can be presented by coequations. Adámek introduced a logic of coequations and proved soundness and completeness for all polynomial functors on the category of sets. Here th...

Abstract We prove that for any $\mathbb{P}^n$-functor all the convolutions (double cones) of three-term complex $FHR \xrightarrow{\psi } FR \xrightarrow{{\operatorname{{\epsilon }}}} {\operatorname{Id}}$ defining its $\mathbb{P}$-twist are isomorphic. also introduce a new notion non-split $\mathbb{P}^n$-functor.

To a B-coring and a (B, A)-bimodule that is finitely generated and projective as a right A-module an A-coring is associated. This new coring is termed a base ring extension of a coring by a module. We study how the properties of a bimodule such as separability and the Frobenius properties are reflected in the induced base ring extension coring. Any bimodule that is finitely generated and projec...

This is a report on some recent joint work with Radu Stancu, to appear in [4]. It is an expanded version of a talk given at the RIMS workshop Cohomology of finite groups and related topics, February 18-20, 2015. 1. Cohomological Mackey functors 1.1. Let G be a finite group, and k be a commutative ring. There are many equivalent definitions of Mackey functors for G over k. For the “naive” one, t...

We study recursive types from a syntactic perspective. In particular, we compare the formulations of recurs-ive types that are used in programming languages and formal systems. Our main tool is a new syntactic explanation of type expressions as functors. We also introduce a simple logic for programs with recursive types in which we carry out our proofs.