We start with a warm-up: we explain Nakajima’s construction of the geometric sl2-action on ⊕ v Hmid(T ∗Gr(v, w)). 1.1. Warm-up: Nakajima’s geometric action. The lagrangian subvariety φ−1(0) ⊂ M(v) is easily seen to be the zero section Gr(v, w) ⊂ T ∗Gr(v, w). Indeed, consider the action of C× on M(v) induced from the C×-action on T ∗R by fiberwise dilations. Under the identification ofM(v) with ...

We describe a sequent calculus μLJ with primitives for inductive and co-inductive datatypes and equip it with reduction rules allowing a sound translation of Gödel’s system T. We introduce the notion of a μ-closed category, relying on a uniform interpretation of open μLJ formulas as strong functors. We show that any μ-closed category is a sound model for μLJ . We then turn to the construction o...

We define a theory of Goodwillie calculus for enriched functors from finite pointed simplicial G-sets to symmetric G-spectra, where G is a finite group. We extend a notion of G-linearity suggested by Blumberg to define stably excisive and ρ-analytic homotopy functors, as well as a G-differential, in this equivariant context. A main result of the paper is that analytic functors with trivial deri...

Let p be a prime number. This paper introduces the Roquette category Rp of finite p-groups, which is an additive tensor category containing all finite p-groups among its objects. In Rp, every finite p-group P admits a canonical direct summand ∂P , called the edge of P . Moreover P splits uniquely as a direct sum of edges of Roquette p-groups, and the tensor structure of Rp can be described in t...

Spans of open maps have been proposed by Joyal, Nielsen, and Winskel as a way of adjoining an abstract equivalence, P-bisimilarity, to a category of models of computation M, where P is an arbitrary subcategory of observations. Part of the motivation was to recast and generalise Milner’s well-known strong bisimulation in this categorical setting. An issue left open was the congruence properties ...

Containers are a neat representation of a wide class of set functors. We have previously [1] introduced directed containers as a concise representation of comonad structures on such functors. Here we examine interpreting the opposite categories of containers and directed containers. We arrive at a new view of a di↵erent (considerably narrower) class of set functors and monads on them, which we ...