We introduce the notions of differentiable groupoids and stratified groupoids, generalizations Lie in which spaces objects arrows have structures spaces, respectively compatible with groupoid structure. After studying basic properties these including Morita equivalence, we prove a de Rham theorem for proper locally contractible groupoids. then focus on study inertia associated to groupoid. show...

Abstract We provide foundations for a characteristic free study of foliated varieties in terms infinitesimal actions formal groupoids. The ultimate goal is the bi-rational geometry same, and to this end we prove cone theorem foliations curves, together with structure theorems extremal rays, and, course, minimal model surfaces. All possible wild ramification effects Deligne–Mumford champ are bui...

Abstract The variety of (pointed) residuated lattices includes a vast proportion the classes algebras that are relevant for algebraic logic, e.g., $$\ell $$ ℓ -groups, Heyting algebras, MV-algebras, or De Morgan monoids. Among outliers, one counts orthomodular and other varieties quantum algebras. We suggest ...

Homotopy Type Theory may be seen as an internal language for the ∞category of weak ∞-groupoids which in particular models the univalence axiom. Voevodsky proposes this language for weak ∞-groupoids as a new foundation for mathematics called the Univalent Foundations of Mathematics. It includes the sets as weak ∞-groupoids with contractible connected components, and thereby it includes (much of)...

744 NOTICES OF THE AMS VOLUME 43, NUMBER 7 Introduction Mathematicians tend to think of the notion of symmetry as being virtually synonymous with the theory of groups and their actions, perhaps largely because of the well-known Erlanger program of F. Klein and the related work of S. Lie, which virtually defined geometric structures by their groups of automorphisms. (See, for example, Yaglom’s a...