By homotopy linear algebra we mean the study of linear functors between slices of the ∞-category of ∞-groupoids, subject to certain finiteness conditions. After some standard definitions and results, we assemble said slices into ∞categories to model the duality between vector spaces and profinite-dimensional vector spaces, and set up a global notion of homotopy cardinality à la Baez– Hoffnung–W...

We consider probability measure preserving discrete groupoids, group actions and equivalence relations in the context of general spaces. study for these objects notions cost, $\beta$-invariant some higher-dimensional variants. also propose various convergence results about $\ell^2$-Betti numbers rank gradient sequences actions, groupoids or under weak finiteness assumptions. In particular we co...

We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbif...

0. Introduction. A symplectic groupoid is a manifold T with a partially defined multiplication (satisfying certain axioms) and a compatible symplectic structure. The identity elements in T turn out to form a Poisson manifold To? and the correspondence between symplectic groupoids and Poisson manifolds is a natural extension of the one between Lie groups and Lie algebras. As with Lie groups, und...

Motivated by symplectic geometry, we give a detailed account of differential forms and currents on orbifolds with corners, the pull-back push-forward operations, their fundamental properties. We work within formalism where category corners is obtained as localization etale proper groupoids corners. Constructions proofs are formulated in terms structure maps groupoids, avoiding use orbifold char...