Any diagram group over a +nite semigroup presentation acts properly, freely, and cellularly by isometrices on a proper CAT(0) cubical complex. The existence of a proper, cellular action by isometries on a CAT(0) cubical complex has powerful consequences for the acting group G. One gets, for example, a proof that G satis+es the Baum–Connes conjecture. Any diagram group over a +nite presentation ...

Cubical type theory is an extension of Martin-Löf type theory recently proposed by Cohen, Coquand, Mörtberg and the author which allows for direct manipulation of n-dimensional cubes and where Voevodsky’s Univalence Axiom is provable. In this paper we prove canonicity for cubical type theory: any natural number in a context build from only name variables is judgmentally equal to a numeral. To a...

Reconstructing 3D objects from 2D images is both challenging for our brains and machine learning algorithms. To support this spatial reasoning task, contextual information about the overall shape of an object critical. However, such not captured by established loss terms (e.g. Dice loss). We propose to complement geometrical including multi-scale topological features, as connected components, c...

Can a group be an orthodox rational agent? This requires the group's aggregate preferences to follow expected utility (static rationality) and evolve by Bayesian updating (dynamic rationality). Group rationality is possible, but only preference aggregation rules which achieve it (and are minimally Paretian continuous) linear-geometric rules, combine individual values linearly beliefs geometrica...

This paper gives a uniform-theoretic refinement of classical homotopy theory. Both cubical sets (with connections) and uniform spaces admit classes weak equivalences, special cases appropriate for the respective Lipschitz settings. Cubical additional compatible structures categories (co)fibrant objects. A categorical equivalence between lifts to full faithful embedding from an associated catego...