The aim of this paper is to present a simplified version of the notion of ∞-groupoid developed by Grothendieck in "Pursuing Stacks" and to introduce a definition of ∞-categories inspired by Grothendieck's approach.

We study the foundations of the differential calculus in quantum geometry. The notions of (differential) quantum space and cone are introduced. Generalizing a construction of Manin, to a quantum cone we associate the quantum group of its "linear automorphisms preserving the differentials" and deduce a de Rham complex on this group. We give examples of differential calculi on quantum hyperplanes and quantum linear groups.

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