fixed ideal, universal compact ringed space, ringed space, paracompactness, Algebra and Number Theory, sheaf of smooth real functions, partitions of unity, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Topological spaces with richer structures, Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.), maximal ideal, tangent module of a smooth manifold, Module categories in associative algebras, complete regularity, Differentiable manifolds, foundations, adjoint functors, de Rham cohomology and algebraic geometry, categories of modules, reflection