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Grothendieck's approach to algebraic geometry in particular tells us to treat all rings as rings of functions on some sort of space. This can also be applied outside of scheme theory (e.g., Gelfand-N …

(1) This is a highly productive way of looking at smooth manifolds. It is responsible for synthetic differential geometry and derived smooth manifolds. Both of these subjects heavily rely on this iden …

Another application of stacks is in synthetic differential geometry. Start with the opposite category of germ-determined finitely generated C^∞-rings and equip it with the appropriately defined Groth …

Model categories provide a powerful framework for commuting (homotopy) limits and colimits, and, more generally, for commuting left adjoint functors and (homotopy) limits, as well as right adjoint fun …

Yes, both Theorem A and Theorem B are special cases of a more general construction. Denote by $R$ the category of commutative unital C*-algebras or the category of commutative rings. Denote by $R'$ th …

Stacks are used in complex analysis, for example. See the papers by Finnur Lárusson, in particular, Excision for simplicial sheaves on the Stein site and Gromov's Oka principle, which shows that havi …

One can define an analogue of the crystalline topos for smooth manifolds. This is known as the de Rham stack of $M$. One of the easiest constructions of the de Rham stack embeds smooth manifolds fully …

Given any von Neumann algebra $M$, we can define its noncommutative $\def\L{{\cal L}} \L^p$-spaces $\L^p(M)$ for any $\def\C{{\bf C}} p∈\C$ such that $\Re p≥0$. Here I use the notation $\L^p:={\rm L}^ …

For sufficiently nice topological spaces $X$ (e.g., locally connected for the last two categories to be equivalent, and semilocally simply connected and locally path-connected for all three to be equi …

The étale space construction produces non-Hausdorff and nonparacompact spaces (e.g., smooth manifolds) in many practical examples that have nothing to do with algebraic geometry. The étale space is …

How can we characterize the algebras (at least within all the C^∞(M)'s), that come from compact manifolds? An algebra of the form C^∞(M) corresponds to a compact manifold if and only if all of it …

Is there a construction of the spectrum of a ring, where it is defined as a locale or Grothendieck site generated by the D(f), and where the relation that D(fk)=D(f) is definitional? Yes, the Zarisk …

There are many such results. Consider some smooth manifolds M and N. The internal hom Hom(M,N) is a sheaf on smooth manifolds. We can compute its tangent bundle, and it turns out that the tangent spac …

Is there a Cech-like way of describing the (hyper)cohomology H∙(X,M∙) or, even better, the complex Rf∗M∙ for some map f? Yes, the Verdier hypercovering theorem allows one to compute sheaf cohomology …

Urs Schreiber has written a lot on gerbes and their applications to physics: https://ncatlab.org/nlab/show/Urs+Schreiber See, for instance, the expository works “Differential cohomology in a cohesive …