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How about defining det(M) as ⨁kExtkSym(M*)(O(X),Sym(M*))? Here Sym(M*) acts on O(X) by augmentation map. Ext and Sym are functorial, hence det should also be functorial.

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 …

(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 …

The functor from the category of smooth manifolds to to the category of real algebras that sends a manifold M to C^∞(M) is fully faithful, hence it is an equivalence of categories of smooth manifolds …

Virtually any kind of algebraic structure (e.g., group, ring, module, vector space, affine space, etc.) leads to a stack in categories whose objects are bundles of such structures and morphisms are fi …

This fact admits a much easier proof. To show that for any simplicial fibrant sheaf F and open sets U⊆V the map F(V)→F(U) is a fibration it suffices to show that F(V)→F(U) has a right lifting property …

Since Čech cohomology is mentioned, I presume that $C$ is the category of open subsets of a topological space. More generally, we can assume $C$ to be an arbitrary site. In this case, the answer to bo …

$\underline{G}$ is the homotopy loop space of $BG$. More precisely, the two terminal maps $G\rightarrow pt$ and $G\rightarrow pt$ yield a weak equivalence $\underline{G} \rightarrow pt\times_{BG} pt …

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 …

Question: does the vector bundle generated by this sheaf on X∖Sing is isomorphic to the cotangent bundle on X∖Sing is ? If not what is the sheaf we should define on X to get the cotangent bundle on X …

The following question is motivated by a desire to find a rough analog in algebraic geometry of the usual notion of gluing of smooth bordisms. Suppose k is an algebraically closed field of characteri …

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 …

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 …

The manifold $\def\Hom{\mathop{\rm Hom}}\def\R{{\bf R}}\Hom(\R^{0|2},M)$ is isomorphic to the pullback of the parity-reversed bundle vector bundle $TM⊕TM$ along the projection map $TM→M$. This is Lemm …

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 …