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The curvature form descends to a genuine 2-form on the base space (unlike the connection 1-form). In fact, locally on the base space, we can pick a trivialization of the principal bundle and compute …

Stacks form an (∞,1)-category. The latter informal notion has many equivalent implementations: simplicial category, topological category, quasicategory (also known as ∞-category), Segal category, com …

Yes. If $E→M$ and $F→M$ are vector bundles over a smooth manifold $M$, then differential operators $E→F$ of order less than $k≥0$ can be identified with sections of a finite-dimensional vector bundle …

I presume “group” means “Lie group”. Invariant differential operators on the Lie group $\def\R{{\bf R}}\R$ have the form $∑_{k≥0}a_k {∂^k\over ∂x^k}$, where $a_k∈\R$. Thus, any linear differential dif …

This has already been done, see the article EFC-algebra and references therein. In particular, the paper of Pridham constructs the topos of ∞-sheaves on the site of (derived) Stein spaces and explores …

Smooth manifolds of negative dimension are defined in derived geometry. Recall that if A→M and B→M are two transversal submanifolds of codimension a and b respectively, then their intersection C is ag …

Both $m∘∇$ and $d$ are natural operations from $k$-forms to $(k+1)$-forms. By Palais's theorem, all such operations are proportional to the de Rham differential. That is, $m∘∇=λd$ for some $λ$. By exa …

Substituting $f=f_1f_2$ in the definition of a derivative endomorphism immediately implies that $D_M$ is a derivation, using the fact that $g_1ψ=g_2ψ$ for all vector fields $ψ$ implies $g_1=g_2$, wher …

A $d$-dimensional flat real vector bundle $E→M$ is classified by a map $\def\B{{\sf B}}\def\GL{{\rm GL}}M→\B\GL(d)_δ$, where $\GL(d)_δ$ is the orthogonal group equipped with the discrete topology. Arb …

This is the Duistermaat–Kolk construction of a simply connected Lie group that integrates the given Lie algebra $\def\g{{\frak g}}\g$. The starting observation is that for any simply connected Lie gro …

Would this be of any interest to solve some geometric questions. Is there a notion of "derived stack" in the differential geometry setting. The notion of a derived stack in the setting of differenti …

A gerbe on a manifold $M$ is a morphism of simplicial presheaves $$\def\tB{{\sf B}}\def\U{{\rm U}}\def\cC{{\sf\check C}}\cC(U)→\tB^2\U(1),$$ where $\{U_i\}_{i∈I}$ is an open cover of $M$, $\cC(U)$ is …

After integration we have a number so isn't it a function? Differential $k$-forms can be integrated along a submersion with $d$-dimensional fibers, which yields a differential $(k-d)$-form. Fiberwis …

A folk theorem says that star-shaped open subsets of R^n are diffeomorphic to R^n. Is there a citeable reference for a proof of this result? For the sake of being definite, let's say that “citeable” m …

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 …