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We can give a complete classification of (candidates for) delta-distributions on a smooth manifold $M$ at point $p$. Specifically, a delta-distribution is a smooth linear functional on the space of sm …

This is true for infinitely differentiable curves: if a map sends smooth curves to smooth curves, then it is smooth, by a theorem of Boman from 1967: Jan Boman. Differentiability of a Function and of …

I typed up a proof of this result: Numerable open covers and representability of topological stacks. The result is proved in greater generaility for arbitrary numerable open covers of topological sp …

Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map whose rank at any point of $\R^m$ is at most $d$. Here and below, smooth means infinitely differentiable. Can we fi …

The proof of this fact is available in modern textbooks. For example, see Theorem 7.16 in Jet Nestruev's Smooth Manifolds and Observables (Second Edition, 2020). In fact, the cited book contains a lot …

If C is a cartesian closed category with finite limits, then so is the category of internal groupoids in C. Indeed, the internal hom can be constructed by replicating the usual definitions of a functo …

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 …

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 …

every group can be declared a Lie group if one allows the more general definition since one is permitted to have an uncountable number of connected components. A manifold is paracompact if and o …

A coordinate-free proof of the inverse function theorem in the finite-dimensional case is provided by Theorem 19.6 in "Topological Geometry" by Ian R. Porteous. In general, the cited book is an expos …

On a smooth manifold $M$ one can define various algebraic structures, natural with respect to diffeomorphisms: the differential graded-commutative algebra $\Omega(M)$ of differential forms on $M$; t …

As far as I am aware, there is nothing in the literature that treats Riemannian or pseudo-Riemannian metrics on derived smooth manifolds. However, there is an extensive treatment of symplectic structu …

The constant rank theorem says that if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest …

A classical result by Brown and Gersten says that to verify the homotopy descent property for the Zariski topology it suffices to verify it for Zariski squares and the empty cover of the empty scheme. …

For the purposes of this question, let's say that a blurring of a smooth triangulation $T$ of a smooth manifold $X$ is a smooth homotopy $h\colon [0,1] \times X \to X$ such that $h_0=\operatorname{id} …