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My gut would be no. Let us analyze what happens if this were to be true. Then, between any $z$,$w$ in the orbit of $x$, there is a unique arrow (since any two guys in the same orbit have isomorphic is …

Let $h:E' \hookrightarrow E$ denote the inclusion of vector bundles. Let $p:Coker(h) \to B$ be defined in the obvious way (fiber at $x$ is $E_x/h(E'_x)$). It suffices to construct local trivialization …

Recall that a morphism $f:C \to D$ in a category $\mathscr{C}$ is representable if for all maps $g:E \to D$ in $\mathscr{C},$ the pullback $C \times_{D} E$ exists. Let now $\mathscr{C}$ be the catego …

Given a smooth function $f:\mathbb{R}^n \to \mathbb{R},$ how much is known about the cohomology of the interior of the zero set $f^{-1}\left(0\right)$? I know this can be quite wild, e.g. this amounts …

There is a theorem due to Losik which shows that the category of Frechet manifolds embeds fully-faithfully into diffeological spaces. (Diffeological spaces are concrete sheaves on the site of (Euclide …

To be quick, just like manifolds, orbifolds have a fixed dimension. This does not vary point to point. This is also true of their tangent spaces. This is actually true for any etale differentiable sta …

I recommend looking at Moerdijk and MrCun's "Introduction to Foliations and Lie Groupoids". On page 42, they define the frame bundle and orthonormal frame-bundle of an orbifold and show it's a smooth …

The answer is no. Claim: If $\mathfrak{DiffSt}$ were cocomplete, it would be reflective in $St\left(Mfd\right).$ Proof: Let $i:Mfd \hookrightarrow \mathfrak{DiffSt}$ be the full and faithful inclus …

How about this? Apply the tangent functor $T$ to $M_\bullet$ to get a new simplicial manifold $TM_\bullet,$ that is take the composite $$\Delta^{op} \stackrel{M_\bullet}{\longrightarrow} Mfd \stackre …

Can we characterize when a submersion $F:M \to N$ between two smooth manifolds has connected fibers? If this is too hard, what are some sufficient conditions?

Let $X$ be a Frechet or Banach manifold. We can define tangent vectors by equivalence classes of smooth curves. But, we could also define them as derivations of germs of smooth functions. Do these two …

Suppose that $G$ is a finite group acting faithfully on a topological space $X$. In the smooth setting, one can deduce that for each $x$ in $M$, the induced map $$G_x \to Diff_x\left(M\right)$$ from t …

There is a classical result which says that the assignment $$M \mapsto C^{\infty}\left(M\right)$$ is an embedding of the category of (paracompact Hausdorff) smooth manifolds into the opposite category …

I will show that stacks over diffeological spaces are "the same" (in the sense of equivalence of 2-categories) as ordinary stacks on manifolds. The Grothendieck pre-topology in question is the Grothe …

If $f:M \to N$ is a submersion with connected fibers, then the fibers of $f$ foliate $M$. This is called a simple foliation of $M$ and the leaf space can be identified with $N$. Suppose that a foliati …