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Andres Mejia
Andres Mejia
  • 131
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I'm reading these notes

Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$

If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element is an injective bundle map on the tangent bundles covering an arbitrary map $g:M \to N$.)

I've gathered from context that there is a connection between formal immersions and the sheaf of sections $\mathcal{F}(U):=\Gamma(V_n(TU) \times_{GL_n} \mathrm{Imm}(\mathbb R^n,N))$ (defined for an arbitrary open $\mathbb R^n \to M$. Please see definition 3.2 and the preceeding paragraph in the linked notes for more details.) This is referenced as the linear approximation to the sheaf of immersions, but I don't know why, although I assume it agrees in the sense of functor calculus.

My questions are the followingMy questions are the following: is the topological sheaf of formal immersions isomorphic to $\mathcal{F}$? If not, is there some relationship? If so, is the scanning map of Segal compatible (via some isomorphism) with the ``obvious" maps $\mathrm{Imm}(U,N) \to \mathrm{Imm}^f(U,N),\,\,\, f \mapsto (df,f)$?

I'm reading these notes

Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$

If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element is an injective bundle map on the tangent bundles covering an arbitrary map $g:M \to N$.)

I've gathered from context that there is a connection between formal immersions and the sheaf of sections $\mathcal{F}(U):=\Gamma(V_n(TU) \times_{GL_n} \mathrm{Imm}(\mathbb R^n,N))$ (defined for an arbitrary open $\mathbb R^n \to M$. Please see definition 3.2 and the preceeding paragraph in the linked notes for more details.) This is referenced as the linear approximation to the sheaf of immersions, but I don't know why, although I assume it agrees in the sense of functor calculus.

My questions are the following: is the topological sheaf of formal immersions isomorphic to $\mathcal{F}$? If not, is there some relationship? If so, is the scanning map of Segal compatible (via some isomorphism) with the ``obvious" maps $\mathrm{Imm}(U,N) \to \mathrm{Imm}^f(U,N),\,\,\, f \mapsto (df,f)$?

I'm reading these notes

Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$

If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element is an injective bundle map on the tangent bundles covering an arbitrary map $g:M \to N$.)

I've gathered from context that there is a connection between formal immersions and the sheaf of sections $\mathcal{F}(U):=\Gamma(V_n(TU) \times_{GL_n} \mathrm{Imm}(\mathbb R^n,N))$ (defined for an arbitrary open $\mathbb R^n \to M$. Please see definition 3.2 and the preceeding paragraph in the linked notes for more details.) This is referenced as the linear approximation to the sheaf of immersions, but I don't know why, although I assume it agrees in the sense of functor calculus.

My questions are the following: is the topological sheaf of formal immersions isomorphic to $\mathcal{F}$? If not, is there some relationship? If so, is the scanning map of Segal compatible (via some isomorphism) with the ``obvious" maps $\mathrm{Imm}(U,N) \to \mathrm{Imm}^f(U,N),\,\,\, f \mapsto (df,f)$?

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Andres Mejia
Andres Mejia
  • 131
  • 7

relationship between "linear approximation" to immersions and formal immersions

I'm reading these notes

Here, I am regarding $\mathrm{Imm}(-,N)$ as a presheaf on the open sets of some manifold $M$

If we take $\mathrm{Imm}^f(-,N)$ to be the sheaf of formal immersion (an element is an injective bundle map on the tangent bundles covering an arbitrary map $g:M \to N$.)

I've gathered from context that there is a connection between formal immersions and the sheaf of sections $\mathcal{F}(U):=\Gamma(V_n(TU) \times_{GL_n} \mathrm{Imm}(\mathbb R^n,N))$ (defined for an arbitrary open $\mathbb R^n \to M$. Please see definition 3.2 and the preceeding paragraph in the linked notes for more details.) This is referenced as the linear approximation to the sheaf of immersions, but I don't know why, although I assume it agrees in the sense of functor calculus.

My questions are the following: is the topological sheaf of formal immersions isomorphic to $\mathcal{F}$? If not, is there some relationship? If so, is the scanning map of Segal compatible (via some isomorphism) with the ``obvious" maps $\mathrm{Imm}(U,N) \to \mathrm{Imm}^f(U,N),\,\,\, f \mapsto (df,f)$?