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On Hirsch I came across this:

"A differential structure $\Phi$ on $M$ is often obtain by collation of differential structures $\Phi_i$ on open sets $U_i$ covering $M$. This means that $$ \left.\Phi_i\right|U_i\cap U_j= \left.\Phi_j\right|U_i\cap U_j, $$ for all $i,j$; and $\Phi$ is the unique differential structure on $M$ containing each $\Phi_i$ as a subset."

Collation is a term I've only come across in his textbook, and I cannot decipher what it means operationally. Also I do not understand the notation used in the equation he uses.

Specifically: what does it mean $\Phi_i$? Does he consider a family of differential structures? What does it mean to restrict a family of differential structures on $U_i\cap U_j$? Is that even possible given that the class of differential structures is immensely "big"?

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  • $\begingroup$ $\Phi_i$ is the collection of all coordinate systems on the open set $U_i$ and accordingly the formula above holds (note that the intersection of two open sets is open so a coordinate system lying in the intersection also lies in the ambient open sets). $\endgroup$
    – user555729
    Apr 28, 2019 at 20:23

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You know that a differential structure on $M$ is is a maximal differential atlas $\Phi$ and that each open $U \subset M$ inherits a differential structure $\Phi \mid_U$ from $M$ by taking all charts in $\Phi$ with domain contained in $U$.

Now let $\{ U_i \}_{i \in I}$ be an open covering of $M$. Assume that for each $i$ we are given a differential structure $\Phi_i$ on $U_i$. Then for any two $i,j \in I$ we get two differential structures $\Phi_i \mid_{U_i \cap U_j}$ on $U_i \cap U_j \subset U_i$ and $\Phi_j \mid_{U_i \cap U_j}$ on $U_i \cap U_j \subset U_j$. The assertion is that if for any two pairs $i,j \in I$ the differential structures $\Phi_i \mid_{U_i \cap U_j}$ and $\Phi_j \mid_{U_i \cap U_j}$ agree, then there exists a unique differential structure $\Phi$ on $M$ such that $\Phi \mid_{U_i} = \Phi_i$ for all $i$.

Of course this requires a proof, but I believe it is fairly obvious.

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