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Bugs Bunny
Bugs Bunny
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All sorts of things can be reconstructed from their "linear representations". One example is Tannaka (Deligne, Tannaka-Krein, etc.) reconstruction where a group is recovered from its category of representations. Another example includes various versions of Bondal-Orlov theorem, where an algebraic variety (not every) is recovered from its (derived or even usual) category of (quasi)-coherent sheaves. There are further examples of this phenomena.

Question: How does one reconstruct a homotopy type?

Talking through my hat, ofif $X$ is a homotopy type, Ione can reconstruct $X$ by Yoneda Lemma from the functor $hom (X, -)$ from the homotopy types to sets. Now vectors bundles on $X$ "remember" part of this: they are roughly morphisms from $X$ to Grassmannians $Gr_n (R^{\infty})$. Somehow, maybe, if you get "enough" spaces: add some nice spaces (like $BG$ for all compact groups, etc) to Grassmannians, then one can reconstruct $X$ from the category of "nice bundles", which are homotopy classes of maps from $X$ to nice spaces.

All sorts of things can be reconstructed from their "linear representations". One example is Tannaka (Deligne, Tannaka-Krein, etc.) reconstruction where a group is recovered from its category of representations. Another example includes various versions of Bondal-Orlov theorem, where an algebraic variety (not every) is recovered from its (derived or even usual) category of (quasi)-coherent sheaves. There are further examples of this phenomena.

Question: How does one reconstruct a homotopy type?

Talking through my hat, of $X$ is a homotopy, I can reconstruct $X$ by Yoneda Lemma from the functor $hom (X, -)$ from the homotopy types to sets. Now vectors bundles on $X$ "remember" part of this: they are roughly morphisms from $X$ to Grassmannians $Gr_n (R^{\infty})$. Somehow, maybe, if you get "enough" spaces: add some nice spaces (like $BG$ for all compact groups, etc) to Grassmannians, then one can reconstruct $X$ from the category of "nice bundles", which are homotopy classes of maps from $X$ to nice spaces.

All sorts of things can be reconstructed from their "linear representations". One example is Tannaka (Deligne, Tannaka-Krein, etc.) reconstruction where a group is recovered from its category of representations. Another example includes various versions of Bondal-Orlov theorem, where an algebraic variety (not every) is recovered from its (derived or even usual) category of (quasi)-coherent sheaves. There are further examples of this phenomena.

Question: How does one reconstruct a homotopy type?

Talking through my hat, if $X$ is a homotopy type, one can reconstruct $X$ by Yoneda Lemma from the functor $hom (X, -)$ from the homotopy types to sets. Now vectors bundles on $X$ "remember" part of this: they are roughly morphisms from $X$ to Grassmannians $Gr_n (R^{\infty})$. Somehow, maybe, if you get "enough" spaces: add some nice spaces (like $BG$ for all compact groups, etc) to Grassmannians, then one can reconstruct $X$ from the category of "nice bundles", which are homotopy classes of maps from $X$ to nice spaces.

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Bugs Bunny
Bugs Bunny
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Tannaka reconstruction for homotopy types

All sorts of things can be reconstructed from their "linear representations". One example is Tannaka (Deligne, Tannaka-Krein, etc.) reconstruction where a group is recovered from its category of representations. Another example includes various versions of Bondal-Orlov theorem, where an algebraic variety (not every) is recovered from its (derived or even usual) category of (quasi)-coherent sheaves. There are further examples of this phenomena.

Question: How does one reconstruct a homotopy type?

Talking through my hat, of $X$ is a homotopy, I can reconstruct $X$ by Yoneda Lemma from the functor $hom (X, -)$ from the homotopy types to sets. Now vectors bundles on $X$ "remember" part of this: they are roughly morphisms from $X$ to Grassmannians $Gr_n (R^{\infty})$. Somehow, maybe, if you get "enough" spaces: add some nice spaces (like $BG$ for all compact groups, etc) to Grassmannians, then one can reconstruct $X$ from the category of "nice bundles", which are homotopy classes of maps from $X$ to nice spaces.