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Try to replace a structure on an object with a map to a clasifying object.

E.g., replace a cohomology class of a space with a map to an Eilenberg-Mclane Eilenberg-MacLane space. Replace a vector/general bundle on a manifold with a map to the Grassmanian/other classifying space.

There must also be plenty of examples outside algebraic topology, though this technique seems to be most popular there...
show/hide this revision's text 2 spelling

Try to replace a structure on an object witha with a map to a clasifying object.

E.g., replace a cohomology class of a space with a map to an Eilenberg-Mclane space. Replace a vector/general bundle on a manifold with a map to the Grassmanian/other classifying space.

There must also be plenty of examples outside algebraic topology, though this technique seems to be most popular there...
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Try to replace a structure on an object witha map to a clasifying object.

E.g., replace a cohomology class of a space with a map to an Eilenberg-Mclane space. Replace a vector/general bundle on a manifold with a map to the Grassmanian/other classifying space.

There must also be plenty of examples outside algebraic topology, though this seems to be most popular there...