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Locally surjective maps of presheaves. #

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Let X be a topological space, and 𝒢 presheaves on X, T : ℱ ⟶ 𝒢 a map.

In this file we formulate two notions for what it means for T to be locally surjective:

  1. For each open set U, each section t : 𝒢(U) is in the image of T after passing to some open cover of U.

  2. For each x : X, the map of stalks Tₓ : ℱₓ ⟶ 𝒢ₓ is surjective.

We prove that these are equivalent.

A map of presheaves T : ℱ ⟶ 𝒢 is locally surjective if for any open set U, section t over U, and x ∈ U, there exists an open set x ∈ V ⊆ U and a section s over V such that $T_*(s_V) = t|_V$.

See is_locally_surjective_iff below.


An equivalent condition for a map of presheaves to be locally surjective is for all the induced maps on stalks to be surjective.