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Let $X$ be a topological space. Let $X^\delta$ denote the space whose elements are the points of $X$, and which is equipped with the discrete topology. There is a continuous map $i : X^\delta\to X$ which is the identity on elements.

With regards to the category of sheaves over $X$ and $X^\delta$, $i$ induces an adjunction $i^{-1} \dashv i_{\ast}$, where $i^{-1}$ is the inverse image functor $Shv(X)\to Shv(X^{\delta})$ and $i_\ast$ is the direct image functor $Shv(X^\delta)\to Shv(X)$.

The composition $i_\ast \circ i^{-1}$ is a monad on $Shv(X)$, which I denote $T$.

It is clear that sheaves in the image of $i^\ast$ are flasque. So the unit of the adjunction $\eta: \mathcal{F}\to T\mathcal{F}$ is an embedding of the sheaf $\mathcal{F}$ into a flasque sheaf.

  • If $\mathcal{F}$ is flasque, does this map necessarily admit a retraction? Conversely, if this map admits a retraction, is $\mathcal{F}$ necessarily flasque?
  • Can we characterize nicely the sheaves for which the unit admits a retraction?
  • Can we characterize nicely the sheaves over $X$ which are retracts of sheaves in the image of $i^{\ast}$?
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    $\begingroup$ In the category of sheaves of sets on a topological space, or more generally any localic topos, a sheaf of sets is flasque if and only if it is an injective object with respect to monomorphisms. Therefore if $F$ is a flasque sheaf of sets then $\eta_F : F \to T F$ has a retraction. The situation with sheaves of abelian groups etc. is more complicated because flasque is different from injective there. $\endgroup$
    – Zhen Lin
    Sep 26, 2020 at 22:56
  • $\begingroup$ Do you have a recommended source for this and related facts about sheaves? I believe I can I can show this from your comment (does it use the fact that given an injection A -> B, each s in B(U) has a unique maximal open subset V where it agrees with an element of A(V)?) but I would be interested in seeing this and similar results proven in context. $\endgroup$
    – Patrick Nicodemus
    Sep 26, 2020 at 23:11
  • $\begingroup$ Do I use Zorn's lemma on subsheaves of B admitting an a map into F? $\endgroup$
    – Patrick Nicodemus
    Sep 26, 2020 at 23:26
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    $\begingroup$ I do not. It is straightforward to prove, anyway. A sheaf of sets is flabby if and only if it has the right lifting property with respect to the set of class of monomorphisms $A \to B$ where $A$ and $B$ are subobjects of the terminal sheaf, so it has the rlp wrt the closure of this class under retracts, pushouts, and transfinite composition. One may verify directly that this closure is the class of all monomorphisms. I do not use Zorn's lemma in this proof but I do use the axiom of choice. $\endgroup$
    – Zhen Lin
    Sep 26, 2020 at 23:28

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