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 $i^{\ast}$?
