Let $X$ be a manifold, $i: Z\to X$ is a closed embedding.


For a sheaf $S$ (of abelian groups) on a manifold $X$ and each $\varepsilon>0$ we denote by $Z\_\varepsilon$ the set of points of $X$ that lie at distance $<\varepsilon$ from $X$. Consider the sheaf $S\_\varepsilon:U\mapsto S(U\cap Z\_\varepsilon)$, and also the section-wise limit $S\_Z=\injlim\_{\varepsilon\to 0} S\_\varepsilon$.


I would like to understand the functor $W: S\to S\_Z$ (I do not want to describe it section-wisely). Unfortunately, $S\_Z$ is only an in-sheaf; it does not have to be a sheaf (in the 'usual' sense; note that the topological space $X$ is not noetherian). In particular, the stalk of $S\_Z$ at any point $x\notin Z$ is easily seen to be $0$ (since $S\_Z(U)=0$ for any $U$ such that its closure is disjoint from $Z$). On the other hand, the section $S\_Z(X\setminus Z)$ does not seem to be $0$ (for example, if $S$ is constant); note that $X\setminus Z$ is not compact! See here the answer of algori to a previous version of this question (yesterday I believed that one can call $S\_Z$ a sheaf).


Any hints (or references) for dealing with my $W$ would be very welcome! Note that $W$ is the limit of $i\_{\varepsilon\ast}i\_{\varepsilon}^\ast$. Unfortunately, the adjunctions that I wrote about previously probably do not hold (in any sense).


P.S. Some observations that do not seem to help me.


1. My definition of $S\_Z$ extends (without any changes) to presheaves.
2. $W':P\mapsto P\_Z$ sends ind-sheaves to ind-sheaves.
3. It seems that ind-sheaves are (exactly) sheaves for the Grothendieck topology that admits only 'finite' covers.


So, perhaps one should use (somehow) the interplay between sheaves, ind-sheaves, and presheaves (and the corresponding topologies). 



Restriction of a sheaf to an infinitely small neighbourhood of a closed submanifold: how to work with this ind-sheaf?
Let me show that $i^{-1}$ can't have a left adjoint when $X$ is a connected topological space and $Z\neq X$ is a point. From the remark by Denis-Charles Cisinski it would follow that $i\_\* i^{-1}$ can't have a left adjoint either.


Suppose $Z=\{x\}$ and $i^{-1}$ had a left adjoint $J$. Then we would have $$Hom (JF,G)=Hom (F,i^{-1} G)$$ for any $F$ a sheaf on $Z$ and $G$ a sheaf on $X$. Take a non-zero sheaf $F\in Sh(Z)$, i.e. a non-zero abelian group. Note that the stalk $(JF)\_x\neq 0$ (this can be seen by taking $G$ to be the constant sheaf with stalk $F$). Let us show that for any $F$ the sheaf $JF$ must be supported at $x$. Suppose there is a $y\neq x$ such that $(JF)\_y\neq 0$. Then take $G=i'\_\* i'^{-1}JF$ where $i'$ is the inclusion $\{y\}\to X$. We have $i^{-1} G=0$, and so the right hand side of the above adjunction formula is zero. The left hand side part is non-zero since the canonical map $JF\to i'\_\* i'^{-1}JF$ is non-zero.


So $JF$ must be supported on $x$. Take $G$ to be the constant sheaf with stalk $F$. This time ii is the left hand side of the formula that is zero (here we use that $X$ is connected and $Z\neq X$, so $JF=i\_\*i^{-1} JF$ can't map into the constant sheaf $G=\underline{F}\_X$ in a non-zero way) and the right hand side that isn't.


