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In section 2.3 of Sheaves on Manifolds, Kashiwara and Schapira define the following endofunctor on abelian sheaves over a space $X$.

For $i: Z \hookrightarrow X$ the inclusion of a closed subspace, they define $F_Z = i_*i^{-1}F$.

For $U$ an open set, they define instead $F_U = \ker(F \to F_{X \setminus U})=j_!j^{-1}F$ where the map is the canonical unit map of the inverse image-direct image adjunction.

For $A = U \cap Z$ a locally closed set, where $U$ is open and $Z$ is closed, they define $F_A = (F_U)_Z=i_*i^{-1}j_!j^{-1}F$.

Thus, we have a functor $F \mapsto F_A$ for any locally closed subset $A$ in $X$, which turns out to be exact.

I have two questions:

  1. Is there a standard name for this functor in the literature? It comes up fairly often in Kashiwara and Schapira, and in other texts I've looked at, but I haven't seen a name for it yet.
  2. Is there a better notation for this functor in use? Its a bit annoying to use when I'm writing $R_X$ for the locally constant sheaf with stalk $R$ on a space $X$, or $\mathcal{O}_X$ for the structure sheaf. I'm considering using the prescript notation $_A{F}$, though it looks a bit funny.
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    This is not an answer, just a comment on the notation. Let $h:A\rightarrow X$ be the inclusion, then $F_A=h_!h^{-1}F$. I often prefer this "6-functors" notation so that I know how the functor $F\mapsto F_A$ behaves with base-change, duality... – 2017-02-18
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    It's interesting because Kashiwara and Shapira introduce this $j_!$ functor later in the text as well, but it seems that they coincide only when the base is locally compact, if I remember correctly. – 2017-02-18
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    Ok maybe, I don't remember. But it works in other context as well in algebraic geometry (for example, with étale sheaves). The point is $i_!=i_*$ for a closed immersion and has a right adjoint $i^!$, $j^{-1}=j^!$ for an open embedding, and the base change property interchanges $i_!$ and $j^{-1}$. If you have these properties, then $h_!h^{-1}$ is a well behaved functor (it is exact, it preserves the stalk in $A$, but annihilates them outside...) – 2017-02-18

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