This is a question concerning Exercise II.6 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, it is new to MSE.
The Details:
I'm not going to relay all of $\S II.5$ and its references to the $\Lambda$ in question. The definition of $\Lambda$ seems spread across the pages 84 to 87, ibid., bringing together concepts like germs, stalks, and bundles, all of which I have little to no experience of (beyond a brief reading of Goldblatt's, "Topoi [. . .]").
Quoting page 87 of Mac Lane and Moerdijk,
The left adjoint functor $$\Gamma\Lambda:{\rm Sets}^{\mathcal{O}(X)^{{\rm op}}} \to{\rm Sh}(X)$$ is known as the associated sheaf functor, or the sheafification functor.
The Question:
Just what is the functor $\Lambda$ from $\S II.5$?
Further Context:
I need to understand $\Lambda$ in order to complete Exercise II.6.
I'm reading the book recreationally.
Check my recent questions here to get a rough idea of my abilities.
This is not a question I think I can answer myself.
The kind of answer I'm looking for is, roughly speaking, a detailed description of $\Lambda$ with an eye to the solution of Exercise II.6.
Please help 🙂
Best Answer
According to §5, $\Lambda$ is the functor taking a presheaf $P$ on $X$ and returning the corresponding étale space $\Lambda_P$, which is a topological space with a natural map $p:\Lambda_P\to X$, which is a local homeomorphism.
Explicitly, as a set $\Lambda_P$ is the disjoint union $\coprod_{x\in X}P_x$ of the germs $P_x$ at each point $x\in X$, and the map $p$ just sends all elements in $P_x$ to $x$. As a topological space, well you should read the corresponding section of §5, it is explained rather well I think, but basically it is the topology such that the continuous sections $s:U\to \Lambda_P$ of $p$ are exactly the sections of the sheafification of $P$.
Now you say you have "little experience" with germs, but I don't know how you are expecting to solve exercises from a book on sheaves without diving into those notions.