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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 :)

Shaun
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  • I don't think it is, @DanielSchepler. Perhaps I should edit the question to include more details for those who do not have the book handy. – Shaun Feb 27 '20 at 17:51
  • I don't think so, @DanielSchepler. Quoting page 87, "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." Does that help? – Shaun Feb 27 '20 at 18:27
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    Oh, OK, the sheafification functor is definitely something I'm familiar with. (Though there are a couple ways to construct it that I'm familiar with: one involves $F^+(U)$ being a subset of $\prod_{x\in U} F_x$, and the other involves $F^+(U)$ being a direct limit over covers of $U$ of an equalizer. Then there's also a construction involving defining a "closure" operator of subpresheaves and applying that closure operator several times, which generalizes to "sheaves wrt a topology on a topos", but I'm guessing that's not what would be in an earlier chapter.) – Daniel Schepler Feb 27 '20 at 18:38
  • But the functor is defined in the previous pages... You cite pages 84 to 87, but what don't you understand more precisely in these pages? Here is an other explanation: https://stacks.math.columbia.edu/tag/007X – Dabouliplop Feb 27 '20 at 18:51
  • Oh. I've definitely read the preceding material, so I must have forgotten it. Chapter II's exercises are challenging for me and so have taken up a lot of my time - time I'm investing minimally as I've not much of it spare. Exercise II.6 refers to its $\S 5$, so, naturally, I assumed the definition of $\Lambda$ would be there. What $\Lambda$ does to the arrows of its domain, I suppose, @Idéophage, is my main issue, besides understanding germs properly and the like. – Shaun Feb 27 '20 at 19:17
  • Thank you for the other explanation, too, @Idéophage. – Shaun Feb 27 '20 at 19:18

1 Answers1

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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.

Captain Lama
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