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This is about $\S II.5$ of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]" and is a follow-up to this: Just what is Mac Lane & Moerdijk's $\Lambda$ from $\S II.5$?

The Question:

What is Mac Lane and Moerdijk's $\Gamma$ in $\S II.5$?

Thoughts:

I'm at a loss on:

  • What the domain of $\Gamma$ is.

  • What the codomain of $\Gamma$ is.

(I'm guessing it's ${\rm Sh}(X)$.)

  • What $\Gamma$ does to the objects in its domain.

  • What $\Gamma$ does to the arrows in its domain.

So . . . Basically everything. (Sorry!)

When I read $\S II.5$ for the first time, I had an illusory sense of understanding that has now buckled under further investigation.

It appears that $\Gamma$ is defined in terms of its relationship to $\Lambda$, something I have only a rudimentary grasp of.

I need to understand $\Gamma$ in order to do Exercise II.6.

Please help :)

Shaun
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    Based on what $\Lambda$ turned out to be, and the fact that $\Gamma \Lambda$ is the sheafification functor, I'm guessing $\Gamma$ has domain $\mathbf{Top}_{/ X}$ and on objects, it takes $f : Y \to X$ to the sheaf of continuous sections $s : X \to Y$ of $f$ (that is, the right inverses of $f$). – Daniel Schepler Feb 29 '20 at 05:04

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