Timeline for When does the direct image functor nicely push past the power/exists functor?

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Mar 18, 2013 at 17:43	comment	added	David Spivak		Now that I finally understand Zhen's answer, I realize how cool the perspective of Mike's answer is. I don't think I could have easily started there, but it looks like a very powerful approach, and I'm about to sink my teeth into Johnstone B.3 to get a piece of it. Thanks guys!
Mar 17, 2013 at 16:00	comment	added	Zhen Lin		If we have a geometric morphism $f : \mathcal{D} \to \mathcal{E}$, then $f^$ makes $\mathcal{D}$ into an $\mathcal{E}$-indexed topos $\mathbb{D}$, whose fibre over an object $E$ is the slice $\mathcal{D}_{/ f^ E}$. Johnstone explains this in detail in Chapter B3 of Sketches of an elephant. A less high-tech version of this is to simply consider $\mathcal{D}$ as an $\mathcal{E}$-enriched category, with $\underline{\mathcal{D}}(X, Y) = f_(Y^X)$; then $f_ : \mathcal{D} \to \mathcal{E}$ becomes a enriched-representable functor in an obvious way.
Mar 17, 2013 at 14:39	comment	added	David Spivak		Looks neat, but how do we formalize our ability to "pretend that $f_{\ast}$ is the global sections functor..."? Maybe a reference for understanding a morphism $f_{\ast}\colon D\to E$ of topoi in terms of the internal logic of $E$ would help me understand this.
Mar 17, 2013 at 14:29	history	edited	David Spivak	CC BY-SA 3.0	typo
Mar 17, 2013 at 1:52	history	answered	Mike Shulman	CC BY-SA 3.0