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In chapter I, section 9, proposition 5 in Mac Lane and Moerdijk's Sheaves in Geometry and Logic, it is stated that if $f : B' \to B$ is a morphism in a complete category $\mathcal{C}$ and the category $\mathcal{C}$ and the slice categories $\mathcal{C}/B$ and $\mathcal{C}/B'$ are (all three) cartesian closed, then pullback along $f$ preserves all colimits which exist in $\mathcal{C}/B$.

  1. Why should $\mathcal{C}$ be complete? It must have pullbacks (along $f$), but is this completeness hypothesis needed anywhere?
  2. This proposition holds because if $\mathcal{C}/B$ is cartesian closed then pullback along $f$ has a right adjoint and then preserves colimits (this is a theorem appearing right before this proposition), so why are also $\mathcal{C}$ and $\mathcal{C}/B'$ assumed to be cartesian closed?

Thank you.

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    $\begingroup$ 1. Yes, completeness is superfluous – you just need finite limits. 2. One must be clear about which pullback functor has a right adjoint. If $\mathcal{C}_{/ B}$ is cartesian closed then $B' \times_B (-) : \mathcal{C}_{/ B} \to \mathcal{C}_{/ B}$ has a right adjoint, but that's not the claim. $\endgroup$ – Zhen Lin Apr 13 '16 at 9:55
  • $\begingroup$ @ZhenLin it isn't? Isn't the functor you describe pullback along $f$? $\endgroup$ – Abel Apr 13 '16 at 10:06
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    $\begingroup$ @Abel No, functor which the proposition is about is $B' \times_B (-) : \mathcal{C}_{/B} \to \mathcal{C}_{/B'}$. Note "$B'$" instead of "$B$" at the end. $\endgroup$ – Ingo Blechschmidt Apr 13 '16 at 10:22
  • $\begingroup$ @IngoBlechschmidt I fail to see the difference between what I call "pullback along $f$" and $B' \times_B (-)$. To me, the proposition talks about the "change of base" functor or "pullback along $f$" functor $f^*$, which sends an object $X \to B$ in $\mathcal{C}/B$ to the pullback of the cospan $B' \to B \leftarrow X$ with the canonical projection to $B'$, so that this is an object in $\mathcal{C}/B'$, and I understand that if $\mathcal{C}/B$ is cartesian closed then $f^*$ has a right adjoint. I would call this functor also $B' \times_B (-)$. Where am I wrong? $\endgroup$ – Abel Apr 13 '16 at 10:40
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    $\begingroup$ @Abel The difference between the functor Zhen Lin mentioned and your functor (which is also the functor the proposition is about) is only in the specification of what the target category is. Zhen Lin referred to a functor which takes values in $\mathcal{C}_{/B}$. You referred to a functor which takes values in $\mathcal{C}_{/B'}$. In the notation "$B' \times_B (-)$" the target category is not explicitly specified. This is what Zhen Lin refers to when he says "One must be clear about which pullback functor has a right adjoint". $\endgroup$ – Ingo Blechschmidt Apr 13 '16 at 11:54

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