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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 4th 2012

    It is asserted at internally projective object that a projective object in a topos is internally projective. I’m having a hard time seeing this. Can anyone help?

    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 4th 2012

    Isn’t the statement that the object xx is internally projective equivalent to the statement that xx is projective in the stack semantics? If so then the assertion follows quite quickly, I think.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 4th 2012

    If it’s easy to see, would you be able to write down a proof, written at a level understandable to a reader of Mac Lane-Moerdijk? I confess that I’m not following your suggestion.

    Wait: I knew this topic seemed familiar! See this discussion. Why wasn’t the page internally projective object ever corrected?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 4th 2012

    Maybe I was just jumping to the assumption that the page was in fact correct :-) I’m glad you found the old discussion so quickly.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 4th 2012

    This comment also relates to the ongoing COSHEP/PAx thread.

    I think I might have an example that shows that not even presheaf toposes need satisfy the internal PAx. This example would serve two purposes: to show the need for care in the examples under the topos section, and to give an example of an externally projective object which is not internally projective, which is the topic of this thread.

    Let CC be the poset whose objects belong to the set {a,b}\mathbb{N} \cup \{a, b\}, and whose non-identity morphisms consist of those of \mathbb{N} under its usual linear order, and a morphism from each nn in \mathbb{N} to aa, and a morphism from each nn to bb. I’m going to try to argue that C(,a)C(-, a) is not internally projective in the presheaf topos Set C opSet^{C^{op}}. (Same for C(,b)C(-, b) of course.)

    Let FF be the presheaf defined by F(a)=F(b)=F(a) = F(b) = \emptyset and F(n)={n,n1,n2,}F(n) = \{-n, -n-1, -n-2, \ldots\}, with transition map F(n+1)F(n)F(n+1) \to F(n) the obvious inclusion. Let GG be the presheaf defined by G(a)=G(b)=G(a) = G(b) = \emptyset and G(n)=1G(n) = 1 for all nn. The unique morphism FGF \to G is clearly epic. I am going to try to argue that the induced map F C(,a)G C(,a)F^{C(-, a)} \to G^{C(-, a)} is not epic, more particularly that the function F C(,a)(b)G C(,a)(b)F^{C(-, a)}(b) \to G^{C(-, a)}(b) is not surjective.

    By the usual formula for the exponential (internal hom) F C(,a)F^{C(-, a)}, we have

    F C(,a)(b)=Nat(C(,a)×C(,b),F)F^{C(-, a)}(b) = Nat(C(-, a) \times C(-, b), F)

    Let θ:C(,a)×C(,b)F\theta: C(-, a) \times C(-, b) \to F be a natural transformation. The components at aa and bb are fairly uninteresting: they just amount to identity maps \emptyset \to \emptyset. The component at nn amounts to a map θ n=1×1F(n)\theta_n = 1 \times 1 \to F(n). The naturality condition means that we have a matching family where the element θ n+1\theta_{n+1} maps to θ n\theta_n under the inclusion F(n+1)F(n)F(n+1) \to F(n). In other words, a natural transformation θ\theta amounts to an element of the inverse limit lim F(n)\lim_{\leftarrow} F(n), in other words the intersection of the F(n)F(n), which is empty.

    Thus we calculate F C(,a)(b)F^{C(-, a)}(b) to be empty. On the other hand, it’s pretty easy to calculate that G C(,a)(b)=1G^{C(-, a)}(b) =1. Obviously F C(,a)(b)G C(,a)(b)F^{C(-, a)}(b) \to G^{C(-, a)}(b) is not surjective.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 15th 2013

    I have a new question about the notion of “internally projective”. For a notion to be sensibly regarded as “internal” (and in particular, for it to correspond to some statement in the stack semantics), it ought to be stable under pullback. Is it true that if EE is internally projective in 𝒯\mathcal{T}, then E×XE\times X is internally projective in 𝒯/X\mathcal{T}/X for any XX? If so, then right now I’m having a hard time seeing why.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeAug 24th 2013

    The answer to my question in #6 is yes, see this answer. I’ve added this and a more explicit stack-semantics characterization to the page internally projective object.