Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
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?
Isn’t the statement that the object $x$ is internally projective equivalent to the statement that $x$ is projective in the stack semantics? If so then the assertion follows quite quickly, I think.
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?
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.
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 $C$ be the poset whose objects belong to the set $\mathbb{N} \cup \{a, b\}$, and whose non-identity morphisms consist of those of $\mathbb{N}$ under its usual linear order, and a morphism from each $n$ in $\mathbb{N}$ to $a$, and a morphism from each $n$ to $b$. I’m going to try to argue that $C(-, a)$ is not internally projective in the presheaf topos $Set^{C^{op}}$. (Same for $C(-, b)$ of course.)
Let $F$ be the presheaf defined by $F(a) = F(b) = \emptyset$ and $F(n) = \{-n, -n-1, -n-2, \ldots\}$, with transition map $F(n+1) \to F(n)$ the obvious inclusion. Let $G$ be the presheaf defined by $G(a) = G(b) = \emptyset$ and $G(n) = 1$ for all $n$. The unique morphism $F \to G$ is clearly epic. I am going to try to argue that the induced map $F^{C(-, a)} \to G^{C(-, a)}$ is not epic, more particularly that the function $F^{C(-, a)}(b) \to G^{C(-, a)}(b)$ is not surjective.
By the usual formula for the exponential (internal hom) $F^{C(-, a)}$, we have
$F^{C(-, a)}(b) = Nat(C(-, a) \times C(-, b), F)$Let $\theta: C(-, a) \times C(-, b) \to F$ be a natural transformation. The components at $a$ and $b$ are fairly uninteresting: they just amount to identity maps $\emptyset \to \emptyset$. The component at $n$ amounts to a map $\theta_n = 1 \times 1 \to F(n)$. The naturality condition means that we have a matching family where the element $\theta_{n+1}$ maps to $\theta_n$ under the inclusion $F(n+1) \to F(n)$. In other words, a natural transformation $\theta$ amounts to an element of the inverse limit $\lim_{\leftarrow} F(n)$, in other words the intersection of the $F(n)$, which is empty.
Thus we calculate $F^{C(-, a)}(b)$ to be empty. On the other hand, it’s pretty easy to calculate that $G^{C(-, a)}(b) =1$. Obviously $F^{C(-, a)}(b) \to G^{C(-, a)}(b)$ is not surjective.
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 $E$ is internally projective in $\mathcal{T}$, then $E\times X$ is internally projective in $\mathcal{T}/X$ for any $X$? If so, then right now I’m having a hard time seeing why.
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.
1 to 7 of 7