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This is a crosspost of this MSE question.

In Johnstone's Topos Theory appears the following lemma.

7.41 Lemma. Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two distinct element of $P(X)$. Let $(V_j\to V)$ be a finite $J$-covering family in $\mathsf C$, and $v$ and element of $P(h_V)$. There there exists a refinement $Q$ of $P$ such that (a) the images of $x,y$ under the natural map $P(X)\to Q(X)$ are distinct, and (b) the image of $v$ in $Q(h_V)$ is in $\bigcup _j\; \operatorname{im}(Q(h_{V_j})\to Q(h_V))$.

So the refinement continues to "separate" $x,y$, which have to do with the sheaf $X$ (but not the covering family $(V_j\to V)$), but has the additional property (b), which seems to be only about representables associated to the covering family $(V_j\to V)$ (and not the sheaf $X$).

Questions.

  1. What is a nice conceptual intuition for the notion of a pseudo-point? Is there a geometric intuition?
  2. What's the idea behind property (b)?

This lemma seems to be topological in nature, but I just can't seem to unpack the content of property (b) of the refinement.

This is a crosspost of this MSE question.

In Johnstone's Topos Theory appears the following lemma.

7.41 Lemma. Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two distinct element of $P(X)$. Let $(V_j\to V)$ be a finite $J$-covering family in $\mathsf C$, and $v$ and element of $P(h_V)$. There there exists a refinement $Q$ of $P$ such that (a) the images of $x,y$ under the natural map $P(X)\to Q(X)$ are distinct, and (b) the image of $v$ in $Q(h_V)$ is in $\bigcup _j\; \operatorname{im}(Q(h_{V_j})\to Q(h_V))$.

So the refinement continues to "separate" $x,y$, which have to do with the sheaf $X$ (but not the covering family $(V_j\to V)$), but has the additional property (b), which seems to be only about representables associated to the covering family $(V_j\to V)$ (and not the sheaf $X$).

Questions.

  1. What is a nice conceptual intuition for the notion of a pseudo-point? Is there a geometric intuition?
  2. What's the idea behind property (b)?

This lemma seems to be topological in nature, but I just can't seem to unpack the content of property (b) of the refinement.

In Johnstone's Topos Theory appears the following lemma.

7.41 Lemma. Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two distinct element of $P(X)$. Let $(V_j\to V)$ be a finite $J$-covering family in $\mathsf C$, and $v$ and element of $P(h_V)$. There there exists a refinement $Q$ of $P$ such that (a) the images of $x,y$ under the natural map $P(X)\to Q(X)$ are distinct, and (b) the image of $v$ in $Q(h_V)$ is in $\bigcup _j\; \operatorname{im}(Q(h_{V_j})\to Q(h_V))$.

So the refinement continues to "separate" $x,y$, which have to do with the sheaf $X$ (but not the covering family $(V_j\to V)$), but has the additional property (b), which seems to be only about representables associated to the covering family $(V_j\to V)$ (and not the sheaf $X$).

Questions.

  1. What is a nice conceptual intuition for the notion of a pseudo-point? Is there a geometric intuition?
  2. What's the idea behind property (b)?

This lemma seems to be topological in nature, but I just can't seem to unpack the content of property (b) of the refinement.

Source Link
Arrow
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  • 10.5k
  • 1
  • 27
  • 71

Intuition for pseudo-points and the inductive step in Johnstone's proof of Deligne's completeness theorem

This is a crosspost of this MSE question.

In Johnstone's Topos Theory appears the following lemma.

7.41 Lemma. Let $P$ be a pseudo-point of $\mathsf C$, $X$ a $J$-sheaf on $\mathsf C$, and $x,y$ two distinct element of $P(X)$. Let $(V_j\to V)$ be a finite $J$-covering family in $\mathsf C$, and $v$ and element of $P(h_V)$. There there exists a refinement $Q$ of $P$ such that (a) the images of $x,y$ under the natural map $P(X)\to Q(X)$ are distinct, and (b) the image of $v$ in $Q(h_V)$ is in $\bigcup _j\; \operatorname{im}(Q(h_{V_j})\to Q(h_V))$.

So the refinement continues to "separate" $x,y$, which have to do with the sheaf $X$ (but not the covering family $(V_j\to V)$), but has the additional property (b), which seems to be only about representables associated to the covering family $(V_j\to V)$ (and not the sheaf $X$).

Questions.

  1. What is a nice conceptual intuition for the notion of a pseudo-point? Is there a geometric intuition?
  2. What's the idea behind property (b)?

This lemma seems to be topological in nature, but I just can't seem to unpack the content of property (b) of the refinement.