I'm using Goldblatt's "Topoi: A categorical analysis of logic" as an introduction to topos logic. At the end of chapter 6, he defines the validity of propositional formulas (I'm not concerned with first-order) in a pretty straightforward way: a valuation $V$ assigns to each propositional variable $x_i$ a "truth-value", i.e., an arrow $V(x_i) : 1 \rightarrow \Omega$. He then extends this notion inductively to every formula in the following way:
- $V(\neg \varphi) = \neg \circ V(\varphi)$
- $V(\varphi_1 \lor \varphi_2) = \cup \circ <\varphi_1,\varphi_2>$$V(\varphi_1 \lor \varphi_2) = \cup \circ \langle\varphi_1,\varphi_2\rangle$
- $V(\varphi_1 \land \varphi_2) = \cap \circ <\varphi_1,\varphi_2>$$V(\varphi_1 \land \varphi_2) = \cap \circ \langle\varphi_1,\varphi_2\rangle$
- $V(\varphi_1 \supset \varphi_2) = \Rightarrow \circ <\varphi_1,\varphi_2>$$V(\varphi_1 \supset \varphi_2) = \Rightarrow \circ \langle\varphi_1,\varphi_2\rangle$
A formula $\varphi$ is said to be valid in a topos $\mathcal{C}$ if, for every possible valuation $V$, $V(\varphi) = \top$. If $\varphi$ is a propositional formula with $n$ variables, this construction gives us a way of attributing to $\varphi$ one and only one arrow $F(\varphi) : \Omega^n \rightarrow \Omega$, by "interpreting" the logical operators ($\lor,\land,\neg,\supset$) but not the variables.
Further, in section 7.4, it appears to me that he states that the validity of a formula $\varphi$ is equivalent to the commutativity of the following diagram: $$ \require{AMScd} \begin{CD} \Omega^n @>{id_{\Omega^n}}>> \Omega^n\\ @V{!}VV @VV{F(\varphi)}V\\ 1 @>>{\top}> \Omega \end{CD} $$
Now, if $\mathcal{C}$ is a well-pointed category, I can prove that this is in fact an equivalence, but I'm not being able to do it for a general topos. In fact, because his statements in section 7.4 are so vague, I'm not even sure this is true in general.
Has anyone thought about this?
Thanks in advance!
