Topos $M_{2}$ from Goldblatt's book

Question

Why in Topoi Goldblatts' book, on page 123, is said that the topos $M_{2}$ has only two truth-values and in the following, page 140 (example 4), we have truth tables for truth-morphisms in $M_{2}$ with three truth-values?

Answer 1

The next three paragraphs are mostly a recapitulation of what Goldblatt is saying albeit taking a different approach to get there. You can skip them if you want.

$M_2$-$\mathbf{Set} \simeq [M_2,\mathbf{Set}]$ where $[C,D]$ is the category of functors from $C$ to $D$, and $M_2$ is the monoid viewed as a one-object category. Every presheaf category, i.e. category of functors $[C^{op},\mathbf{Set}]$, is a topos. Let $\text{Sub} : [C^{op},\mathbf{Set}]^{op}\to\mathbf{Set}$ be the Sub functor, i.e. it takes a functor to the set of equivalence classes of monomorphisms into the functor, i.e. the set of subfunctors. The subobject classifier is then $\Omega \equiv \text{Sub}\circ\mathcal{Y}$ where $\mathcal{Y} = \text{Hom}(=,-) : C \to [C^{op},\mathbf{Set}]$ is the Yoneda embedding.

Specializing these general results to $M_2$-$\mathbf{Set}$, first set $C=M_2^{op}$ and call the arbitrary object of the category $M_2^{op}$ $\star$. A monomorphism into an action $F$ is an injective function $\iota : G(\star)\to F(\star)$ that satisfies $F(m)(\iota(g)) = \iota(G(m)(g))$. Next, we build an action $F|\text{Im}(\iota)$ which is just the action of $F$ on the subset $\text{Im}(\iota)$. $G\cong F|\text{Im}(\iota)$ and $\iota$ factors as that isomorphism and the inclusion of $Im(\iota)$ into $F(\star)$. So every monomorphism is isomorphic to a subset action which we can use as representatives of the equivalence classes of monomorphisms, i.e. a subfunctor is represented by one of these subset actions. However, not every subset of $F(\star)$ gives rise to an subset action; only ones that are closed under the action of $F$. Summarizing, the subfunctors of $F$ are the subsets of $F(\star)$ that are closed under the action of $F$. $\mathcal{Y}(\star) : M_2\to\mathbf{Set}$ is the monoid $M_2$ acting on itself on the left. So $\Omega(\star) = \text{Sub}(\mathcal{Y}(\star))$ is the set of subsets of $M_2$ that are closed under multiplication on the left, the left ideals. $\Omega(m)$ is the inverse image of multiplication on the right by $m$, which is what you gave. The ideals of $M_2$ are $\{\}$, $\{0\}$, and $\{0,1\}$. To specify a function of $\Omega(\star)$ you need to specify what it does for these three elements like any other function in $\mathbf{Set}$.

What Goldblatt is saying, though, is that $\Omega$ (the action) only has two points (or global elements, arrows from the terminal object), i.e. $|[M_2,\mathbf{Set}](1,\Omega)| = 2$ where $1(\star) = \{\star\}$ and $1(m) = id$. An arrow $1\to\Omega$ is an element $b$ of $\Omega(\star)$ such that $\Omega(0)(b) = b$. Only $\{\}$ and $\{0,1\}$ satisfy that. So, as Goldblatt says, $M_2$-$\mathbf{Set}$ is not well-pointed, i.e. we can't see if two homomorphisms of actions are equal by seeing if they are equal at each point (global element) as Goldblatt also demonstrates.

Your confusion probably revolves around (lack of) well-pointedness and generally the distinction between an internal and external point of view. Let's start looking at things through an internal logic lens. First off, since this is a topos, we can immediately do basically anything you're used to in set theory except we have to use an intuitionistic logic. For our particular topos, we know for each object $F$ there is a set function $F(0) : F(\star)\to F(\star)$ which, since $0$ commutes with all of $M_2$, lifts to a homomorphism $F(0) : F \to F$. (Note, at $\Omega$ this is Goldblatt's $f_\Omega$ function.) As a logic rule we have: $x:A\vdash \bar{x}:A$ writing $\overline{(-)}$ for the homomorphism. Here's where it gets interesting. We have for any (closed) term $t$, $\vdash t = \bar{t}$ but $x:A \nvdash x = \bar{x}$. We do have $x:A\vdash \bar{x} = \bar{\bar{x}}$. I'll write $M_2$ for the functor $\mathcal{Y}(\star)$ since it internalizes $M_2$. Notice that there is only one closed term (i.e. global element) of type $M_2$, $\vdash 0 : M_2$ because $\overline{1} = 0$. Note also that only two values of $\Omega$ satisfy $x = \bar x$.

Here's a less abstract example. Equip the rationals with the action that is the floor function at $0$, call it $\lfloor\mathbb{Q}\rfloor$. The only closed terms we can write of type $\lfloor\mathbb{Q}\rfloor$ are the integers. Nevertheless, we can make statements like $r:\lfloor\mathbb{Q}\rfloor,s:\lfloor\mathbb{Q}\rfloor\vdash \bar{s} = 1 \land 2r=1 \Rightarrow s + r > 1$ but note that if the consequent had been $s - r < s$, that would not have held which shows that $2r=1$ is not always false.

Another, perhaps more interesting example. Let $R = \mathbb{R}[\alpha]/(\alpha^2)$ with the action at $0$ being evaluation of the polynomial at the real number $0$. The only closed terms of this type are the real numbers. We can internally construct the type $D : \{ x : R\ |\ x^2 = 0\}$. We can prove for analytic functions $f$ and $g$, $$x:R\vdash(\forall d:D. f(x+d)-f(x) = g(x)d) \implies f'(x) = g(x)$$

The general theme for $M_2$-$\mathbf{Set}$ is the types have a "classical" part, namely where $x = \bar{x}$ (and this can be internalized as the subset type $\{ x:A\ |\ x = \bar x\}$), and a "non-classical shadow". Note that the "classical" part really is classical (if your underlying $\mathbf{Set}$ is classical); the subset type can be made into a functor which factors through the inclusion of $\mathbf{Set}$ into $M_2$-$\mathbf{Set}$ and whose image is equivalent to $\mathbf{Set}$. The "non-classical shadow" is only non-classical internally; externally everything is as classical as your meta-logic.