Elementary topos theory

Consider presheaves on the chain $0 < 1 < 2$ (one arrow $i \to j$ whenever $i \le j$). How many sieves are there on the object $2$, i.e. how many elements does $\Omega(2)$ have?

A presheaf on $\{0 < 1\}$ has $A_1 = \{a, b, c\}$, $A_0 = \{x, y\}$ and $\mathrm{res}(a) = \mathrm{res}(b) = x$, $\mathrm{res}(c) = y$. How many subpresheaves $(S_1, S_0)$ does it have? (Recall: $S_1 \subseteq A_1$, $S_0 \subseteq A_0$ and $\mathrm{res}(S_1) \subseteq S_0$.)

In the Sierpiński topos the truth values form the chain $0 < \tfrac12 < 1$, with $a \Rightarrow b = 1$ if $a \le b$ and $a \Rightarrow b = b$ otherwise, and $\neg a = a \Rightarrow 0$. Which formulas take the value $1$ for every truth value of $p$? (Choose all that apply.)

Why is the category $\cat{Grp}$ of groups not an elementary topos?

Let $G = \Z/2$ and consider the topos $G\text{-}\cat{Set}$ of sets with a $G$-action. Which statement is true?

In the lesson's Lean file, chi S : Hom A Omega is the characteristic map of a subpresheaf S : Sub A, and pullbackTrue φ : Sub A pulls true back along φ. Which statement expresses the full subobject-classifier property (existence and uniqueness of characteristic maps)?

In the uniqueness part of Theorem 3 we wrote, for $x \in X(c)$ and $f : d \to c$: $f \in \phi_c(x) \iff f^*\phi_c(x) = t_d \iff \phi_d(x \cdot f) = t_d \iff x \cdot f \in U(d)$. What justifies the middle equivalence $f^*\phi_c(x) = t_d \iff \phi_d(x \cdot f) = t_d$?

Which statements about ETCS (Lawvere's Elementary Theory of the Category of Sets) are true? (Choose all that apply.)

The truth values of a presheaf topos on a finite poset are its down-sets. Represent them as a list opens of frozensets (closed under unions and intersections). Write negation(u, opens), returning the Heyting negation $\neg u$ — the largest member of opens disjoint from u — and is_boolean(opens, top), returning True exactly when u | negation(u, opens) == top for every u in opens.