Covariant Isotropy of Grothendieck Toposes and Extensive Categories

Abstract

We provide an explicit characterization of the covariant isotropy group of any Grothendieck topos, i.e. the group of (extended) inner automorphisms of any sheaf over a small site. In order to do so, we first extend previous techniques for computing covariant isotropy from locally finitely presentable categories to locally presentable categories. As a consequence, we also obtain an explicit characterization of the centre of a Grothendieck topos, i.e. the automorphism group of its identity functor. We conclude by providing a more categorical approach to show that these characterizations also extend to any extensive category.

Appendix

Lemma (3.4)

Let $(\mathbb {C}, \mathcal {J})$ be a small subcanonical site in which no object is covered by the empty sieve, and let $F \in \mathsf {Sh}(\mathbb {C}, \mathcal {J})$. If $f, g : D \rightarrow C$ are parallel morphisms in $\mathbb {C}$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _f(\mathsf {x}_C) = \alpha _g(\mathsf {x}_C)$, then $f = g$.

Proof

Assume the hypotheses, let $\mathsf {y}: \mathbb {C}\rightarrow \mathsf {Set}^{\mathbb {C}^\mathsf {op}}$ be the Yoneda embedding, and let $\mathsf {a}: \mathsf {Set}^{\mathbb {C}^\mathsf {op}} \rightarrow \mathsf {Sh}(\mathbb {C}, \mathcal {J})$ be the associated sheaf functor, defined in terms of the plus-construction as $\mathsf {a}(G) := G^{++}$ for any presheaf G (cf. e.g. [11, Section III.5]). Since $(\mathbb {C}, \mathcal {J})$ is subcanonical, we know that $\mathsf {y}C \in \mathsf {Sh}(\mathbb {C}, \mathcal {J})$, from which it follows by Lemma 3.3 that the coproduct presheaf $F + \mathsf {y}C$ is separated. By [11, Lemma III.5.5], it then follows that $(F + \mathsf {y}C)^+ \in \mathsf {Sh}(\mathbb {C}, \mathcal {J})$. Recall that for any $D \in \mathsf {Ob}(\mathbb {C})$, the set $(F + \mathsf {y}C)^+(D)$ is the set of matching families in the presheaf $F + \mathsf {y}C$ for covers in $\mathcal {J}(D)$ modulo the equivalence relation which identifies two such matching families $(x_f)_{f \in R}$ and $(y_g)_{g \in S}$ when there is a cover $T \in \mathcal {J}(D)$ with $T \subseteq R \cap S$ and $x_h = y_h$ for all $h \in T$. We then have a canonical natural transformation $\eta : F \rightarrow (F + \mathsf {y}C)^+$ defined as follows: for any $D \in \mathsf {Ob}(\mathbb {C})$, the function $\eta _D : F(D) \rightarrow (F + \mathsf {y}C)^+(D)$ sends $d \in F(D)$ to the equivalence class of the matching family $(F(f)(d))_{f \in t_D}$, where $t_D \in \mathcal {J}(D)$ is the maximal sieve.

Since $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _f(\mathsf {x}_C) = \alpha _g(\mathsf {x}_C)$ by assumption and $\eta : F \rightarrow (F + \mathsf {y}C)^+$ is a natural transformation and hence a morphism in $\mathsf {Sh}(\mathbb {C}, \mathcal {J})$, it follows by (the bounded infinitary version of) [14, Lemma 3.1.2] that

$$\begin{aligned} (F + \mathsf {y}C)^+(f) = (F + \mathsf {y}C)^+(g) : (F + \mathsf {y}C)^+(C) \rightarrow (F + \mathsf {y}C)^+(D). \end{aligned}$$

The equivalence class $\left[ (h)_{h \in t_C}\right]$ of the matching family $(h)_{h \in t_C}$ in $\mathsf {y}C$ belongs to $(F + \mathsf {y}C)^+(C)$, and hence we have

$$\begin{aligned} (F + \mathsf {y}C)^+(f)\left( \left[ (h)_{h \in t_C}\right] \right) = (F + \mathsf {y}C)^+(g)\left( \left[ (h)_{h \in t_C}\right] \right) , \end{aligned}$$

i.e.

$$\begin{aligned} \left[ (fh)_{h \in t_D}\right] = \left[ (gh)_{h \in t_D}\right] \in (F + \mathsf {y}C)^+(D). \end{aligned}$$

So there is a cover $T \in \mathcal {J}(D)$ with $f \circ h = g \circ h$ for every $h \in T$. We also have a matching family $(fh)_{h \in T}$ for $T \in \mathcal {J}(D)$ in $\mathsf {y}C$. Since $\mathsf {y}C$ is a sheaf and in particular separated, there is at most one morphism $k : D \rightarrow C$ in $\mathbb {C}$ with $k \circ h = f \circ h$ for every $h \in T$. But f and g both satisfy this property, so that we must have $f = g$, as desired. $\square$

Lemma (3.5)

Let $(\mathbb {C}, \mathcal {J})$ be a small subcanonical site in which no object is covered by the empty sieve, and let $F \in \mathsf {Sh}(\mathbb {C}, \mathcal {J})$ and $C \in \mathsf {Ob}(\mathbb {C})$. For any pure closed term $t \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t \downarrow$ and t : D for some $D \in \mathsf {Ob}(\mathbb {C})$, there is some morphism $f : D \rightarrow C$ in $\mathbb {C}$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t = \alpha _f(\mathsf {x}_C)$.

Proof

Assume the hypotheses, and let us prove the claim by induction on the structure of $t \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ with t pure and $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t \downarrow$.

• If $t \equiv \mathsf {x}_C : C$, then $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t \downarrow$ and we clearly have $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \mathsf {x}_C = \alpha _{\mathsf {id}_C}(\mathsf {x}_C)$.

• Suppose $t \equiv \alpha _g(t') : D$ for some morphism $g : D \rightarrow D'$ in $\mathbb {C}$ and some pure $t' \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ of sort $D' \in \mathsf {Ob}(\mathbb {C})$, and suppose moreover that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _g(t') \downarrow$, so that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t' \downarrow$. Then by the induction hypothesis, there is some morphism $h : D' \rightarrow C$ in $\mathsf {Ob}(\mathbb {C})$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t' = \alpha _h(\mathsf {x}_C)$, and hence we obtain

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t = \alpha _g(t') = \alpha _g(\alpha _h(\mathsf {x}_C)) = \alpha _{h \circ g}(\mathsf {x}_C), \end{aligned}$$

  as desired.

• Suppose that $t \equiv \sigma _J\left( \left( t_h\right) _{h \in J}\right) : D$ for some $D \in \mathsf {Ob}(\mathbb {C})$ and $J \in \mathcal {J}(D)$ and pure $t_h \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ of sort $\mathsf {dom}(h)$ for every $h \in J$, and suppose also that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \sigma _J\left( \left( t_h\right) _{h \in J}\right) \downarrow$, so that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_h \downarrow$ for every $h \in J$. Moreover, it follows that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _g\left( t_h\right) = t_{h \circ g}$ for every $h \in J$ and $g \in \mathsf {Arr}(\mathbb {C})$ with $\mathsf {cod}(g) = \mathsf {dom}(h)$, and that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _h(t) = t_h$ for every $h \in J$. By the induction hypothesis, we know for every $h \in J$ that there is some morphism $f_h : \mathsf {dom}(h) \rightarrow C$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_h = \alpha _{f_h}(\mathsf {x}_C)$. So for any $h \in J$ and $g \in \mathsf {Arr}(\mathbb {C})$ with $\mathsf {cod}(g) = \mathsf {dom}(h)$, we have

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _{f_h \circ g}(\mathsf {x}_C) = \alpha _g\left( \alpha _{f_h}(\mathsf {x}_C)\right) = \alpha _g(t_h) = t_{h \circ g} = \alpha _{f_{h \circ g}}(\mathsf {x}_C). \end{aligned}$$

  By Lemma 3.4, it then follows that $f_h \circ g = f_{h \circ g}$ for every $h \in J$ and $g \in \mathsf {Arr}(\mathbb {C})$ with $\mathsf {cod}(g) = \mathsf {dom}(h)$. This means that the family of morphisms $\left( f_h\right) _{h \in J}$ is matching in $\mathsf {y}C = \mathbb {C}(-, C)$ for the cover $J \in \mathcal {J}(D)$. Since $\mathsf {y}C$ is a sheaf by assumption, it follows that there is a unique morphism $f : D \rightarrow C$ in $\mathbb {C}$ with $f \circ h = f_h$ for every $h \in J$. We now show that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \sigma _J\left( \left( t_h\right) _{h \in J}\right) = \alpha _f(\mathsf {x}_C)$, completing the proof. By the axiom of $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}$ expressing the uniqueness of amalgamations of matching families, it suffices to show that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _h\left( \alpha _f(\mathsf {x}_C)\right) = t_h$ for every $h \in J$. But we have

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _h\left( \alpha _f(\mathsf {x}_C)\right) = \alpha _{f \circ h}(\mathsf {x}_C) = \alpha _{f_h}(\mathsf {x}_C) = t_h, \end{aligned}$$

  as desired.

$\square$

Lemma (3.6)

Let $(\mathbb {C}, \mathcal {J})$ be a small site in which no object is covered by the empty sieve, and let $F \in \mathsf {Sh}(\mathbb {C}, \mathcal {J})$ and $C \in \mathsf {Ob}(\mathbb {C})$. For any morphism $f \in \mathsf {Arr}(\mathbb {C})$ with $\mathsf {cod}(f) = C$, there is no $a \in F(\mathsf {dom}(f))$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _f(\mathsf {x}_C) = c_a$.

Proof

Assume the hypotheses, let $f : D \rightarrow C$, and suppose towards a contradiction that there were some $a \in F(D)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _f(\mathsf {x}_C) = c_a$. Then for any $G \in \mathsf {Sh}(\mathbb {C}, \mathcal {J})$ and natural transformation $\gamma : F \rightarrow G$, it follows by [14, Lemma 3.1.2] that the function $G(f) : G(C) \rightarrow G(D)$ is constant on $\gamma _{D}(a) \in G(D)$. Let $\mathbb {1} : \mathbb {C}^\mathsf {op}\rightarrow \mathsf {Set}$ be the terminal sheaf constant on the singleton $\{*\}$, and consider the coproduct presheaf $F + \mathbb {1}$ (assuming without loss of generality that F and $\mathbb {1}$ are pointwise disjoint). By Lemma 3.3, it follows that $F + \mathbb {1}$ is separated, so that $(F + \mathbb {1})^+$ is a sheaf by [11, Lemma III.5.5]. We have a natural transformation $\eta : F \rightarrow (F + \mathbb {1})^+$ defined as follows: for any $X \in \mathsf {Ob}(\mathbb {C})$, the function $\eta _X : F(X) \rightarrow (F + \mathbb {1})^+(X)$ sends $x \in F(X)$ to the equivalence class of the matching family $(F(h)(x))_{h \in t_X}$, where $t_X \in \mathcal {J}(X)$ is the maximal sieve. To obtain a contradiction and complete the proof, it suffices to show that $(F + \mathbb {1})^+(f) : (F + \mathbb {1})^+(C) \rightarrow (F + \mathbb {1})^+(D)$ is not constant on $\eta _D(a) = \left[ (F(h)(a))_{h \in t_D}\right] \in (F + \mathbb {1})^+(D)$. We have $\left[ (*)_{h \in t_C}\right] \in (F + \mathbb {1})^+(C)$. If we had

$$\begin{aligned} (F + \mathbb {1})^+(f)\left( \left[ (*)_{h \in t_C}\right] \right) = \left[ (F(h)(a))_{h \in t_D}\right] , \end{aligned}$$

so that $\left[ (F(h)(a))_{h \in t_D}\right] = \left[ (*)_{h \in t_D}\right]$, then there would be some cover $J \in \mathcal {J}(D)$ with $F(k)(a) = *$ for all $k \in J$, which is impossible, since $J \ne \varnothing$ and F and $\mathbb {1}$ are disjoint. So $(F + \mathbb {1})^+(f)\left( \left[ (*)_{h \in t_C}\right] \right) \ne \left[ (F(h)(a))_{h \in t_D}\right]$, as desired. $\square$

Lemma (3.7)

Let $(\mathbb {C}, \mathcal {J})$ be a small site, and let $F \in \mathsf {Sh}(\mathbb {C}, \mathcal {J})$ and $C \in \mathsf {Ob}(\mathbb {C})$. For any closed term $t \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t \downarrow$ and t : D for some $D \in \mathsf {Ob}(\mathbb {C})$, there is some cover $J \in \mathcal {J}(D)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t = \sigma _J\left( \left( t_h\right) _{h \in J}\right)$, where for any $h \in J$ either $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_h = c_a$ for some $a \in F(\mathsf {dom}(h))$ or $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_h = \alpha _f(\mathsf {x}_C)$ for some morphism $f : \mathsf {dom}(h) \rightarrow C$ in $\mathbb {C}$.

Proof

Assume the hypotheses. The (final case of the) following inductive proof is essentially a syntactic version of the proof of [11, Lemma III.5.5]. We prove the desired result by induction on the structure of $t \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t \downarrow$.

• If $t \equiv \mathsf {x}_C : C$, so that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t \downarrow$, then we have the maximal sieve $t_C \in \mathcal {J}(C)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _f(\mathsf {x}_C) \downarrow$ for every $f \in t_C$, and for any $f \in t_C$ and $g \in \mathsf {Arr}(\mathbb {C})$ with $\mathsf {cod}(g) = \mathsf {dom}(f)$ we have $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _g(\alpha _f(\mathsf {x}_C)) = \alpha _{f \circ g}(\mathsf {x}_C)$. It then follows that

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \sigma _{t_C}\left( \left( \alpha _f(\mathsf {x}_C)\right) _{f \in t_C}\right) \downarrow \end{aligned}$$

  and

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _f\left( \sigma _{t_C}\left( \left( \alpha _f(\mathsf {x}_C)\right) _{f \in t_C}\right) \right) = \alpha _f(\mathsf {x}_C) \end{aligned}$$

  for every $f \in t_C$, so that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \mathsf {x}_C = \sigma _{t_C}\left( \left( \alpha _f(\mathsf {x}_C)\right) _{f \in t_C}\right)$ by the uniqueness of amalgamations, as desired.

• If $t \equiv c_a : D$ for some $D \in \mathsf {Ob}(\mathbb {C})$ and $a \in F(D)$, then the reasoning is identical to the first case.

• Suppose that $t \equiv \alpha _k(s) : E$ for some morphism $k : E \rightarrow D$ in $\mathbb {C}$ and some term $s \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ with s : D, and suppose that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t \downarrow$, which implies that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash s \downarrow$. By the induction hypothesis, there is some cover $J \in \mathcal {J}(D)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash s = \sigma _J\left( \left( s_h\right) _{h \in J}\right)$, where for any $h \in J$ either $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash s_h = c_a$ for some $a \in F(\mathsf {dom}(h))$ or $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash s_h = \alpha _f(\mathsf {x}_C)$ for some morphism $f : \mathsf {dom}(h) \rightarrow C$ in $\mathbb {C}$. We then have the pullback sieve $k^*J \in \mathcal {J}(E)$ consisting of those morphisms f with codomain E for which $k \circ f \in J$, with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _f(\alpha _k(s)) \downarrow$ for every $f \in k^*J$, and for any $f \in k^*J$ and $g \in \mathsf {Arr}(\mathbb {C})$ with $\mathsf {cod}(g) = \mathsf {dom}(f)$ we have $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _g(\alpha _f(\alpha _k(s))) = \alpha _{f \circ g}(\alpha _k(s))$. It then follows that

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \sigma _{k^*J}\left( \left( \alpha _f(\alpha _k(s))\right) _{f \in k^*J}\right) \downarrow \end{aligned}$$

  and

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _f\left( \sigma _{k^*J}\left( \left( \alpha _f(\alpha _k(s))\right) _{f \in k^*J}\right) \right) = \alpha _f(\alpha _k(s)) \end{aligned}$$

  for every $f \in k^*J$, so that

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _k(s) = \sigma _{k^*J}\left( \left( \alpha _f(\alpha _k(s))\right) _{f \in k^*J}\right) = \sigma _{k^*J}\left( \left( \alpha _{k \circ f}(s)\right) _{f \in k^*J}\right) \end{aligned}$$

  by the uniqueness of amalgamations. But for any $f \in k^*J$ we have $k \circ f \in J$, so that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _{k \circ f}(s) = s_{k \circ f}$, and hence by the induction hypothesis it follows that $t \equiv \alpha _k(s)$ is provably equal to a term of the desired form.

• Lastly, suppose that $t \equiv \sigma _J\left( \left( t_h\right) _{h \in J}\right) : D$ for some $D \in \mathsf {Ob}(\mathbb {D})$ and some cover $J \in \mathcal {J}(D)$, with $t_h \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ of sort $\mathsf {dom}(h)$ for every $h \in J$. Assume that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t \downarrow$, so that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_h \downarrow$ and $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _h(t) = t_h$ for every $h \in J$, and $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _g(t_h) = t_{h \circ g}$ for every $h \in J$ and $g \in \mathsf {Arr}(\mathbb {C})$ with $\mathsf {cod}(g) = \mathsf {dom}(h)$. By the induction hypothesis, we know for every $h \in J$ that there are some cover $J_h \in \mathcal {J}(\mathsf {dom}(h))$ and terms $t_{h, k} \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ of sort $\mathsf {dom}(k)$ for every $k \in J_h$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_h = \sigma _{J_h}\left( \left( t_{h, k}\right) _{k \in J_h}\right)$, and each $t_{h, k}$ is provably equal to an object constant of F or a term of the form $\alpha _\ell (\mathsf {x}_C)$ for some morphism $\ell : \mathsf {dom}(k) \rightarrow C$ of $\mathbb {C}$. By the reasoning in the previous case, for every $h \in J$ and $g \in \mathsf {Arr}(\mathbb {C})$ with $\mathsf {cod}(g) = \mathsf {dom}(h)$ we have that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)$ proves the equations

  $$\begin{aligned}&\ \ \ \ \sigma _{J_{h \circ g}}\left( \left( t_{h \circ g, k}\right) _{k \in J_{h \circ g}}\right) \\&= t_{h \circ g} \\&= \alpha _g(t_h) \\&= \alpha _g\left( \sigma _{J_h}\left( \left( t_{h, k}\right) _{k \in J_h}\right) \right) \\&= \sigma _{g^*J_h}\left( \left( t_{h, g \circ k}\right) _{k \in g^*J_h}\right) . \end{aligned}$$

  Considering the cover $T_{h, g} := J_{h \circ g} \cap g^*J_h \in \mathcal {J}(\mathsf {dom}(g))$, for any $k \in T_{h, g}$ we then have that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)$ proves

  $$\begin{aligned} t_{h \circ g, k} = \alpha _k\left( \sigma _{J_{h \circ g}}\left( \left( t_{h \circ g, k}\right) _{k \in J_{h \circ g}}\right) \right) = \alpha _k\left( \sigma _{g^*J_h}\left( \left( t_{h, g \circ k}\right) _{k \in g^*J_h}\right) \right) = t_{h, g \circ k}. \end{aligned}$$

  Now consider the sieve $K := \left\{ h \circ k :h \in J, k \in J_h\right\}$ on D. Since $J \in \mathcal {J}(D)$ and $J_h \in \mathcal {J}(\mathsf {dom}(h))$ for all $h \in J$, it follows by the transitivity axiom for a Grothendieck topology that $K \in \mathcal {J}(D)$. For any $h \in J$ and $k \in J_h$, we have $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_{h, k} \downarrow$. We now claim that if $h, h' \in J$ and $k \in J_h, k' \in J_{h'}$ with $h \circ k = h' \circ k'$, then $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_{h, k} = t_{h', k'}$. Indeed, for any $g \in T_{h, k} \cap T_{h', k'}$ we have

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _g\left( t_{h, k}\right) = t_{h, k \circ g} = t_{h \circ k, g} = t_{h' \circ k', g} = t_{h', k' \circ g} = \alpha _g\left( t_{h', k'}\right) . \end{aligned}$$

  Since $T_{h, k} \cap T_{h', k'} \in \mathcal {J}(\mathsf {dom}(k)) = \mathcal {J}(\mathsf {dom}(k'))$, we then conclude from the fourth group of axioms for $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}$ that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_{h, k} = t_{h', k'}$, as desired. For any $h \in J$ and $k \in J_h$, we now set $s_{h \circ k} := t_{h, k}$, which is well-defined up to provable equality in $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)$ by what we just showed. For any $g \in \mathsf {Arr}(\mathbb {C})$ with $\mathsf {cod}(g) = \mathsf {dom}(h \circ k)$ we also have

  $$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _g(s_{h \circ k}) = \alpha _g(t_{h, k}) = t_{h, k \circ g} = s_{h \circ (k \circ g)} = s_{(h \circ k) \circ g}, \end{aligned}$$

  which entails that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \sigma _K\left( \left( s_{h \circ k}\right) _{h \circ k \in K}\right) \downarrow$. Given that each $s_{h \circ k} \equiv t_{h, k}$ has the desired form by the induction hypothesis, it remains to prove that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \sigma _J\left( \left( t_h\right) _{h \in J}\right) = \sigma _K\left( \left( s_{h \circ k}\right) _{h \circ k \in K}\right)$. By the uniqueness of amalgamations, it suffices to show for any $h' \in J$ that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_{h'} = \alpha _{h'}\left( \sigma _K\left( \left( s_{h \circ k}\right) _{h \circ k \in K}\right) \right)$, i.e. that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \sigma _{J_{h'}}\left( \left( t_{h', k}\right) _{k \in J_{h'}}\right) = \alpha _{h'}\left( \sigma _K\left( \left( s_{h \circ k}\right) _{h \circ k \in K}\right) \right)$. And to show this, it again suffices by uniqueness of amalgamations to show for any $k \in J_{h'}$ that $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_{h', k} = \alpha _k\left( \alpha _{h'}\left( \sigma _K\left( \left( s_{h \circ k}\right) _{h \circ k \in K}\right) \right) \right)$, which just follows because $h' \circ k \in K$ and $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash s_{h' \circ k} = t_{h', k}$. This completes the proof.

$\square$

Lemma (3.8)

Let $(\mathbb {C}, \mathcal {J})$ be a small subcanonical site in which no object is covered by the empty sieve, and let $F \in \mathsf {Sh}(\mathbb {C}, \mathcal {J})$ and $C \in \mathsf {Ob}(\mathbb {C})$. For any closed term $t \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t \downarrow$ and t : C, if there is some term $s \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash s \downarrow$ and s : C and $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t[s/\mathsf {x}_C] = \mathsf {x}_C$, then there is a pure term $t' \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t = t'$.

Proof

Assume the hypotheses. By Lemma 3.7, there is a cover $J \in \mathcal {J}(C)$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t = \sigma _J\left( \left( t_h\right) _{h \in J}\right)$ for some terms $t_h \in \mathsf {Term}^c\left( \Sigma ^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C)\right)$ of sort $\mathsf {dom}(h)$ for all $h \in J$ satisfying the conditions of Lemma 3.7. Now fix $h \in J$. By assumption, we have

$$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _h(\mathsf {x}_C) = \alpha _h\left( t[s/\mathsf {x}_C]\right) = \alpha _h\left( \sigma _J\left( \left( t_h[s/\mathsf {x}_C]\right) _{h \in J}\right) \right) = t_h[s/\mathsf {x}_C]. \end{aligned}$$

We now show that there can be no object $a \in F(\mathsf {dom}(h))$ with $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_h = c_a$. For if there were, then we would have $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash \alpha _h(\mathsf {x}_C) = t_h[s/\mathsf {x}_C] = c_a[s/\mathsf {x}_C] = c_a$, contradicting Lemma 3.6. Then by Lemma 3.7 we must have $\mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t_h = \alpha _{f_h}(\mathsf {x}_C)$ for some morphism $f_h : \mathsf {dom}(h) \rightarrow C$ in $\mathbb {C}$ (for every $h \in J$). We therefore have

$$\begin{aligned} \mathbb {T}^{(\mathbb {C}, \mathcal {J})}(F, \mathsf {x}_C) \vdash t = \sigma _J\left( \left( t_h\right) _{h \in J}\right) = \sigma _J\left( \left( \alpha _{f_h}(\mathsf {x}_C)\right) _{h \in J}\right) , \end{aligned}$$

with the latter term being pure. $\square$