Characteristic Function for Subobject Classifier in the Topos of Presheaves

category-theorytopos-theory

Let $\mathbb{C}$ be a locally small category with pullbacks and $\hat{\mathbb{C}} = \mathrm{Set}^{\mathbb{C}^{\mathrm{op}}} = [\mathbb{C}^{\mathrm{op}}, \mathrm{Set}]$ the category of presheaves over $\mathbb{C}$. EDIT: Actually, it turns out that the correct requirement is that $\mathbb{C}$ is a small category.

In order to prove, that $\hat{\mathbb{C}}$ is an elementary topos one needs that $\hat{\mathbb{C}}$ has a subobject classifier $\Omega$. I am having a hard time understanding the working of the characteristic function as it is presented in Cecilia Flori's "First Course in Topos Quantum Theory". Before I state my problem, let me recall the definition of the subobject classifier as it is given there.

Background

Using sieves, $\Omega$ is defined as the functor $\mathbb{C}^{\mathrm{op}} \rightarrow \mathrm{Set}$ that sends

  • every $A \in \mathbb{C}$ to the set $\Omega(A)$ of sieves on $A$, i.e. to the set of all sets $S \subset \mathrm{Mor}(\mathbb{C})$ such that $S$ contains only morphisms with codomain $A$ and such that $S$ is closed under precomposition,
  • every $f: A \leftarrow B$ to the map $\Omega(f): \Omega(A) \rightarrow \Omega(B), \quad S \mapsto \{g:C \rightarrow B \, |\; f \circ g \in S\}$.

Then the "true" function $t: 1 \rightarrow \Omega$ is defined componentwise by $$t_A: \{\star\} \rightarrow \Omega(A), \quad \star \mapsto (\downarrow A)$$
for every $A \in \mathbb{C}$, where $(\downarrow A)$ is the principal sieve on $A$, which contains every morphism with codomain $A$.

Problem

Now, consider a monic arrow $\sigma: F \rightarrow X$ in $\hat{\mathbb{C}}$, where every component of $\sigma$ is a set-inclusion. What is then the corresponding characteristic function $\chi^\sigma: X \rightarrow \Omega$? For a given $A \in \mathbb{C}$, Flori presents the diagram below:

Diagram

Then she says:

From the above diagram we can see that $\chi^\sigma_A$ assigns to each element $x \in X(A)$ a sieve $\chi^\sigma_A(x)$ on $A$ such that $$ \chi^\sigma_A(x) = \{f: B \rightarrow A \, | \; X(f)(x) \in F(B)\}.$$

However, I cannot see how this statement follows directly from the diagram.

What I did so far

In order to get $F(B)$ into the picture, I chose a fixed $f: B \rightarrow A$ and looked at the following diagram:

My_Diagram

From that I was at least able to conclude $$\{f: B \rightarrow A \, | \; X(f)(x) \in F(B) \} \subset \chi^\sigma_A(x), $$
by assuming for some $f: B \rightarrow A$ that $X(f)(x) \in F(B)$ holds. Then the commuting diagram in the background gives $\chi^\sigma_B(y) = (\downarrow B)$ for $y = X(f)(x)$. Hence, $\mathrm{id}_B$ is in the sieve $\Omega(f) (\chi^\sigma_A(x))$ and by definition of $\Omega(f)$ it follows directly that $f$ is in $\chi^\sigma_A(x)$. But what about the other inclusion?

Any help is appreciated!

Best Answer

Rather than requiring $\mathbb{C}$ to be a locally small category with pullbacks, the correct requirement here is actually that $\mathbb{C}$ be small. If $\mathbb{C}$ is not small, then we should not expect $\Omega(A)$ to form a set as defined.

For instance, let $\mathbb{C}$ be the ordinals, considered as a category under the reverse order. Then $\mathbb{C}$ has pullbacks and is locally small. But each ordinal $\beta \geq \alpha$ gives rise to a sieve $S_\beta = \{\gamma \mid \gamma \geq \beta\}$ on $\alpha$, and $S$ is injective, so the class of sieves on $\alpha$ is a proper class and not a set. So $\Omega$ is not well-defined in this instance.

Let's move on to the substance of the question. We start out by supposing that $\sigma$ is the pullback of $\top$ along some morphism $\chi^\sigma$.

Recall that a diagram is a pullback square in $\hat{\mathbb{C}}$ if and only if the diagram, evaluated at each $A \in \mathbb{C}$, is a pullback square in $\mathbb{C}$. So we have, for each $A$, a pullback diagram as pictured in the question.

Let's begin by noting that

Lemma: For all $A \in \mathbb{C}$, for all $x \in X(a)$, $\downarrow A = \chi^\sigma_A(x)$ if and only if $x \in F(A)$.

Proof: this follows immediately from the pullback diagram evaluated at $A$. $\square$

Corollary: For all $A \in \mathbb{C}$, for all $x \in X(a)$, $1_A \in \chi^\sigma_A(x)$ if and only if $x \in F(A)$.

Proof: $1_A \in \chi^\sigma_A(x)$ if and only if $\chi^\sigma_A(x) = \downarrow A$ if and only if $x \in F(A)$. $\square$

Let's introduce some notation.

Notation. For some presheave $P$, some objects $A, B \in \mathbb{C}$, some morphism $f : B \to A$, and some $x \in P(A)$, write $x \cdot f = P(f)(x)$. Note that $x \cdot 1_A = x$ and that $x \cdot (f \circ g) = (x \cdot f) \cdot g$. $\square$

Now, we're ready to move on to the major theorem.

Thm. Consider objects $A, B \in \mathbb{C}$, arrow $f : B \to A$, and $x \in X(A)$. Then $f \in \chi^\sigma_A(x)$ if and only if $x \cdot f \in F(A)$.

Proof: note that $\chi^\sigma_B(x \cdot f) = \chi^\sigma_A(x) \cdot f$ by naturality. Now using the definition of $\cdot f$ on $\Omega$, we see that $f \in \chi^\sigma_A(x)$ if and only if $1_B \in \chi^\sigma_A(x) \cdot f$. And of course, $1_B \in \chi^\sigma_A(x) \cdot f = \chi^\sigma_B(x \cdot f)$ if and only if $x \cdot f \in F(B)$ by the corollary. $\square$

Thus, we can write $\chi^\sigma_A(x) = \{f : B \to A \mid x \cdot f \in F(B)\}$, which is exactly what we desired to prove.

Note that what we have shown so far is

If $\sigma$ is the pullback of $\top$ along $\chi^\sigma$, then for all $A$, for all $x \in X(A)$, $\chi^\sigma_A(x) = \{f : B \to A \mid x \cdot f \in F(B)\}$.

What we then need to show is the flip side:

Consider the family of functions defined by $\chi^\sigma_A(x) := \{f : B \to A \mid x \cdot f \in F(B)\}$. This family is a natural transformation $X \to \Omega$, and the pullback of $\top$ along $\chi^\sigma$ is $\sigma$.

This gives us both existence and uniqueness.

Related Question