# Todd Trimble Monic endomorphisms on the subobject classifier

###### Proposition

Let $\Omega$ be the subobject classifier in a topos, and let $\phi: \Omega \to \Omega$ be a monomorphism. Then $\phi$ is an involution.

###### Proof

Introduce pullbacks

$\array{ U & \stackrel{i}{\hookrightarrow} & 1 & & & V & \hookrightarrow & U\\ \mathllap{k} \downarrow & & \downarrow \mathrlap{t} & & & \downarrow & & \downarrow \mathrlap{k} \\ \Omega & \underset{\phi}{\to} & \Omega, & & & 1 & \underset{t}{\hookrightarrow} & \Omega }$

and observe

• Fact 1: $\phi t = \chi_V: 1 \to \Omega$ classifies the inclusion $V \to 1$.

From the composite pullback

$\array{ V & \stackrel{1_V}{\to} & V & \hookrightarrow & 1 \\ \mathllap{mono} \downarrow & & \downarrow & & \downarrow \mathrlap{t} \\ U & \underset{i}{\to} & 1 & \underset{\chi_V}{\to} & \Omega }$

we deduce

• Fact 2: $\chi_V i = k$.

We therefore have a composite pullback

$\array{ U & \stackrel{1_U}{\to} & U & \to & 1 \\ \mathllap{i} \downarrow & & \downarrow \mathrlap{k} & & \downarrow \mathrlap{t} \\ 1 & \underset{\chi_V}{\to} & \Omega & \underset{\phi}{\to} & \Omega }$

so that

• Fact 3: $\phi \chi_V = \chi_U$.

Next, we have pullbacks

$\array{ V & \hookrightarrow & U \Rightarrow V & \to & 1 \\ \mathllap{mono} \downarrow & & \downarrow & & \downarrow \mathrlap{\chi_U} \\ U & \underset{i}{\to} & 1 & \underset{\chi_V}{\to} & \Omega }$

using the inclusion $V \hookrightarrow U$. Now using fact 2, this gives a pullback

$\array{ V & \hookrightarrow & U \\ \downarrow & & \downarrow \mathrlap{k} \\ 1 & \underset{\chi_U}{\to} & \Omega }$

which is equivalent to the equation $\phi \chi_U = \phi t$. Since $\phi$ is monic, this establishes

• Fact 4: $\chi_U = t$.

Combining facts 3 and 4, we deduce

• Fact 5: $\phi \chi_V = t$.

From facts 1 and 5, we therefore have $\phi^2 t = t$. The pullback of the monic $\phi^2$ along $t$ is a monic $W \to 1$, but from $\phi^2 t = t$ this monic has a section $1 \to W$, so this section is an isomorphism and the two subobjects $t$ and $j$ in the diagram

$\array{ 1 & \to & W & \to & 1 \\ & \mathllap{t} \searrow & \downarrow \mathrlap{j} & pb & \downarrow \mathrlap{t} \\ & & \Omega & \underset{\phi^2}{\to} & \Omega }$

coincide, proving $\phi^2 = 1_\Omega$.

Created on February 25, 2013 at 03:37:00 by Todd Trimble