Todd Trimble
Monic endomorphisms on the subobject classifier

Proposition

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

Proof

Introduce pullbacks

U i 1 V U k t k Ω ϕ Ω, 1 t Ω\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: ϕt=χ V:1Ω\phi t = \chi_V: 1 \to \Omega classifies the inclusion V1V \to 1.

From the composite pullback

V 1 V V 1 mono t U i 1 χ V Ω\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: χ Vi=k\chi_V i = k.

We therefore have a composite pullback

U 1 U U 1 i k t 1 χ V Ω ϕ Ω\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: ϕχ V=χ U\phi \chi_V = \chi_U.

Next, we have pullbacks

V UV 1 mono χ U U i 1 χ V Ω\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 VUV \hookrightarrow U. Now using fact 2, this gives a pullback

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

which is equivalent to the equation ϕχ U=ϕt\phi \chi_U = \phi t. Since ϕ\phi is monic, this establishes

  • Fact 4: χ U=t\chi_U = t.

Combining facts 3 and 4, we deduce

  • Fact 5: ϕχ V=t\phi \chi_V = t.

From facts 1 and 5, we therefore have ϕ 2t=t\phi^2 t = t. The pullback of the monic ϕ 2\phi^2 along tt is a monic W1W \to 1, but from ϕ 2t=t\phi^2 t = t this monic has a section 1W1 \to W, so this section is an isomorphism and the two subobjects tt and jj in the diagram

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

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

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