Todd Trimble Monic endomorphisms on the subobject classifier

Skip the Navigation Links | Home Page | All Pages |
Proposition

Let Ω\Omega be the subobject classifier in a topos, and let ϕ:Ω→Ω\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:1→Ω\phi t = \chi_V: 1 \to \Omega classifies the inclusion V→1V \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 ↪ U⇒V → 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 V↪UV \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 W→1W \to 1, but from ϕ 2t=t\phi^2 t = t this monic has a section 1→W1 \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