Every Map X \xrightarrow{f} \Omega is the Characteristic Map of a Monomorphism

 

Proposition

 Let \mathcal{E} be a topos with subobject classifier \Omega.  If f: X \to \Omega, then f = \chi_m for some mono m: Y \rightarrowtail X.

Proof

A topos is closed under taking pullbacks.  We already have the diagram,

Taking its pullback, we get a diagram,

But a pullback of \text{true} along any morphism f is always monic,

By the definition of subobject classifier, namely the part about uniqueness of \chi_m, we must have that \chi_m = f.

\blacksquare

Comments

Post a Comment

Popular posts from this blog

90% visual proof of Contravariant Yoneda Lemma

By the fact that \alpha_X : \text{Hom}_C(X,X) \to AX we have that \alpha_X(\text{id}_X) =: u \in AX and we're done with mapping any natural map \alpha : \text{Hom}_C(\cdot, Y) \to A to an element u \in AX. The sides of the triangular prism in the bottom diagram need to commute for each f:Y\to X, g : Z \to Y in C. For one, the triangular endcaps must commute because A, \text{Hom}_C(\cdot, X) are both contravariant functors. Next, the three square sides commute by naturality. We must then have that any time \alpha_Y(f) = A(f)\circ u, the diagram commutes.
Read more

Definition of Injective Object

 Let C be a category.  An object I\in C is injective if we have the following: English: For every mono m : X \rightarrowtail Y and map f: Y \to I, there exists a map g:Y\to I such that the triangle commutes, or in other words gm = f. Equivalently, for each mono m: X \rightarrowtail Y the induced map m^*: \text{Hom}(Y, I) \to \text{Hom}(X,I) given by m^*(g) = gm is surjective.  This is actually a nice way to handle the inherent \exists g logic, that I have not considered before.
Read more