Lifting subobject classifier of a category to its arrow category

category-theorytopos-theory

I'm reading Goldblatt's Topoi and trying to practice categorical reasoning, generalizing the example of $\mathbf{Set}^\rightarrow$ being a topos.

So, let $\mathcal{C}$ be a category with a subobject classifier $\Omega, \top$ where $\top : \mathbf{1} \rightarrow \Omega$. Consider the arrow category $\mathcal{C}^\rightarrow$: does it have a subobject classifier?

I think that it does: $\text{id}_\Omega, (\top, \top)$ seems like a good candidate. In particular, if $f, g$ are some monic arrows in $\mathcal{C}$ with characters $\chi_f, \chi_g$, then $(\chi_f, \chi_g)$ is the character of $(f, g)$ in $\mathcal{C}^\rightarrow$.

The construction I'm having in mind seems to follow from the structure of the proof I've done earlier that if $\mathcal{C}$ has pullbacks, then so does $\mathcal{C}^\rightarrow$ (namely, "gluing" two pullbacks of $\mathcal{C}$ in a cube-like diagram naturally compatible with the arrow category structure produces a pullback in $\mathcal{C}^\rightarrow$), but said proof is quite lengthy, so I'm omitting it here. The uniqueness of characters follows from the uniqueness of characters in $\mathcal{C}$.

So, my questions:

  1. If $\mathcal{C}$ has S.C.s, then does $\mathcal{C}^\rightarrow$ also have them?
  2. If it does, is the above indeed a S.C.?
  3. If it is, why Goldblatt is using a seemingly more involved construct for the specific example of $\mathbf{Set}^\rightarrow$?

Best Answer

$\require{AMScd}$If $\cal E$ is an elementary topos, then so is the functor category ${\cal E}^C$ for every small category $C$; the classifying monomorphism is just the image of ${\sf true}:1 \hookrightarrow \Omega$ along the diagonal functor $\Delta : {\cal E} \to {\cal E}^C$.

To see this, you can consider the obvious map ${\cal E}^C(G, \Delta\Omega) \to \text{Sub}(G)$ that sends a morphism $g : G \to \Delta\Omega$ to the pullback of the square $$ \begin{CD} S(g) @>>> \Delta1 \\ @Vm_gVV @VVV \\ G @>>g> \Delta\Omega \end{CD} $$ Limits and monics in functor categories are defined objectwise, so this just means taking the pullback objectwise; explicitly, for each $x\in\cal E$ $$ \begin{CD} S(g)x @>>> 1 \\ @Vm_{g,x}VV @VVV \\ Gx @>>g> \Omega \end{CD} $$ is a pullback in $\cal E$. Such pullback $S(g) \hookrightarrow G$ exists, because $\cal E$ is a topos, and it is a monic.

It remains to show that every monic arises in this way. Given such a monic $\alpha : F \Rightarrow G$, each $\alpha_x : Fx \to Gx$ in $\cal E$ corresponds to a certain $\chi_x^\alpha : Gx \to \Omega$ by virtue of the bijection $\text{Sub}(Gx)\cong {\cal E}(Gx,\Omega)$; since now the collection $\chi_x^\alpha : Gx \to \Omega$ forms a cocone for $G$, it corresponds to a unique natural transformation $\bar\chi^\alpha : G \Rightarrow \Delta\Omega$.

It is now just a matter of uwinding the definition to see that the pullback $m_{\bar\chi^\alpha} : S(\bar \chi^\alpha) \Rightarrow G$ coincides with $\alpha$, and conversely that if $g : G \Rightarrow \Delta\Omega$ is a morphism, $\bar\chi^{m_g} = g$.

Related Question