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Stream: learning: questions

Topic: Epis in a topos

Samuel Steakley (Sep 26 2022 at 19:23):

The other day, I read on the nLab page on Lawvere's fixed point theorem that a map $f : A \to B$ in a topos is an epi if and only if the direct image map $\exists_f : \Omega^A \to \Omega^B$ retracts the inverse image map $\Omega^f : \Omega^B \to \Omega^A$. I had never studied topos theory before, and I wanted to understand why this is the case. I started digging, and now I have learned some basics for toposes.

I've managed to determine that the structure of a topos $\mathcal{E}$ provides for a natural isomorphism $\textrm{Sub} \cong \mathcal{E}(-,\Omega) \cong \mathcal{E}(1, \Omega^{(-)})$. Of course, $\textrm{Sub} : \mathcal{E}^{op} \to \textbf{Set}$ is the (contravariant) functor sending each object to its set of subobjects, and $\Omega$ is the object of "truth values" which is obtained from the subobject classifier. It jumps out at me that the "inverse image map" must be intimately connected to the value of a map $f$ under $\textrm{Sub}$, but I am now stuck, and I fear that I am suffering from not having closely studied the underlying adjunction of the cartesian closed structure of a topos.

Can anyone tell me if I am headed in roughly the right direction, or if I am way off track? I could be off track either in focusing on the natural isomorphism I displayed above, or in the idea that I may be needing more background on the cartesian closed structure. Any hints or tips would be greatly appreciated.

Morgan Rogers (he/him) (Sep 27 2022 at 10:40):

Consider a subobject $m: S \hookrightarrow A$ corresponding to a morphism $s: 1 \to \Omega^A$. From this perspective, a reason that $\Omega^f$ is called the inverse image map is that the subobject of $B$ corresponding to $\Omega^f \circ s$ is precisely the pullback of $m$ along $f$. The direct image map, meanwhile, sends a subobject $n: T \hookrightarrow B$ to the image of $f \circ n$.