Complement Toposes are dual in a sense to (elementary) Toposes and are expected to have typed higher paraconsistent logic as its internal language (as dual intuitionistic logic is paraconsistent).
Now one of the key examples of Toposes are Presheaf & Sheaf Toposes. Is there a corresponding analogue for Complement Toposes?
Addressing points raised above, complement topoi are just topoi, and this is basically trivial. The real point of the alternate definition is that it provides a way to associate to each topos an "internal paraconsistent logic," dually in an appropriate sense to the usual internal intuitionistic logic. The corresponding "sheafy" notion is closed set sheaves.
All of this is treated in Chris Mortensen's book Inconsistent mathematics (incidentally, Mortensen is also the author of the above-mentioned SEP article).
I want to also respond to the negative reaction to this material above. Basically, the point I want to make is that there is indeed "genuine" mathematics here, but serious sloppiness (both expository and mathematical) does a number on its image - and I wouldn't be surprised if it also makes some fairly trivial results appear nontrivial.
Consider for example the following passage in Mortensen's book (from page $105$):
It is clear that, if $E$ is a complement-topos and $E'$ is the category obtained by renaming $F$ as $T$ and each $\overline{\chi}_f$ as $\chi_f$ then $E'$ is a topos
(and vice-versa, although he doesn't state this explicitly). Of course this is silly: $E$ literally is $E'$, the difference is in what we're doing with that category. Mortensen I think wants the reader to separate the two purposes mentally by using two different terms, but $(i)$ I don't actually think that's a good idea and $(ii)$ it certainly isn't helped by treating the "interpretation data" above as intrinsic to the topos itself.
Personally, I think the right way to describe the situation would be the following:
Define "complement-topos" and describe the associated construction of a paraconsistent logic and how complement-topoi can be built from closed set sheaves.
Prove that complement-topoi and topoi coincide.
Summarize the situation as "We have shown that each topos has an associated paraconsistent logic as well as the usual intuitionistic logic."
To make matters worse, Mortensen's book also has genuine mathematical errors. For example - in my opinion, the most egregious example - chapter $3$ talks about "the classical denumerable nonstandard model of arithmetic," which is clearly bonkers.
Now this doesn't affect the material Mortensen actually cares about, and indeed as far as I can tell there are no real errors in the results he presents. However, the situation really doesn't do the subject any favors.
I took a second glimpse into the paper. Because I am not a mathematician I cannot judge the paper on the mathematical level.
However, as a computer scientist, I can say that it is really strange if a meaning is not invariant under syntax. Specifically, the author defines a subobject classifier to be an object $\Omega$ together with a morphism $\mathit{true} \colon 1 \rightarrow \Omega$, such that for every mono $m \colon S \rightarrow A$ there exists a unique morphism $\chi_m \colon S \rightarrow \Omega$ such that the diagram: $$\require{AMScd} \begin{CD} A @>{m}>> S\\ @V{!}VV @VV{\chi_m}V \\ 1 @>{\mathit{true}}>> \Omega \end{CD}$$ is a pullback and claims that a topos is a category with finite limits, finite colimits, exponents and subobject classifier. Then, in section 3, he introduces the concept of a "complement topos" as a category with finite limits, finite colimits, exponents and a "complement-classifier". Where, according to the text, a "componnet-classifier" is an object $\Omega$ together with a morphism $\mathit{false} \colon 1 \rightarrow \Omega$, such that for every mono $m \colon S \rightarrow A$ there exists a unique morphism $\overline{\chi_m} \colon S \rightarrow \Omega$ such that the diagram: $$\require{AMScd} \begin{CD} A @>{m}>> S\\ @V{!}VV @VV{\overline{\chi_m}}V \\ 1 @>{\mathit{false}}>> \Omega \end{CD}$$ is a pullback (yes, I obtained this paragraph by copy-paste-rename).
On the other hand, from Theorem 1 from the paper one may infer that the author uses some inconsistent variant of the meta-logic (i.e. every statement is true in the logic) --- which is quite reasonable --- taking into consideration that his work, as he admits, is built on the foundations of "inconsistent mathematics".
I also consulted the paper "Bi-Heyting algebras, toposes and modalities" by Reyes and Zolfaghari. Here is their characterisation of toposes with co-Heyting internal (thus, bi-Heyting) logic:
Proposition 3.2: A topos $\mathcal{E}$ is bi-Heyting iff there is a Boolean topos $\mathcal{B}$ and a surjective geometric morphism $\Gamma \colon \mathcal{E} \rightarrow \mathcal{B}$ such that the canonical $\delta \colon \Omega_\mathcal{E} \rightarrow \Omega_\mathcal{B}$ has a left lax adjoint.
BTW, showing that presheaf toposes are bi-Heyting without using the above proposition is a good exercise.
ps. My surname is Przybylek, not "Przyblek".
