# What is the analogue for the category of presheafs for complement toposes?

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?

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What you have quoted does not look like a serious paper. However, if you are interesting in toposes with co-Heyting internal logic (thus, bi-heyting internal logic) then the obvious examples are presheaf toposes. In fact, geometrically, Grothendieck toposes with co-Heyting internal logic look similar to presheaf toposes --- there is a simple characterisation of such toposes due to (if I recall correctly; if not --- apologise) due to Reyes and Zolfaghari. Perhaps the relevant paper is "Bi-Heyting algebras, toposes and modalities" though I do not have access to it at the moment. – Michal R. Przybylek Jun 21 '13 at 8:25
I meant --- "interested". – Michal R. Przybylek Jun 21 '13 at 21:16
I would have guessed categories of sheaves on the poset of closed sets of a space, but I haven't checked. – David Roberts Jun 22 '13 at 0:42
@Roberts: that is what I would have guessed, but then I found that paper. – Mozibur Ullah Jun 23 '13 at 0:22
@Przyblek: It is quoted in the SEP, so it ought to have a reasonably serious content. – Mozibur Ullah Jun 23 '13 at 0:23

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".

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I've only just glanced at the paper, but saw and also found very strange this definition of "complement classifier", since it differs not at all from the definition of subobject classifier, save for the fact that the morphism most people name "true" is what he renames as "false". – Todd Trimble Jun 30 '13 at 0:34