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 this is indeed "genuine" mathematics, 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.

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.

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 this is indeed "genuine" mathematics, but serious sloppiness (both expository and mathematical) does a number on its image.

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.

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 this is indeed "genuine" mathematics, 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.