Add Barr-exact, Barr-coexact, pretopos properties#103

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dschepler wants to merge 15 commits intoScriptRaccoon:mainfrom

dschepler:barr-exact

Commits on Apr 18, 2026

Initial addition of Barr-exact, Barr-coexact, pretopos properties and some implications
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Add a few assignments for Barr-exactness
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Resolve Barr-coexact for Set*
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Revert addition of Barr-coexact implication of elemetary topos - it turns out the proof on ncatlab for categories monadic over Set requires the external axiom of choice on Set, so it will only gene…
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Add manual resolution of Barr-coexact for Set
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Add a couple cases which were unresolved by removing the faulty implication topos->Barr-coexact, but are still cases of topoi with external axiom of choice
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Resolve Barr-coexact for Top and Top*
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Update expected properties
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Add monadically concrete -> cocomplete implication
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Add implication pretopos -> balanced
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Add some related properties, and a bit of background on the relevance of regularity to Barr-exactness
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Add local essential smallness consequence of monadically concrete
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Add not Barr-coexact data for Prost
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Fix garbled language around kernel pairs in some of the proofs
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Commits on Apr 19, 2026

Merge branch 'ScriptRaccoon:main' into barr-exact
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