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Prove remaining properties of the category of locally ringed spaces - #443

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@ScriptRaccoon ScriptRaccoon commented Oct 28, 2026 •

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The category LRS of locally ringed spaces (over a commutative ring) has 8 unknown properties left, as of writing. In this PR, all of them will be decided. The proofs come (more or less) from #403.

  • The monomorphisms and regular epimorphisms of LRS are described (the other types of special morphisms already had descriptions).
  • LRS has an extremal cogenerator (which also settles 3 related properties).
  • LRS is well-powered (this is reduced to a lemma showing that the category of local commutative rings is well-copowered).
  • From these results and completeness, we also deduce that LRS is cototal (which automatically implies that LRS is cocomplete, but we keep this assignment, of course).
  • LRS does not have filtered-colimit-stable monomorphisms.
  • LRS does not have a regular subobject classifier (and neither does Sch). The proof differs from the one in the mentioned PR, as it is more closely aligned with the proof that CAlg(R) does not have a regular quotient object classifier.

As a result, LRS has no unknown properties left. 🎉

Other adjustments for locally ringed spaces:

  • Two redundant assignments have been removed (cartesian closed, concretizable).
  • The notation has been unified and made more precise: $X$ denotes a locally ringed space, $|X|$ its underlying topological space, and $\mathcal{O}_X$ its structure sheaf. For a morphism $f : X \to Y$, the underlying continuous map is denoted $|f| : |X| \to |Y|$. In particular, locally ringed spaces are no longer written as tuples, and their morphisms are no longer confused with their underlying continuous maps.

@ScriptRaccoon ScriptRaccoon added the data additions and updates to the database label Oct 28, 2026
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ScriptRaccoon force-pushed the locally-ringed-spaces-properties branch 2 times, most recently from f659192 to f5c97cb Compare October 30, 2026 08:17
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ScriptRaccoon force-pushed the locally-ringed-spaces-properties branch from f5c97cb to 146e1ea Compare October 30, 2026 09:01
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ScriptRaccoon marked this pull request as ready for review October 30, 2026 14:02
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