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The category of metric spaces has no regular subobject classifier - #378

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ScriptRaccoon merged 2 commits into
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regular-subobjects-metric-spaces
Sep 17, 2026
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ScriptRaccoon merged 2 commits into
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regular-subobjects-metric-spaces

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@ScriptRaccoon ScriptRaccoon commented Sep 17, 2026

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This PR implements #304 and shows that

  • Met and Met have no regular subobject classifier
  • PMet has a regular subobject classifier
  • the regular monomorphisms PMet are the injective isometric maps

The regular monomorphisms in Met and Met are still not understood.

The number of unknown (category, property)-pairs goes down from 72 to 69.

Thanks @dschepler for providing the proofs!


After merging, I realized that it also brings down the number of missing combinations from 409 to 399. Namely, PMet (or its dual) newly witnesses:

- regular subobject classifier ∧ ¬binary copowers
- regular subobject classifier ∧ ¬binary coproducts
- regular subobject classifier ∧ ¬cokernel pairs
- regular subobject classifier ∧ ¬equalizers of cokernel pairs
- regular subobject classifier ∧ ¬pushouts
- regular quotient object classifier ∧ ¬binary powers
- regular quotient object classifier ∧ ¬binary products
- regular quotient object classifier ∧ ¬kernel pairs
- regular quotient object classifier ∧ ¬coequalizers of kernel pairs
- regular quotient object classifier ∧ ¬pullbacks

The initial goal of this milestone is reached, but I think we can do better.

@ScriptRaccoon ScriptRaccoon linked an issue Sep 17, 2026 that may be closed by this pull request
also, classify the regular monos
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ScriptRaccoon force-pushed the regular-subobjects-metric-spaces branch from b4dcc33 to 1e35cc7 Compare September 17, 2026 06:35
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ScriptRaccoon merged commit daf4893 into main Sep 17, 2026
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ScriptRaccoon deleted the regular-subobjects-metric-spaces branch September 17, 2026 06:38
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Met, Met_oo do not have regular subobject classifiers

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