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The smooth category has many problems. One is that pullbacks of regular epimorphisms need not exist.

However, pullbacks along submersions always exist. It also seems that submersions are universal (i.e pullback stable). Surjective submersions are moreover regular epis. Since surjectivity is also stable under base change, surjective submersions are a universal subclass of the class of regular epis.

Is this the largest universal subclass of the regular epimorphisms in the smooth category?

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