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The property that every split monomorphism is an isomorphism appears already implicitly in several places. It should be added separately.
It has been mentioned in comments to the implications right_cancellative_kernel_pairs and trivial_coreflexive_equalizer_preservation and thin_dominant_functor. But there are more places that can be refactored.
The property is self-dual, so maybe it is better to choose a name that reflects this. Maybe "splitting-trivial"?
I think it is also equivalent to the property that every split idempotent endomorphism is the identity.
Claude has suggested the name "retract-trivial", which I like.
Closely related notion: Dedekind-finite monoids (https://en.wikipedia.org/wiki/Dedekind-finite_ring)
So maybe such categories can also be called Dedekind-finite, because $M$ is Dedekind-finite if and only if $BM$ has the property.
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ScriptRaccoon commentedon Oct 27, 2026
Claude has suggested the name "retract-trivial", which I like.
Closely related notion: Dedekind-finite monoids (https://en.wikipedia.org/wiki/Dedekind-finite_ring)$M$ is Dedekind-finite if and only if $BM$ has the property.
So maybe such categories can also be called Dedekind-finite, because