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Some time ago, I read an essay-like page (linked from SE) that listed perhaps a dozen structures outside of category theory (rings, vector spaces, etc), emphasizing how they were defined by rules with exceptions (like division by 0) and contrasting this with structures defined by universal rules. But I could never find the page again, despite searching nLab (and other sites) many times. Does anyone know I’m referring to? Or recognize the page I’m describing?
Maybe it was variety of algebras. And the issue is that fields, which are non-zero commutative rings such that every non-zero element is invertible do not form a variety of algebras.
There’s the concept of an essentially algebraic theory.
Doesn’t exactly fit your description but maybe https://ncatlab.org/nlab/show/too+simple+to+be+simple
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