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I'm not sure how well this fits the bill, but in algebraic geometry and number theory, the notion of mixed motives is still undefined, although people have a fairly good idea of what properties they want the category of mixed motives to have.

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I'm not sure how well this fits the bill, but in algebraic geometry and number theory, the notion of mixed motives is still undefined, although people have a fairly good idea of what properties they want the category of mixed motives to have.