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Some say motive as in motivation, some say motif as in recurring theme in music. In some concrete cases it is actually not so hard to see what is going on: the category is defined from algebraic correspondences by splitting idempotents (Karoubi construction), which is not so terrible. The explicit examples such as Fermat varieties tend to fall in the motif class, while the broad categorical properties hoped for in the motive class. The flamboyant Nick Katz talk ihes.fr/jsp/site/… shows why these ideas live on.
