I just found out that Marius Furter has a whole series of lectures on enriched category theory! This is really fantastic.
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This looks wonderful. There is a real shortage of good introductory material on enriched category theory.
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To me the subject of enriched categories jumps from being “trivial” (perhaps mindblowing, but really a straightforward generalization of ordinary category theory) to “nontrivial” when it reaches the subject of limits and colimits. Because there’s usually not a “diagonal” enriched functor \Delta: C \to C^D where C is the enriched category we’re trying to take limits or colimits in and D is the “diagram” enriched category, the usual trick of defining limits and colimits as right and left adjoints to \Delta: C \to C^D doesn’t work, nor even the whole idea of a cone or cocone. So we must turn to weighted limits and weighted colimits. So it would be really nice to have a gentle, expository introduction to enriched category theory that covers these.
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