Finding the "right" questions and notions is certainly a prominent theme in category theory, perhaps more prominently than in other fields. Still, just like in other fields, solving open problems was always part of the agenda. For example, half a century ago people asked whether every "standard construction" (=monad) is induced by an adjunction, and it took a few years to have two interesting answers. And there is ceratinly a string of examples leading all the way to today. I don't know whether there are any >famous< unsolved problems in ordinary category theory, but there are certainly non-trivial questions. Here is one that we formulated in an article with Reinhard B"orger (Can. J. Math 42 (1990) 213-229) two decades ago: A category A is total (Street-Walters) if its Yoneda embedding A ---> Set^{A^{op}} has a left adjoint. Then 1. A has small colimits, and 2. any functor A-->B that preserves all existing colimits of A has a right adjoint. Do properties 1 and 2 imply totality for A? I must admit that, after formulating the question we never considered it again, so there may well be a known or quick answer. So don't hold back please, especially since I plan to incorporate several questions of this type in my CT09 talk. Walter. Quoting Michael Shulman <shulman@uchicago.edu>:
Probably people are going to jump on me for saying this, but it seems to me that category theory is different from much of mathematics in that often the difficulty is in the definitions rather than the theorems, and in the questions rather than the answers. Thus, there are probably many unsolved problems in category theory, but we don't know what they are yet, because figuring out what they are is the main aspect of them that is unsolved. (-:
Mike
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