Re: Famous unsolved problems in ordinary category theory

[tholen@mathstat.yorku.ca]

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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