Early CT problems that are still open

Early CT problems that are still open

[Vaughan Pratt]

What early problems in category theory remain open today?  Who first
posed them, and when and where?

Context: In December the Journal of the AMS rejected the solution of the
half-century-old lattice theory problem of whether the congruence
lattices of lattices are exactly the distributive algebraic lattices.
Since these two classes coincide for all algebraic lattices with up to
aleph_1 compact generators [Huhn 1989], one would imagine that surely
the equivalence must extend to all cardinalities.  Fred Wehrung recently
showed it doesn't, refined by Pavel Ruzicka to show that they diverge
exactly at aleph_2.  For two such naturally defined classes it's very
unusual to first diverge at such a high cardinal.  (For elementary
classes it's impossible: if they're still together at aleph_0 they're
the same.)

Now JAMS doesn't ordinarily cater to either lattice theory or category
theory.  Yet its mission statement declares JAMS to be "devoted to ...
all areas of pure and applied mathematics."  The general feeling among
lattice theorists is that, whatever JAMS might think of those areas it
is unaccustomed to serving, rejecting the solution to so celebrated an
open problem is beyond the pale given their mission statement.  There is
no likelihood of their being overwhelmed with such so space can't be a
reason.  More on this at http://clp.stanford.edu .

My interest here in early open CT problems is to get a sense of how
comparable CT's situation is with lattice theory's.  On the one hand it
might seem insane for either category theorists or lattice theorists to
bother the JAMS audience if they're not interested.  On the other, if
it's a really neat result then why hide it under a bushel?  The
mathematical community at large ought to be sufficiently open-minded as
to appreciate such an achievement.  If a category theorist were to
publish the solution to a long-standing CT problem in JAMS, it would
reflect well on CT, and it would lend support to JAMS's mission statement.

Vaughan