@Mark: perhaps the "emperor's clothes" went a bit too far (although I have to say that IMO quite a lot of Indiscrete Thoughts, which I have read and reread a number of times, is similarly both exasperatingly vague and polemical at the same time). To put this in balance: I am a huge admirer of Rota when he is behaving like a professional mathematician; he writes mathematics beautifully and most thought-provokingly. But in this particular case, nobody here seems to know what he's talking about exactly, and I'd honestly like to know if there's anything there. Can one of his students explain?

@Yemon: that's true, but if really he had something important to say, then maybe he (or one of his students) should have published it or made a note of it somewhere, in a responsible way. Instead, he whines (and coyly hints at secret knowledge possessed by lattice theorists). It looks kind of passive-aggressive to me, and I eagerly await for someone to prove that the emperor wore some beautiful clothes in this case.

What annoys me is: if Rota knowingly asserts that lattice theory will contribute new insights if only algebraic geometers would pay more attention, then why doesn't he just spit it out? Given his stature, algebraic geometers would have paid attention. Why cloak it all in mystery and coyness?

No, it's not insane. In fact I don't agree with Andre that this notion is "wrong" (although I understand why he says that) -- it's just a natural extension of defining a 2-category as a Cat-enriched category: a strict n-category is an (n-1)-Cat-enriched category. Strict n-categories are still a technically useful notion; for example, Batanin's notion of weak oo-category starts with the monad on globular sets whose algebras are strict oo-categories, and develops an associated notion of operad. Weak oo-categories are then algebras over certain contractible such operads.

I'd like to see this stay open. As Ben says, the speculation (and tendentiousness, for that matter) is Rota's, not Mariano's. I've often wondered myself what the heck Rota meant here, and suspect: not all that much. (The example of the Chinese remainder theorem doesn't exactly bowl me over.)

"Category theory should be seen as a language." To me, category theory is far, far more than "just a language" -- it points to new ways of looking at the world, uncovering cross-connections between disparate areas and speaking of them in a precise, controlled way. It is in fact a fantastically valuable branch of mathematics. And yet, it is poorly supported, and the passion with which some very intelligent people pursue it is poorly understood. Why? Maybe because of an enduring trope that it is "just a language". (Maybe you don't mean that, Colin, but other people do. It makes me sad.)

Since you mention below that you are interested in the analytic category, let me add that there is a very general notion of manifold based on the notion of pseudogroup of homeomorphisms, which covers many special cases: topological, smooth, analytic, complex, foliated, elliptic, hyperbolic, etc. (reference: Thurston's Three-Dimensional Geometry and Topology). It seems to me a really good answer to your question ought to work in this generality.

Colin, yes there are open maps which are nonidentity idempotents (take the projection of the disjoint union of two lines onto one, and then follow with one of the inclusions of one line into two). There are no nontrivial idempotents that are open inclusions however. I'm inclined to return to the question in the full category of manifolds (and similar question for the category of analytic manifolds). My instinct is that idempotents split in either category, and that this would be an easy local computation. Hopefully more time for this later (unless someone else gets there first).

I think another problematic example would be trying to take the coequalizer of two maps $\mathbb{R} \to \mathbb{R}$, where one is the identity and the other is smooth, increasing, and equal to the identity everywhere but an open line segment. I'd have to think hard to see if type of example is really problematic, but on the face of it, it looks as though one is forced to make some uncomfortable identifications. All that being said, it could be that the category of manifolds and all smooth maps does have some significant colimits. An interesting test case is whether idempotents split.

Countable coproducts exist, and it seems plausible to me that (countable) filtered colimits of open embeddings would exist. I would imagine coequalizers are problematic.

Yes. My head must be in topos theory too much, where we sometimes have difficulty making even just one choice. ;-)

Joel, I'm not following you when you say, "which provides for us a canonical way of picking points from the space". It sounds like you need to choose a bijection with $\mathbb{N}$ in order to proceed.

Nice! Well, that's that!

I'd be interested if there were any good answers to this. A related question is mathoverflow.net/questions/4044/logical-endofunctors-of-set where even in the case $E = Set$, existence of non-trivial logical endofunctors involve fairly large cardinal hypotheses.