Thank you for your comments; oh indeed there are some misprints. I'll update the post with some explanations and corrections.

Thanks; Yes, I think it does imply that $\kappa$ is inaccessible: I am talking about a sheaf of functions, i.e. $T(alpha)$ is a set of functions $\alpha\longrightarrow 2$. If $2^{<\kappa}\geq\kappa$ then there is a sheaf of functions on $\kappa$ that has no global section i.e. no global branch: just make sure $f_{\alpha}\not\in T(\alpha+1)$ where $f_i$'s are some enumeration of functions in $2^{<\kappa}$.

There is a paper by Drinfeld (and related papers by A.Besser and D.Grayson) arxiv.org/abs/math/0304064 where he dwells on the notion of the interval..i quote: We reformulate the definitions so that the following facts become obvious: (i) geometric realization commutes with finite projective limits (e.g., with Cartesian products); (ii) the geometric realization of a simplicial set (resp. cyclic set) is equipped with an action of the group of orientation preserving homeomor- phisms of the segment I := [0, 1] (resp. the circle S1 )

David White: not quite, a small posetal category may all small limits and colimits, and may have a non-trivial model category structure. (Non-trivial in the technical sense that it is not one of the three trivial model category structures where one of the three classes is the class of all morphisms).

Model category structures on partial orders are usually not cofibrantly generated as well (for class-sized partial orders, of course).

Will Savin: yes, it is exactly the point of the question to "go around" that counterexample. I should perhaps add that the reason for the question is that one can prove very partial positive results, for some very small subcategories of Schemes.

Will Savin: the topological fundamental group is intended to be an example. it works for K=C or K⊂C and an appropriate subcategory of Schemes. (Well, the topological fundamental group of a smooth projective complex algebraic variety is not necessarily residually finite, so you have to do something about that). In general, I do not know; perhaps my definition is too naive to work for general Schemes. Also, in char p, I do not know; but possibly it is not too hard to construct by hand.

Thanks. Yes, this kind of question is probably indirectly related...But I do not see a direct relation.

Tom Leinster: in fact, I am confused. What definition of a 2-category you are using; is it in your survey of definitions of n-categories? E.g. what is the 'specified invertible 2-cell' in CAT/T ? I assume T is the codomain category T of H:S-->T. Sorry for bothering you about this.