Steve Vickers asked by email whether in constructive mathematics a (-1)-category should be allowed to be an arbitrary "subsingleton", i.e. a set with at most one element. I think that yes, these should be allowed - but maybe other things too! Indeed, a space has homotopy dimension -1 iff 1) given 2 points in the space there exists a path joining them, and 2) given 2 paths joining them, there exists a path of paths joining *them* and 3) given 2 paths of paths joining *them*, there exists a path of paths of paths joining THEM, and so on ad infinitum. As a nonconstructivist, I would say that either the space is empty in which case clause 1) is vacuous, or it's nonempty in which case we go on and read the resulting infinite list of clauses. But a constructive mathematician would have to proceed differently here, not being allowed to use the excluded middle. Do the remaining clauses provide any extra challenges for the constructivist? Then there's the bit where, having gotten a space that's either empty or contains one arc-component with vanishing homotopy groups, I conclude that if it's *nice* (e.g. a CW complex) it's homotopy equivalent to a space that's either empty or one point. I don't know how this reasoning (which uses the Whitehead theorem) gets affected by constructivism. Steve's idea sounds interesting, for this reason. If you plow through the detailed exchange between James Dolan and Toby Bartels, you'll see that (-1)-categories secretly represent TRUTH VALUES. In any approach to math where "truth values" are more interesting than merely 0 or 1, (-1)-categories will be correspondingly more interesting than merely sets with 0 or 1 elements. I guess this is familiar from topos theory. But I don't know if there's an extra twist due to all the "higher-dimensional" stuff going on in my reasoning above. Are truth values for an omega-categorical constructivist still more interesting than for an ordinary constructivist? The ordinary constructivist may not know whether two things are equal. The omega-categorical constructivist may not know whether all morphisms between two things are related by a 2-morphism, or whether all such 2-morphisms are related a 3-morphism, and so on....