An -category is -simply connected (or just -connected) if any two parallel -morphisms are equivalent for . You may interpret this definition as weakly or strictly as you like, by starting with weak or strict notions of -category. Note that we include the case to mean that the -category is inhabited when .
The delooping hypothesis says that a -tuply monoidal -category is the same thing as a pointed -simply connected -category; more generally, a -tuply monoidal, -simply connected -category is the same as a -tuply monoidal, -simply connected -category.
One might want a stricter definition for -categories, but this is certainly correct for -groupoids. Indeed, we can say that an -groupoid is -simply connected if and only if its fundamental n-groupoid? is trivial for . In particular, an -simply connected -groupoid is contractible.
This discussion on terminology occurred when the page was at k-tuply connected n-category.
Mike: I’m pretty sure that in algebraic topology, 1-connected means simply connected, and 0-connected means connected. So your definitions make a space 0-connected when its fundamental -groupoid is 1-tuply connected. I would prefer that we retain the topologists’ numbering and call this a 0-connected -groupoid (the ‘tuply’ sounds weird to me for connectedness), with the off-by-one shift happening in the delooping: the -fold delooping of a -tuply monoidal -category would be a -connected -category.
Toby: If the topologists have a system for this, then we should probably use it, even if it is a poor system. (I don't suppose that any topologists say ‘-simply connected’ instead?) The previous usage on the Lab was inconsistent, although I didn't check whether that inconsistency was all my fault.
In general, a -foo should always be the same as a foo, as with -category, -poset, -groupoid, -group, -stuff, etc. Trying to match the ‘right’ dimension isn't going to work consistently and runs into too many other existing terms. I'm only sorry that the topologists did it otherwise here.
(Not to mention that most homotopy-theoretic dimension counting is off by anyway. But that's another topic, indeed a topic that always making a -foo a foo will safely avoid.)
I have no opinion about ‘tuply’; it just came naturally to me since I was thinking about -tuple monoidality.
Mike: Yeah, I agree that in general it is better if a 1-foo is the same as an unadorned foo, but not everyone has adhered to that. Wikipedia agrees with me about the meaning of -connected in topology.
Toby: All right, I'll change the numbering and move the page. But what do you think about ‘-simply connected’? This doesn't seem to have any conflicts, although search for ‘-simply connected’ to see what might be potential conflicts —or might actually agree with me (I haven't been able to check).