Thanks, I'm still a novice at understanding how these things work. Incidentally, in view of the original question, I have no intention of retiring any time soon.

Just thank you to Qiaochu Yuan and Shripad. Answer has been edited.

This answer, a cell complex, not only has the merit of brevity, but also it makes sense and applies prior to any knowledge of model category. This is standard, and helpful while proving the model axioms. More substantially, it applies in many general situations where one may or may not have a model structure.

There is a 1957 paper ``Supercomplexes'' by Victor Gugenheim that if I remember rightly goes towards combining simplices and cubes.

No, reduced is taken: means the 0th object is a point, not the first. That terminology is the one that sticks.

Everything Justin says is correct. I like model categories too, nowadays, but, as a practical matter, I know of nothing of real use in iterated loop space theory that actually requires use of a model structure on the category of topological operads. Admittedly, the use of products of operads feels like a cheap trick, but it is nice to be able to prove some things cheaply, without the slightest use of categorical homotopy theory. Younger mathematicians should understand that a huge amount of important algebraic topology that is still current predates the widespread use of model categories.

Michael, if you know my work, then you know that in recent years I have erred on the side of the conceptual (I find it easier). But if you want to learn a subject, you have to learn its actual content before you learn how to abstract away from that content in order to prove things that are not as accessible as you might hope within the traditional methodology. I do not think $\infty$ categories can be appreciated without solid prior grounding in algebraic topology or algebraic geometry or at least homological algebra, preferably all three.