For some time I had the impression that infinity-categories are just a generalization of higher order categories: categories whose arrows have arrows among them, and they have further arrows etc. Essentially – every category should be an infinity-category, possibly “degenerate”, because arrows-among-arrows end to exists anymore at some sufficiently high $n$. I imagined that I could take any category and add arrows-among-arrows and get a higher or even infinity-category. Similarly, I could take any infinity-category, delete some arrows-among-arrows and get a lower order category.
But I am now trying to study this subject using “Introduction to Infinity-Categories” by Markus Land and it gives such definition:
A simplicial set is called an $∞$-category if it has the extension property for all inner horn inclusions $\Lambda^n_j \to \Delta_n$, $n ≥ 2$, $0 < j < n$.
A simplicial set, by definition, is a functor which assigns to every ordered set of numbers $[0, \dotsc, n]$ a set of $n$-simplices (possibly, degenerate) $X_n = \{ [0, 0, 0, \dotsc], [0, 1, 2, 2, \dotsc] \}$. It describes some simplicial complex and of course, there exists special unions of faces, called horns and they can be mapped to $n$-simplices in this complex. That is fine.
But I cannot grasp – where is the category in this definition of infinity-category? How is this functor a category as well?
And is there a correspondence between the two notions of higher-order/infinity-categories which we have – 1) the intuitive one as a category with arrows-among-arrows; 2) the exact one – as simplicial set?