I am currently reading Lurie's - Higher Topos Theory, but am having difficulties from the very beginning. I know that the topic is too broad and different mathematicians may define differently some of the notions that shall be mentioned in this post. Therefore I adapt the notation and terminology, Lurie adapts in his book. In view of that, he defines an $\infty-$category, to be a simplicial set say, $S \in Set_{\Delta}$, which satisfies the horn-condition, i.e., for any $n \geq 0$ and $0 < i < n$, any map $\Lambda^n_i \rightarrow S$, extends to a map $\Delta^n \rightarrow S$. However, some pages above, he explicitly says
"In this book, we consider only ($\infty$,1)-categories: that is, higher categories in which all k-morphisms are assumed to be invertible for $k > 1$."
While later on he writes out
Unless we specify otherwise, the generic term “$\infty$-category” will refer to an ($\infty$,1)-category.
Here is the point where I got stuck. In my understanding so far, these $\infty$-categories are literally functors, $\Delta^{op} \rightarrow Set$, with some additional "descent" condition to exploit the combinatorial nature of them. Moreover, for convenience we think of these $\infty$-categories as kind of generalisation of the usual categories, hence someone might think the $S_0$ as the objects, $S_1$ as the morphisms, $S_2$ as the triabgles and so on we can have in mind the usual nerve of a category as a compass to obtain some intuition. What I don't understand, is where the first quote comes into the stage. What does it mean to have $k$-morphisms in the $\infty$-category setup? And if so, what does it mean to have invertible morphisms in that sense?
Thanks in advance!
The intuitive idea of an $\infty$-category is a category-like structure where you have morphisms between morphisms between morphisms between morphisms and so on. That is, you have ordinary objects and morphisms, but you also have "2-morphisms" between parallel 1-morphisms (think natural transformations between functors, or homotopies between maps), and "3-morphisms" between parallel 2-morphisms, and so on. An $(\infty,1)$-category is then a structure of this sort where all the $k$-morphisms for $k>1$ are invertible. The basic motivating example is the $(\infty,1)$-category of spaces, where objects are (nice) topological spaces, morphisms are continuous maps, 2-morphisms are homotopies between maps, 3-morphisms are homotopies between homotopies, 4-morphisms are homotopies between homotopies between homotopies, and so on. Since every homotopy has an inverse (just reverse it), all the $k$-morphisms for $k>1$ are invertible.
So, Lurie's definition of "$\infty$-category" is actually just modeling this notion of $(\infty,1)$-category, not more general $\infty$-categories in which you can have non-invertible morphisms of all dimensions. You should think of an element of $S_2$ as not just a commuting triangle in the ordinary categorical sense, but a triangle which commutes up to a given homotopy. That is, you have objects $A$, $B$, and $C$, maps $A\to B$, $B\to C$, and $A\to C$, and a $2$-morphism ("homotopy") between the composition $A\to B\to C$ and the morphism $A\to C$. Similarly, higher-dimensional simplices are diagrams which commute up to given higher-dimensional morphisms.
In the case that you have an ordinary category, you consider it as an $(\infty,1)$-category by saying there are no $k$-morphisms for $k>1$ other than identity morphisms. So in this case all the higher morphisms in our simplices are identities, and the diagrams actually literally commute. So in that case, the simplicial set is just the usual nerve of the category.