Understand the internal hom simplicial set

Question

I am currently reading the paper "a short course on $\infty$-categories" by M.Groth. The Theorem 1.18 in this paper states:
A simplicial set $X$ is an $\infty$-category if and only if the restriction map $i^*:Map(\Delta^2,X)\to Map(\Lambda^2_1,X)$ is an acyclic Kan fibration.
The question I want to ask is regarding the understanding of the explanation part of this theorem. The author says, right after the theorem, that "We can think of $Map(\Lambda^2_1,X)$ as the space of composition problems and similarly of $Map(\Delta^2,X)$ as the space of solutionms to composition problems."
I wonder why would such statement hold? Because by definition $$Map(\Lambda^2_1,X)_n=Hom_{sSet}(\Delta^n\times\Lambda^2_1,X)$$ And this is not really the "composable" maps, i.e. $Hom_{sSet}(\Lambda^2_1,X)$. Similarly for $\Delta^2$ situation.
Thank you a lot for your help. I feel like this is a stupid question. Somehow I just cannot convince myself.

Answer 1

The simplices of positive dimension in any simplicial set $X$ serve a role analogous to that of a topology on $X_0:$ they describe how the points of $X$ hang together. For instance, a $1$-simplex is simply a path between two points. If $X$ is a mapping simplicial set, as in your question, then the points are maps and the $1$-simplices are homotopies between maps!

In essence, in this context the higher simplices give you homotopies between homotopies. This would be literally true in a cubical set, but for technical and historical reasons simplicial sets are usually preferred. Both heuristically and rigorously, the overall information is the same. And the fact that simplicial sets (with the Kan model structure) and topological spaces are Quillen equivalent model categories is a formalization of the heuristic above that the higher simplices do the same kind of thing as a topology—specifically, this is literally true for questions that are homotopy-invariant. So for homotopy theory, you are absolutely safe in imagining that the higher simplices give a topology to the $0$-simplices, or that they define paths between paths between…in the set of $0$-simplices.

All this is to justify that something like $\mathrm{Map}(\Lambda^2_1,X)$ should roughly be thought of as the space of 2-D inner horns in $X.$ Then the condition given here says that we can define an $\infty$-category as a simplicial set which has the same space of inner horns as of 2-simplices—that is, you have a homotopy equivalence between those spaces, so that you can not only fill every inner horn, but you can do so uniquely up to a homotopy which is itself unique up to a higher homotopy, which is…

To be completely fair, there is one important refinement to this thinking. An $\infty$-category is not really like a space, but rather like some sort of “directed” space, since paths are not always reversible. However, it is the case that an equivalence of $\infty$-categories really does reduce to a pair of equivalences on the proper spaces, I.e. Kan complexes, of points and of edges. And Kan complexes themselves, which are the simplicial sets most definitively like spaces, are everywhere in the theory, for instance in the mapping sets at issue here.