I am trying to understand what is geometric realisation of a simplicial set is.
I am not able to understand what exactly do they mean in notes I found online.
Sometimes, starting something fresh gives better clarity so I am not writing what I have seen in online notes.
A simplicial set $X$ is a collection of sets $X_n$ indexed by integers $n\geq0$ along with maps $$d_i:X_n\rightarrow X_{n-1}, s_:X_n\rightarrow X_{n+1}$$ called face maps and degeneracy maps respectively satisfying compatibility conditions.
I did not see anywhere explicitly but I guess they are not allowing $X_n$ to be empty sets. Is that correct?
Please help me to build up the notion of geometry realisation from scratch.
There are no maps $S \to \varnothing$ if $S$ is nonempty, but there is a unique map $\varnothing \to \varnothing$, so if $X$ is a simplicial set with $X_n = \varnothing$ for some $n$, then in fact $X_n = \varnothing$ for all $n$, and the faces/degeneracies are all $1_\varnothing$. In fact, this "empty simplicial set" is the initial object in the category of simplicial sets.
To motivate geometric realization, you need a very general statement in category theory. For simplicial sets, it says that the natural map $colim_{\Delta^n \to X} \Delta^n \to X$ is an isomorphism for $X$ any simplicial set. This basically says that if you take all the simplices in a simplicial set and glue them together, you get the original simplicial set back.
The geometric realization takes this as motivation. But now you want a topological space, so instead of gluing simplicial simplices together (i.e. the $\Delta^n$ for $n \geq 0$), you're now gluing their toplogical incarnations, which are the standard topological simplices, denoted $|\Delta^n|$ for $n \geq 0$. Thus the geometric realization is $colim_{\Delta^n \to X} |\Delta^n|$.