Question on colimit over the category of simplices I get slightly confused on the colimit definition:
$$X \cong \lim_{\Delta^n \rightarrow X} \Delta^n $$ where the symbol $\lim$ is actually for denoting colimit.
I know that colimit is defined for a diagram $F: J \rightarrow C$ as the initial object of the category $\text{CoCone}(F)$. What is the index category $J$ for the above-mentioned case?
In the category of simplices, $\Delta \downarrow X$, the objects are given by natural transformations $\Delta^n \rightarrow X$ or $\Delta^m \rightarrow X$ and the morphisms  by natural transformations such as $\Delta^n \rightarrow \Delta^m$ induced by $[n] \rightarrow [m]$. This should serve as a CoCone(?) of something. Is this way of thinking correct?
 A: You are entirely that the index category is the category of simplicies (Grothendiecks category of elements is an alternative name) of $X$.
I think you have a misconception in what the data is of a colimit depends on. Here we are missusing notation. A colimit depends on both the index category and functor.
Now the category of simplicies $Elt(X)$ has forgetful functor $U:Elt(X)\to \mathrm{Set}_{\Delta}$ which is given on objects by $$
(\Delta^n \to X) \mapsto \Delta^n.
$$
Now lets explore what a cocone for this functor is. By definition a cocone on $U$ is an object $Y$ and natural transformation $U\to \delta(Y)$, where $\delta$ is the functor $\mathrm{Set}_\Delta\to \mathrm{Fun}(Elt(X),\mathrm{Set}_\Delta)$. Using the Yoneda this can be seen to be the same as an $n$-simplex of $Y$ for all $n$-simplicies of $X$. Satisfying some naturality condition, which heuristically says that this collection defines a simplicial subset of $Y$.
Now it is clear that $X$ defines a cocone for the forgetful functor. So suppose $(Y,\alpha)$ is another cocone for $U$. We want to build map $X\to Y$ making everything commute. We then basicly build this by $x \in X_n$ maps to $\alpha_{x}$ seen as a $n$-simplex in $Y$ by the Yoneda lemma. I leave it to you to show that this is welldefined.
