Which is the functor of this limit?

Question

In the book "Simplicial Homotopy Theory" of Goerss and Jardine in the lemma 2.1 says that there is and isomophism

\begin{align} X\cong&\varinjlim\Delta^n \\ &\Delta\rightarrow X \\ &in\quad \mathbf{\Delta}\downarrow X \end{align}

where $X$ is a simplicial set and $\mathbf{\Delta}\downarrow X$ is the category which objects are simplicial set morphisms $\Delta^n\rightarrow X$ for some natural $n$. Also $\Delta^n=Hom_\Delta($_$,[n]) $ where $[n]={0,1,...,n}$. So my question is, which functor or directed system it is taken in the limit?

Answer 1

The functor goes from the category $\mathbf{\Delta} \downarrow X$ to the category of simplicial sets, and is defined as follows: it sends an object $\sigma : \Delta^n \to X$ to the simplicial set $\Delta^n$ (ignoring any detail $\sigma$ other than its domain), and sends a morphism $\alpha : \sigma \to \sigma'$ to $\alpha$ itself (remember that in $\mathbf{\Delta} \downarrow X$ a morphism $\alpha$ is really just a morphism in simplicial sets from the domain of $\sigma$ to the domain of $\sigma'$ such that $\sigma' \circ \alpha = \sigma$).

Answer 2

The functor is $(\Delta^n\to X)\mapsto \lvert \Delta^n\rvert.$

Notice that by Yoneda, the simplicial maps between $\Delta^n$ and $X$ are just the $n$-simplices of $X$. (this is mentioned on page 6). The functor we're taking the colimit of just assigns to each $n$-simplex of $X$ the standard topological $n$-simplex $\lvert \Delta^n\rvert$ (example 1.1).

Any coface (or codegeneracy) map $n\to m$ will induce a face (or degeneracy) map $\Delta^m\to\Delta^n$, and this will commute with simplicial maps into $X$ (and hence be a morphism in the comma category) only when the $m$-simplex in $X$ corresponding to $\Delta^m$ has its corresponding face (or degeneracy) the selected $n$-simplex in $X$. Because this is a colimit, we are taking a coproduct of all these topological simplices $\lvert \Delta^n\rvert$, one copy for each simplex in $X$, and identifying the faces and degeneracies according to the rules of simplicial sets.