# Understanding the definition and notation of geometric realization

I had trouble making sense of the definition of geometric realization of a simplicial set. Let $\Delta^n$ be the standard n-simplex defined as the functor $\hom_\Delta(-,$*n*$): \Delta \rightarrow$ Set, and $\left|\Delta^{n}\right|$ the topological standard n-simplex, let $X$ be a simplicial set, the realization of $X$ is defined by the colimit:

$\left| X \right| = \underrightarrow{\lim} \: \:\: \left| \Delta^{n} \right|$

$\Delta^{n} \rightarrow X$

in $\Delta\downarrow X$ (the simplex category of $X$).

Frankly I don't understand the notation. Look at the diagram of the colimit in $\Delta\downarrow X$:

$X \cong \underrightarrow{\lim} \: \Delta^{n}$

$\Delta^{n} \rightarrow X$:

Is the geometric realization of $X$ then the geometric realization of the colimit $L = \left| L \right|$? So then $L$ must be standard n-simplex $\Delta^{p}$ for some $p$ no? I want to make sure I'm understanding this right.

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Why would the colimit just be an $n$-simplex? In a typical simplicial set you are taking the colimit over an infinite diagram with no terminal object. – Zhen Lin Oct 27 '12 at 6:42
This is how I see it: So far I know what the geometric realization of a standard n-simplex is, I want to know what's the geometric realization of a simplicial set that is NOT an standard n-simplex. Is the geometric realization of a simplicial set $X$ (that is NOT a standard n-simplex) the geometric realization of the colimit $L$? If that is so, and $L$ is not a standard n-simplex, then $\left| L \right|$ is the geometric realization of a simplicial set that is NOT an standard n-simplex, which is the very thing I want to know in the first place – Mario Carrasco Oct 27 '12 at 17:50
Your notation makes no sense. The geometric realisation of a simplicial set $X$ is defined to be a certain colimit over the diagram of shape $(\Delta^\bullet \downarrow X)$. – Zhen Lin Oct 27 '12 at 17:53
I think this isn't the easiest way to see what geometric realization is trying to do.  We're trying to build a $\Delta$-complex-looking thing, and a simplicial set $X_\bullet$ is first and foremost a list of sets $\{X_n\}_{n \geq 0}$ whose elements are precisely the sets of maps in from the standard $n$-simplices.  So to build $|X_\bullet|$, start with a vertex for every element of $X_0$.  Then, give yourself edges for every element of $X_1$... [cont.] – Aaron Mazel-Gee Nov 3 '12 at 5:15
However, be sure to (a) collapse down those that are degenerate (i.e. they arise from maps $\Delta^1 \rightarrow |X_\bullet|$ of $\Delta$-complexes that take $\Delta^1$ to a single point), and (b) attach all edges to their boundary vertices using the face maps.  Then, continue on up.  The advantage of all this is that there's a lot of power in naturality, so you can do a lot by manipulating a "space" as "maps into the space" (from some particular set of objects).  The main disadvantage is that in this framework you have no choice but to carry around all these degenerate simplices. – Aaron Mazel-Gee Nov 3 '12 at 5:18
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