# 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.

• 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. Oct 27, 2012 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 Oct 27, 2012 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 $(\Delta^\bullet \downarrow X)$. Oct 27, 2012 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.] Nov 3, 2012 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. Nov 3, 2012 at 5:18

I think that $\Delta\downarrow X$ is a slight abuse of notation, as the simplex category of a simplicial set $X$ is the same as the comma category $$\Delta^{(-)}\downarrow X$$ where $\Delta^{(-)}:\Delta\rightarrow\mathbf{sSet}$ is the covariant functor given by $\phi\mapsto\hom_\Delta(-,\phi)$ for an object or morphism $\phi$ of $\Delta$. Note that for each object $[n]$ of $\Delta$, the image through $\Delta^{(-)}$ of $[n]$ is $$\hom_\Delta(-,[n])=\Delta^n$$ the standard $n$-simplex. This functor $\Delta^{(-)}$ is known as the Yoneda embedding for $\Delta$.
The simplex category of $X$, which has exactly one object for each $n$-simplex of $X$, serves as the index when taking a colimit to find $|X|$. For each $n$-simplex $x$ of $X$, let's say that $\rho(x)$ is the corresponding object of $(\Delta\downarrow X)$. Furthermore, for a morphism $\mu:[m]\rightarrow[n]$ of $\Delta$, let's use $\mu_*$ to denote the corresponding map from $X_n$ to $X_m$ (i.e. the image of $\mu$ through $X$). Given an $m$-simplex $y$ of $X$, there is one arrow $\rho(y)\rightarrow\rho(x)$ in $(\Delta\downarrow X)$ for each morphism $\mu$ in $\Delta$ such that $\mu_*(x)=y$.
There is a canonical functor $F:(\Delta\downarrow X)\rightarrow\mathbf{Top}$ defined as follows: for an $n$-simplex $x$ of $X$, its image $$F~\rho(x)=|\Delta^n|$$ is the standard topological $n$-simplex. For a morphism $\eta:\rho(y)\rightarrow\rho(x)$ corresponding to some $\mu:[m]\rightarrow[n]$ in $\Delta$, the map $F\eta$ is given by the continuous function $|\Delta^\mu|:|\Delta^m|\rightarrow|\Delta^n|$.
This brings us finally to the formal definition $$|X|=\varinjlim F$$ of the geometric realization of $X$. This is not a constructive way to find the geometric realization of a simplicial set. The comments above are much more helpful in that regard. There is a useful approach to the construction of geometric realizations detailed in chapter 10.5 of Peter May's book Finite spaces and larger contexts, available on the internet here.