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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 $\Delta\downarrow X$:

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

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

enter image description here

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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  • $\begingroup$ 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. $\endgroup$
    – Zhen Lin
    Oct 27, 2012 at 6:42
  • $\begingroup$ 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 $\endgroup$ Oct 27, 2012 at 17:50
  • $\begingroup$ 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 $(\Delta^\bullet \downarrow X)$. $\endgroup$
    – Zhen Lin
    Oct 27, 2012 at 17:53
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    $\begingroup$ 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.] $\endgroup$ Nov 3, 2012 at 5:15
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    $\begingroup$ 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. $\endgroup$ Nov 3, 2012 at 5:18

1 Answer 1

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From the wikipedia page on simplicial sets, "Δ denotes the simplex category whose objects are finite strings of ordinal numbers..."

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.

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