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In the process of learning about simplical sets, I came across their geometric realization, but not quite understanding it, and wondered a few things.

  1. What the geometrical realization is, a more intuitive explanation.
  2. What a geometrical intuition of a Hausdorff space is.

I understand the basic meaning of simplices and simplical complexes, and in general understand the definition of simplical sets, but don't follow the geometric intuition.


Here are my notes if that's helpful. The definition of the geometric realization of a simplical set is:

There is a functor $|•|: \mathbf{sSet} \to \mathbf{CGHaus}$ called the geometric realization taking a simplicial set $X$ to its corresponding realization in the category of compactly-generated Hausdorff topological spaces.

From my understanding, a Hausdorff space is a space where every point is disconnected from every other point. (I think I pretty much understand functors). So to me this says that it's really not much of a geometric realization, since everything is just an isolated point.

Wondering if one could provide an intuitive understanding of the geometric realization of a simplical set.

The reason is, simplical sets generalize directed multigraphs, and abstract simplical complexes generalize undirected simple graphs, so there is a connection of simplical sets to graph theory. If there is a geometric realization of a simplical set, then there is a geometric realization of a directed multigraph. This seems like an interesting property, in that you could somehow create a geometric rendering or image of a graph.

From wikipedia:

A simplicial set $X$ is a collection of sets $X_n, n=0,1,2,...,$ together with certain maps between these sets: the face maps $d_n,i:X_n\to X_{n−1} (n=1,2,3,...\ and\ 0\leq i\leq n)$ and degeneracy maps $s_n,i:X_n\to X_{n+1} (n=0,1,2,...\ and\ 0\leq i\leq n)$.

The elements of $X_n$ are n-simplices of $X$.

Let $Δ$ denote the simplex category. The objects of $Δ$ are nonempty linearly ordered sets of the form

$$[n] = {0, 1, ..., n}$$

with $n \geq 0$.

Simplicial sets are therefore nothing but presheaves on $Δ$.

A presheaf on a category $C$ is a functor $F\colon C^\mathrm{op}\to\mathbf{Set}$

Given a partially ordered set (S,≤), we can define a simplicial set NS, the nerve of S.

Given that a sheaf allows you to attach data to open sets of a topological space, then there is some relation between graphs and data (sheafs) through simplical sets. Combine this with a geometric realization of the simplical set, and it seems you have a geometric representation of the data.

Looking for help on my understanding of the geometric realization of a simplical set, and it's connection to sheafs/data and graph theory. It would be helpful to learn what a picture might look like for an example simplical set / representation.

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    $\begingroup$ Your understanding of Hausdorff spaces is wrong. A Hausdorff space is a space equipped with a topology rich enough to provide, for any two points $x$ and $y$, open neighbourhoods $U$ of $x$ and $V$ of $y$ such that $U \cap V = \emptyset$. This does not mean "everything is just an isolated point". To get an intuitive understanding of the geometric realisation of a simplicial set, it might help to start with the notion of simplicial complex that gave rise to the much more abstract notion of a simplicial set. $\endgroup$ – Rob Arthan May 2 '18 at 23:28
  • $\begingroup$ Most spaces we care about (excluding maybe schemes) are Hausdorff. For example, any manifold is Hausdorff. An isolated point, however, in a space $X$, is a point that is open. Since in a Hausdorff space every point is closed, if every point were isolated then the space would be discrete. $\endgroup$ – leibnewtz May 3 '18 at 6:32

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