Double bars around poset?

Question

Given a poset X that is contractible, I have a document that says:

"If $H : \|X\| \times I \rightarrow \|X\|$ is a homotopy to the constant map"

What does the $\|-\|$ signify? I have no idea if this is usual notation, so I thought I would give it a shot here.

Answer 1

The phrase

If $H : \|X\| \times I \rightarrow \|X\|$ is a homotopy to the constant map

makes clear that $\lVert X \rVert$ must be some topological space associated to $X$. In fact, as already noticed in paul blart math cop's comments, it is the geometric realization of the poset $X$. I recommend to read this. Quotations:

From a geometric simplicial complex $K$, one gets an abstract simplicial complex $∆(K)$ by letting the faces of $∆(K)$ be the vertex sets of the simplices of $K$. Every abstract simplicial complex $∆$ can be obtained in this way, i.e., there is a geometric simplicial complex $K$ such that $∆(K) = ∆$. Although $K$ is not unique, the underlying topological space, obtained by taking the union of the simplices of $K$ under the usual topology on $\mathbb R^n$, is unique up to homeomorphism. We refer to this space as the geometric realization of $∆$ and denote it by $\lVert \Delta \rVert$.

To every poset $P$, one can associate an abstract simplicial complex $∆(P)$ called the order complex of $P$. The vertices of $∆(P)$ are the elements of $P$ and the faces of $∆(P)$ are the chains (i.e., totally ordered subsets) of $P$.

Perhaps it is more usual to write $\lvert \Delta \rvert$ instead of $\lVert \Delta\rVert$, but notation is always a matter of taste. Anyway,

$$\lVert X \rVert = \lVert \Delta(X) \rVert .$$