nLab ordinal subdivision

Idea

The most usual method of subdivision for a simplicial complex as used in elementary algebraic and geometric topology is the barycentric subdivision. There is however another very well structured subdivision construction encountered which can be useful. The basic geometric construction involves chopping up a geometric $n$-simplex by $n$-planes parallel to a face and halfway between that face and the opposite vertex.

Categorical descriptions

We first define the ordinal subdivision on the simplices, $\Delta[n]$. (Here $\oplus$ denotes the ordinal sum

The ordinal subdivision of $\Delta[n]$, the standard $n$-simplex in simplicial sets, is denoted by $Sd(\Delta[n])$, and is defined as follows:-

$Sd(\Delta[n]) := \int^{p,q} \Delta([p]\oplus[q],[n]) \cdot (\Delta[p] \times \Delta[q]).$

For a general simplicial set we can then define:

the ordinal subdivision of a simplicial set, $X$, is denoted by $Sd(X)$ and is defined by

$$Sd(X) := \int^{n} X_n \cdot Sd(\Delta [n]).$$

This expands to

$$Sd(X) := \int^{n} X_n \cdot \left(\int^{p,q} \Delta([p]\oplus[q],[n]) \cdot (\Delta[p] \times \Delta[q]) \right).$$

This is related to the total décalage of $X$ by

$$Sd(X) = \cong \int^{p,q} DEC(X)_{p,q} \cdot (\Delta[p] \times \Delta[q]).$$

References

• P. J. Ehlers and Tim Porter, Ordinal subdivision and special pasting in quasicategories, Advances in Math. 217 (2007), No 2. pp 489-518.Delta