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 nn-simplex by nn-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, Δ[n]\Delta[n]. (Here \oplus denotes the ordinal sum

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

Sd(Δ[n]):= p,qΔ([p][q],[n])(Δ[p]×Δ[q]). 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, XX, is denoted by Sd(X)Sd(X) and is defined by

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

This expands to

Sd(X):= nX n( p,qΔ([p][q],[n])(Δ[p]×Δ[q])). 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 XX by

Sd(X)= p,qDEC(X) p,q(Δ[p]×Δ[q]). 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

Last revised on July 28, 2014 at 23:17:28. See the history of this page for a list of all contributions to it.