nLab Barratt nerve

Definition

Definition

Given a simplicial set $X \in sSet$, let $ndg(X)$ denote its set of non-degenerate simplices, regarded as a partially ordered set (poset) where $\sigma \leq \sigma'$ iff $\sigma$ is a face of $\sigma'$.

With posets regarded (see there) as categories, let $N(ndg(X)) \,\in\, sSet$ be the simplicial nerve of the poset of non-degenerate simplices. This is also called the Barratt nerve of $X$ (in honor of Michael Barratt):

(1)$$Ba(X) \,\coloneqq\, N\big(ndg(X)\big) \,.$$

[Waldhausen, Jahren & Rognes 2013 Def. 2.2.3]

Properties

In general (namely on simplicial sets which are “singular”), the Barratt nerve (1) is not homotopically well-behaved. For instance:

Example

The Barratt nerve of the “simplicial $n$-sphere” $\Delta^n/\partial \Delta^n$ is the 1-simplex and hence contractible:

$$Ba\big(\Delta^n/\partial \Delta^n\big) \;\underset{iso}{\simeq}\; \Delta^1 \,.$$

[WJR13 p 28]

But on “non-singular” simplicial sets such as produced by subdivision $Sd \,\colon\, sSet \xrightarrow{\;} sSet$, the Barratt nerve produces weakly homotopy equivalent types:

Proposition

The natural transformation

$$Ba \circ Sd (X) \xrightarrow{\;} X$$

is a weak homotopy equivalence and in fact a triangulation in that its topological realization is homotopic to a homeomorphism.

[Fjellbo 2020]

Hence the Barratt nerve of the subdivision models the homotopy type of any simplicial set by the nerve of a poset. This composite

$$Ba \circ Sd \,\colon\, sSet \xrightarrow{\;} sSet$$

is called the improvement functor in WJR13 Thm 2.5.2.

References

• Friedhelm Waldhausen, Bjørn Jahren, John Rognes: Spaces of PL Manifolds and Categories of Simple Maps, Annals of Mathematics Studies, Princeton University Press (2013) [jstor:j.ctt24hqsv, pdf]

• Rune Vegard Fjellbo, p. 21 of: Non-singular simplicial sets, PhD thesis (2018) [URN:NBN:no-75277]

• Rune Vegard Fjellbo: Optimal Triangulation of Regular Simplicial Sets [arXiv:2001.04339]