Geometric realisation as the Skorokhod semi-continuous path space endofunctor

TL;DR

This paper offers a new interpretation of geometric realisation as a Skorokhod-like semi-continuous path space, clarifying its properties and functorial behavior in a categorical framework.

Contribution

It introduces a novel categorical perspective on geometric realisation, showing it factors through an endofunctor and explaining key properties like product commutation.

Findings

Geometric realisation viewed as a Skorokhod semi-continuous path space.

Functorial factorization of geometric realisation through an endofunctor.

Clarification of why geometric realisation commutes with products and group actions.

Abstract

We interpret a construction of geometric realisation by [Besser], [Grayson], and [Drinfeld] of a simplicial set as constructing a space of maps from the interval to a simplicial set, in a certain formal sense, reminiscent of the Skorokhod space of semi-continuous functions; in particular, we show the geometric realisation functor factors through an endofunctor of a certain category. Our interpretation clarifies the explanation of [Drinfeld] "why geometric realization commutes with Cartesian products and why the geometric realization of a simplicial set [...] is equipped with an action of the group of orientation preserving homeomorphisms of the segment [0,1]".