On the notion of geometric realization.

*(English)*Zbl 1073.55010This paper gives a description of the geometric realization of a simplicial set \(X\) which facilitates showing that the realization construction preserves finite products. The construction is also used to demonstrate that the realization of a simplicial set (respectively, cyclic set) has an action by the group of orientation-preserving homeomorphisms of the unit interval \(I=[0,1]\) (respectively, the circle).

This model for the realization of \(X\) is defined as a set by \[ D(X) = \varinjlim_{F \subset I} X(\pi_{0}(I-F)), \] where the colimit is indexed over the finite subsets \(F\) of the unit interval \(I\) and the finite directed set \(\pi_{0}(I-F)\) is identified up to unique isomorphism with some finite ordinal number. One can show that \[ D(\Delta^{n}) = \{(x_{1},\dots,x_{n}) \in I^{n}\;|\;x_{1} \leq \dots \leq x_{n}\}. \] In general, \(D(X)\) is a colimit of such spaces, indexed over the simplices of \(X\), and \(D(X)\) acquires its topology from this construction. The functor \(X \mapsto D(X)\) plainly preserves products at the point set level. \smallskip The space \(D(\Delta^{n})\) is naturally homeomorphic to the space \[ | \Delta^{n} | = \{ (t_{0},t_{1}, \dots ,t_{n})\;|\;0 \leq t_{i}, \sum t_{i} = 1 \} \] via the assignment \[ (t_{0},t_{1}, \dots ,t_{n}) \mapsto (t_{0},t_{0}+t_{1}, \dots ,t_{0}+\dots +t_{n-1}). \] and then it follows that \(D(X)\) is naturally homeomorphic to the standard realization \(| X |\) of \(X\).

This model for the realization of \(X\) is defined as a set by \[ D(X) = \varinjlim_{F \subset I} X(\pi_{0}(I-F)), \] where the colimit is indexed over the finite subsets \(F\) of the unit interval \(I\) and the finite directed set \(\pi_{0}(I-F)\) is identified up to unique isomorphism with some finite ordinal number. One can show that \[ D(\Delta^{n}) = \{(x_{1},\dots,x_{n}) \in I^{n}\;|\;x_{1} \leq \dots \leq x_{n}\}. \] In general, \(D(X)\) is a colimit of such spaces, indexed over the simplices of \(X\), and \(D(X)\) acquires its topology from this construction. The functor \(X \mapsto D(X)\) plainly preserves products at the point set level. \smallskip The space \(D(\Delta^{n})\) is naturally homeomorphic to the space \[ | \Delta^{n} | = \{ (t_{0},t_{1}, \dots ,t_{n})\;|\;0 \leq t_{i}, \sum t_{i} = 1 \} \] via the assignment \[ (t_{0},t_{1}, \dots ,t_{n}) \mapsto (t_{0},t_{0}+t_{1}, \dots ,t_{0}+\dots +t_{n-1}). \] and then it follows that \(D(X)\) is naturally homeomorphic to the standard realization \(| X |\) of \(X\).

Reviewer: Rick Jardine (London /Ontario)