## On the notion of geometric realization.(English)Zbl 1073.55010

This 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.
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 $$X$$.

### MSC:

 55U10 Simplicial sets and complexes in algebraic topology 18G30 Simplicial sets; simplicial objects in a category (MSC2010) 19D55 $$K$$-theory and homology; cyclic homology and cohomology

### Keywords:

simplicial set; geometric realization; cyclic set
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