[Paper Review] A simple approach to geometric realization of simplicial and cyclic sets

Amnon Besser|ArXiv.org|May 18, 2003

Homotopy and Cohomology in Algebraic Topology5 references3 citations

TL;DR

This paper presents a geometric realization of simplicial and cyclic sets via order-preserving maps from the unit interval to finite partially ordered sets (posets), offering a natural, topological interpretation of standard simplices. The key contribution is a functorial, product-preserving geometric realization that extends naturally to cyclic sets using periodic posets, with the cyclic realization of $[[n]]$ shown to be homeomorphic to $| abla_n| \times S^1$, unifying the simplicial and cyclic theories through order-theoretic foundations.

ABSTRACT

This is the same version that was previously only on my home page. We give a description of geometric realization which makes it evident that it commutes with products. A similar approach is used to treat cyclic sets. Our approach is similar to those of Drinfeld and Grayson.

Motivation & Objective

• To provide a natural, combinatorially grounded explanation for the geometric realization of standard simplices, which traditionally lacks justification in homotopy theory.
• To establish a geometric realization functor for simplicial sets that commutes with products, using the space of order-preserving maps from the unit interval to finite posets.
• To extend the framework to cyclic sets by introducing periodic posets (ppsets) as the analog of posets in the cyclic setting.
• To show that the cyclic category of Connes is isomorphic to a subcategory of degree-1 periodic posets, unifying the cyclic and simplicial theories.
• To prove that geometric realization of cyclic sets preserves products, using the circle action and topological structure on ppset maps.

Proposed method

• Define the geometric realization of a finite poset $P$ as the space of upper semicontinuous, order-preserving maps from the unit interval $I = [0,1]$ to $P$, equipped with a metric topology.
• Show that this realization functor on posets is naturally isomorphic to the standard simplicial set $\mathcal{S}(P)$, with $\mathcal{S}([n]) = \Delta_n$.
• Introduce periodic posets (ppsets) as posets with a free $\mathbb{Z}^k$-action, where $k$ is the degree, to model cyclic sets.
• Define the cyclic realization of a ppset $P$ as the space of $\mathbb{Z}$-equivariant, order-preserving maps from $\mathbb{R}$ to $P$ with periodicity, modulo the action.
• Construct a homeomorphism between the realization of the standard cyclic $n$-simplex $[[n]]$ and $|\Delta_n| \times S^1$, using fractional and integer parts of real functions.
• Prove that the cyclic realization functor commutes with products by showing $||\mathcal{C}(P) \times \mathcal{C}(Q)|| \cong ||\mathcal{C}(P)|| \times ||\mathcal{C}(Q)||$ via the diagonal circle action.

Experimental results

Research questions

• RQ1 How can the geometric realization of standard simplices be given a natural, order-theoretic interpretation without arbitrary postulates?
• RQ2 Can the geometric realization of simplicial sets be shown to commute with products using only order-preserving maps and topology?
• RQ3 What is the appropriate categorical generalization of posets for cyclic sets, and how does it support a geometric realization theory?
• RQ4 Is the cyclic category of Connes isomorphic to a subcategory of periodic posets of degree 1?
• RQ5 Does the geometric realization of cyclic sets preserve products, and if so, how is this structure encoded?

Key findings

• The geometric realization of a finite poset $P$ is naturally identified with the space of order-preserving, upper semicontinuous maps from $I = [0,1]$ to $P$, with the compact-open topology.
• The simplicial set $\mathcal{S}(P)$ associated to a poset $P$ has a geometric realization isomorphic to the realization of $P$ itself, establishing a natural correspondence.
• The standard cyclic $n$-simplex $\tilde{\Delta}_n$ is geometrically realized as $|\Delta_n| \times S^1$, with the circle acting trivially on $|\Delta_n|$.
• The cyclic category of Connes is isomorphic to the full subcategory of degree-1 periodic posets with positive archimedean structure.
• Any positive archimedean compact ppset $P$ is isomorphic to $[[n]]$ where $n+1 = |\mathbb{Z} \backslash P|$, providing a classification of such objects.
• The geometric realization of cyclic sets commutes with products: $||\mathcal{C}(P) \times \mathcal{C}(Q)|| \cong ||\mathcal{C}(P)|| \times ||\mathcal{C}(Q)||$ via the diagonal circle action.