Categorical aspects of toric topology

TL;DR

This paper advocates integrating category theory into toric topology, demonstrating how algebraic models derived from homotopy colimits of diagrams over face categories can elucidate properties of toric spaces like moment-angle complexes and quasitoric manifolds.

Contribution

It introduces a categorical framework for toric topology, providing algebraic models for toric spaces via homotopy colimits and illustrating their applications through explicit calculations.

Findings

Quasitoric manifolds are rationally formal.

The rational Pontrjagin ring of J(K) is quadratic dual to Q[K].

DJ(K) is coformal if and only if K is a flag complex.

Abstract

We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces X_K, such as moment-angle complexes Z_K, quasitoric manifolds M, and Davis-Januszkiewicz spaces DJ(K). We first exhibit X_K as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)-diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as algebraic models for X_K. Such models encode many standard algebraic invariants, and their existence is assured by the Quillen structure. We provide several illustrative calculations, often over the rationals, including proofs that quasitoric manifolds (and various generalisations) are rationally formal; that the rational Pontrjagin ring of the loop space \Omega DJ(K) is isomorphic to the quadratic dual of the Stanley-Reisner algebra Q[K] for flag complexes K; and that DJ(K) is coformal precisely when K is flag. We conclude by describing algebraic models for the loop space \Omega DJ(K) for any complex K, which mimic our previous description as a homotopy colimit of topological monoids.