Categorical aspects of toric topology
59 citations recorded by the source corpus.
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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/14050cbf36e67d047fb77e03965597faffd477f0
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In Discrete Mathematics and Combinatorics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Toric Topology (2012, methodology): “Our exposition follows closely the introductory sections of the work of Panov and Ray [260].”
- Toric topology (2012, methodology): “We refer the reader to the book ([9]) by Buchstaber-PanovCthe paper ([75]) by Panov-RayCthe lecture note ([8]) by Buchstaber for other viewpoints of toric topology, and to the proceedings “Toric Topology” (Harada et al.”
- The homotopy types of moment-angle complexes for flag complexes (2012, methodology): “The theorem above is formulated in [21] with Q-coefficients, but the argument (using the Adams cobar construction and a result of Fröberg [13] on quadratic duality) works for arbitrary field.”