## Categorical aspects of toric topology

Citations: | 3 - 0 self |

### BibTeX

@MISC{Panov_categoricalaspects,

author = {Taras E Panov and Nigel Ray},

title = {Categorical aspects of toric topology},

year = {}

}

### OpenURL

### Abstract

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 XK, such as moment-angle complexes ZK, quasitoric manifolds M, and Davis-Januszkiewicz spaces DJ(K). First we exhibit every XK 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 XK. 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 ΩDJ(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 ΩDJ(K) for any complex K, which mimic our previous description as a homotopy colimit of topological monoids. 1.

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Citation Context ... obey the left lifting property with respect to acyclic fibrations. The model structure for tmon is originally due to Schwänzl and Vogt [40], and may also be deduced from Schwede and Shipley’s theory =-=[41]-=- of monoids in monoidal model categories; weak equivalences and fibrations are those homomorphisms which are weak equivalences and fibrations in top, and cofibrations obey the appropriate lifting prop... |

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Citation Context ...is-Januszkiewicz space DJ(K). As introduced in [11], DJ(K) is defined by the Borel construction ZK ×T m ET m , and its cohomology ring H ∗ (DJ(K); R) is isomorphic to the Stanley-Reisner algebra R[K] =-=[43]-=- for any coefficient ring R. Following [8] and [35], it may be interpreted as the colimit of a diagram B K , whose value on each face σ of K is the cartesian product BT σ ; then R[K] is isomorphic to ... |

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Citation Context ...ied with its dual polytope P , which is simple. In this case, certain subtori T m−n < T m act freely on ZK, and their quotient spaces M n are the toric manifolds introduced by Davis and Januszkiewicz =-=[11]-=-. In order to avoid confusion with the toric varieties of algebraic geometry, we follow current convention [8] by labelling them quasitoric manifolds. The quotient of any such M by the n–torus T m /T ... |

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Citation Context ...terwoven in the study of categories of diagrams, whose terminology and notation we introduce as we proceed. For more complete background information we refer to the books of Hirschhorn [23] and Hovey =-=[25]-=-. We define our simplicial complexes K on a graded set V of vertices v1, . . . , vm, each of which has dimension 2. So K is a collection of faces σ ⊆ V , closed under the formation of subsets and incl... |

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Citation Context ...ed complexes are studied by Hovey [24], for example. In chR, we assume that the fibrations are epimorphic in positive degrees and the cofibrations are monomorphic with degree-wise projective cokernel =-=[14]-=-. In particular, every object is fibrant. The existence of limits and colimits is assured by working dimensionwise, and functoriality of the factorisations (2.2) follows automatically from the fact th... |

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Citation Context ...as objects of chR. For any object C of dgc0,R, the loop algebra Ω∗C is given by the tensor algebra TR(s −1 C) on the desuspended R-module C = Ker(ε: C → R), and agrees with Adams’s cobar construction =-=[1]-=- as objects of chR. The graded homology algebra H(Ω∗C) is denoted by CotorC(R, R), and is isomorphic to the Pontrjagin ring H∗(ΩX; R) when C is the singular chain complex of a reduced CW-complex X. Ov... |

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Citation Context ...ant, so there is a weak equivalence M[l] → holim Pl by Proposition 4.9(2). In 2-dimensional integral homology, l induces l∗ : Z V → Z n , which extends to a dicharacteristic homomorphism ℓ: T V → T n =-=[9]-=-. The kernel of ℓ is an (m−n)-dimensional subtorus T [l] < T V , which acts freely on ZK with quotient M[l]. Moreover, ℓ maps T σ isomorphically onto its image, which we denote by T (σ) < T n for any ... |

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Citation Context ...py categories (or certain of their full subcategories) in favourable cases. Our first examples of model categories are geometric, as follows. Notation 2.5. • top: pointed k-spaces and continuous maps =-=[45]-=-. • tmon: topological monoids and continuous homomorphisms. We assume that topological monoids are k-spaces and are pointed by their identities, so that tmon is a subcategory of top. The standard mode... |

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Citation Context ...re the rational formality of DJ(K) is established for every simplicial complex K. We extend this result below, to a class of toric spaces that includes quasitoric manifolds and the torus manifolds of =-=[28]-=- as special cases. By way of contrast, we note that calculations of Baskakov [2] and Denham and Suciu [12] confirm that many moment-angle complexes ZK support non-trivial Massey products, and so canno... |

17 |
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Citation Context ...he loop space ΩDJ(K) is modelled geometrically by a homotopy colimit in the category tmon of topological monoids. Developing these themes leads us naturally into the world of Quillen model categories =-=[37]-=-. We therefore work with model categories whenever we are able. As well as being currently fashionable, Quillen’s theory suggests questions that we might not otherwise have asked, and presents challen... |

16 | The homotopy type of the complement of a coordinate subspace arrangement
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Citation Context ...zi = zj = 0} ≃ S 5 \ (S 1 ∪ S 1 ∪ S 1 ) ≃ Σ 3( S 2 \ (S 1 ∪ S 1 ∪ S 1 ) ) 1≤i<j≤3 ≃ Σ 3 (S 0 ∨ S 0 ∨ S 0 ∨ S 1 ∨ S 1 ) ≃ S 3 ∨ S 3 ∨ S 3 ∨ S 4 ∨ S 4 , where the three circles are disjoint in S 5 . By =-=[21]-=-, the complement (10.1) is homotopy equivalent to a wedge of spheres for all m. The loop space ΩDJ(K) is homotopy equivalent to a free product T ∗ · · · ∗ T of m circles [36, Example 6.8], and the abo... |

14 |
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Citation Context ...−→ · · · � ←− Mk = (H(M), 0). Formality only has meaning in an algebraic model category. Sullivan’s approach to rational homotopy theory is based on the PL-cochain functor APL : top → cdga. Following =-=[16]-=-, APL(X) is defined as A ∗ (S•X), where S•(X) denotes the total singular complex of X and A ∗ : sset → cdga is the polynomial de Rham functor of [5]. The PL-de Rham Theorem yields a natural isomorphis... |

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Citation Context ...entials, and are the monoids in chR. A model category structure in dgaR is therefore induced by applying Quillen’s path object argument, as in [41]; a similar structure was first proposed by Jardine, =-=[26]-=- (albeit with cohomology differentials), who proceeds by modifying the methods of [5]. Fibrations are epimorphisms, and cofibrations are determined by the appropriate lifting property.sCATEGORICAL ASP... |

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Citation Context ...flects the fact that (ujvj = v2 j = 0 for j ∈ σ) is an acyclic ideal [3]. These models also work over Z, and are used in [3] to establish an integral version of Corollary 6.12 (which was confirmed in =-=[17]-=- by other methods). The model categorical interpretation must be relaxed in this case; nevertheless, ∧R(U) ⊗ R[K] may still be interpreted as a homotopy limit in dgaR. As shown in [2], the algebra ∧(U... |

6 |
Model Categories, volume 63
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Citation Context ...terwoven in the study of categories of diagrams, whose terminology and notation we introduce as we proceed. For more complete background information we refer to the books of Hirschhorn [22] and Hovey =-=[24]-=-. We define our simplicial complexes K on a graded set V of vertices v1, . . . , vm, each of which has dimension 2. So K is a collection of faces σ ⊆ V , closed under the formation of subsets and incl... |

5 |
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Citation Context ... ≥ 2 in C.s12 TARAS E PANOV AND NIGEL RAY 3.5. Differential graded coalgebras. Model structures on more general categories of differential graded coalgebras have been publicised by Getzler and Goerss =-=[19]-=-, who also work over a field. Once more, we restrict attention to Q. The objects of dgc are comonoids in ch, and the morphisms preserve comultiplication. The cofibrations are monomorphisms, and the fi... |

4 |
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Citation Context ... We extend this result below, to a class of toric spaces that includes quasitoric manifolds and the torus manifolds of [28] as special cases. By way of contrast, we note that calculations of Baskakov =-=[2]-=- and Denham and Suciu [12] confirm that many moment-angle complexes ZK support non-trivial Massey products, and so cannot be formal. A further goal is to place these facts in the context of [35], wher... |

4 |
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Citation Context ...which are necessarily supplemented. Model structure was first defined on the category cdgc1 of simply connected rational cocommutative coalgebras by Quillen [38], and refined to cdgc0 by Neisendorfer =-=[32]-=-. The cofibrations are monomorphisms, and the fibrations are determined by the appropriate lifting property. Limits exist because cdgc has finite products and filtered limits, whereas colimits are cre... |

4 | Colimits, Stanley-Reisner algebras, and loop spaces
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(Show Context)
Citation Context ..., DJ(K) is defined by the Borel construction ZK ×T m ET m , and its cohomology ring H ∗ (DJ(K); R) is isomorphic to the Stanley-Reisner algebra R[K] [43] for any coefficient ring R. Following [8] and =-=[35]-=-, it may be interpreted as the colimit of a diagram B K , whose value on each face σ of K is the cartesian product BT σ ; then R[K] is isomorphic to the limit of the corresponding cat op (K)-diagram o... |

4 |
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Citation Context ...motopy groups, fibrations are Serre fibrations, and cofibrations obey the left lifting property with respect to acyclic fibrations. The model structure for tmon is originally due to Schwänzl and Vogt =-=[40]-=-, and may also be deduced from Schwede and Shipley’s theory [41] of monoids in monoidal model categories; weak equivalences and fibrations are those homomorphisms which are weak equivalences and fibra... |

3 | The homotopy type of the complement of the codimension-two coordinate subspace arrangement
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(Show Context)
Citation Context ...zi = zj = 0} � S 5 \ (S 1 ∪ S 1 ∪ S 1 ) � Σ 3� S 2 \ (S 1 ∪ S 1 ∪ S 1 ) � 1≤i<j≤3 � Σ 3 (S 0 ∨ S 0 ∨ S 0 ∨ S 1 ∨ S 1 ) � S 3 ∨ S 3 ∨ S 3 ∨ S 4 ∨ S 4 , where the three circles are disjoint in S 5 . By =-=[20]-=-, the complement (10.3) is homotopy equivalent to a wedge of spheres for all m. The loop space ΩDJ(K) is homotopy equivalent to a free product T ∗ · · · ∗ T of m circles [35, Example 6.8], and the abo... |

3 | Simplicial Objects in Algebraic Topology. Volume 11 - May - 1967 |

2 |
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Citation Context ...model for the inclusion Dσ → (S ∞ ) σ × T V \σ is the projection ∧(U) ⊗ S(σ) → ∧(U) ⊗ S(σ) � (ujvj = v 2 j = 0 for j ∈ σ), which reflects the fact that (ujvj = v2 j = 0 for j ∈ σ) is an acyclic ideal =-=[3]-=-. These models also work over Z, and are used in [3] to establish an integral version of Corollary 6.12 (which was confirmed in [17] by other methods). The model categorical interpretation must be rel... |

2 |
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Citation Context ... given by d(l ⋆ 1) = dLl ⋆ 1 and d(1 ⋆ w) = z ⋆ 1. For historical reasons, a differential graded Lie algebra L is said to be coformal whenever it is formal in dgl. 3.7. Adjoint pairs. Following Moore =-=[31]-=-, [25], we consider the classifying functor B∗ and the loop functor Ω∗ as an adjoint pair (3.3) Ω∗ : dgc0,R ←−−− −−−→ dgaR :B∗.sCATEGORICAL ASPECTS OF TORIC TOPOLOGY 13 For any object A of dgaR, the c... |

1 |
Cohen-Macaulay rings. Volume 39
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(Show Context)
Citation Context ...y formal, and extend our analysis to generalisations such as the torus manifolds of [28]. For any simplicial convex n–polytope, the Stanley-Reisner algebra of the boundary complex K is Cohen–Macaulay =-=[7]-=-, and Q[K] admits a 2-dimensional linear system of parameters l1, . . . , ln. If the parameters are integral, then an associated quasitoric manifold M[l] may be constructed; up to homotopy, it is the ... |

1 |
Loops spaces on families of quasitoric manifolds
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- 2007
(Show Context)
Citation Context ...ts. Their properties are considerably simplified when K satisfies the requirements of a flag complex, but we postpone discussion of this situation until the following section. Geometric models for ΩM =-=[10]-=- are currently under development. Following [35], we loop the fibrations of Sections 6 and 7 to obtain fibrations (8.1) ΩZK −→ ΩDJ(K) −→ T V and ΩM −→ ΩDJ(K) Ωl −−→ T n . Each of these admits a sectio... |

1 | cobar equivalence - Adams’ - 1992 |

1 | Modules of topological spaces, applications ot homotopy colimits and E∞ structures - Hollender, Vogt - 1992 |

1 |
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(Show Context)
Citation Context ...oal is to place these facts in the context of [35], where the properties of tmon as a geometric model category play an important rôle, but remain implicit. We therefore study the rational coformality =-=[33]-=- of DJ(K), which depends on the rational structure of ΩDJ(K) and is verified for any flag complex K. Our study includes an investigation of the Lie algebra π∗(ΩDJ(K)) ⊗Z Q, and is related to calculati... |

1 |
On Davis Januszkiewicz Homotopy Types. I. Formality and Rationalisation. Algebr Geom Topol
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- 2005
(Show Context)
Citation Context ...a. We also consider other rational models of similar power, such as differential graded coalgebras, Lie algebras, and Pontrjagin rings. The categorical viewpoint has already motivated studies such as =-=[34]-=-, where the rational formality of DJ(K) is established for every simplicial complex K. We extend this result below, to a class of toric spaces that includes quasitoric manifolds and the torus manifold... |

1 |
Vergleich verschiedener Homotopie-Limes- und -Kolimeskonstruktionen in Modellkategorien
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(Show Context)
Citation Context ...veloped to remedy this deficiency. We outline their construction in this section, and discuss basic properties. The literature is still in a state of considerable flux, and we refer to Recke’s thesis =-=[39]-=- for a comparison of several alternative treatments. Here we focus mainly on those of Recke’s statements that are inspired by Hirschhorn, and make detailed appeal to [22] as necessary. With cat(K) and... |

1 |
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- 1997
(Show Context)
Citation Context ... Q) is isomorphic to (9.3) T (U) � (u 2 i = 0, uiuj + ujui = 0 for {i, j} ∈ K) for any flag complex K. Remark 9.4. Algebra (9.3) is the quadratic dual of (9.1). A quadratic algebra A is called Koszul =-=[42]-=- if its quadratic dual coincides with ExtA(Q, Q), so Proposition 9.2 asserts that Q[K] is Koszul whenever K is flag. Theorem 9.5. For any flag complex K, there are isomorphisms H∗(ΩDJ(K); Q) ∼ = T (U)... |

1 |
Rational Homotopy Ccalculus of Functors
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(Show Context)
Citation Context ...enges of independent interest. For example, the existence of homotopy colimits in an arbitrary model category has been known for some time [22], but specific constructions are still under development =-=[46]-=- in several cases that we discuss below; and the advent of [13] raises the possibility of working in the more general framework of homotopical categories.sCATEGORICAL ASPECTS OF TORIC TOPOLOGY 3 We co... |

1 |
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- 1999
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Citation Context ...te. If, however, we can justify interpreting XK as a homotopy colimit [6], such difficulties often evaporate. This possibility seems first to have been recognised by Welker, Ziegler, and ˇ Zivaljević =-=[47]-=-, who show that toric varieties themselves are expressible as homotopy colimits. Our own evidence is provided in [35], where the loop space ΩDJ(K) is modelled geometrically by a homotopy colimit in th... |

1 |
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(Show Context)
Citation Context ...R〈∂∆ m−1 〉) for m ≥ 3. The element w of (10.4) is the higher commutator product; it is the Hurewicz image of the higher Samelson product of 1–dimensional generators ui ∈ π1(ΩDJ(K)) for 1 ≤ i ≤ m, see =-=[48]-=-. It reduces to the ordinary commutator for m = 2. The fact that w is the non-trivial higher commutator product of u1, . . . , um constitutes the additional homology information necessary to distingui... |