The groupoid structure of groupoid morphisms

Chen, Bohui; Du, Cheng-Yong; Wang, Rui

doi:10.1016/j.geomphys.2019.103486

Cited by 7 publications

(5 citation statements)

References 12 publications

(27 reference statements)

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“…Although the statements in [CDW19] are for groupoids, the proof applies to proper étale Lie groupoids. Sometimes, it is helpful to replace the weak fiber product E × C E ′ , where the pullback square is only commutative up to natural transformations, to a smaller model whose pullback square is actually commutative.…”

Section: 41mentioning

confidence: 99%

“…In the following, we will briefly describe how the construction of the orbifold structures on C k (Σ, W ) and W k,p (Σ, W ) goes in [Che06a]. More precisely, we will follow [CDW19] to describe C k (Σ, W ), W k,p (Σ, W ) as orbits spaces of groupoids, and then we use the local constructions in [Che06a] in order to show that the groupoids are proper étale Lie groupoids modeled over Banach manifolds. The following results are only stated and proved for C k maps, as the cases for W k,p , C ∞ are similar.…”

Section: Morphisms Between Orbifoldsmentioning

confidence: 99%

“…Following [CDW19], a full equivalence ψ : C → D is an equivalence where ψ is surjective on the object level. Assume φ, ψ : E, E ′ → C are two full equivalences; the strict fiber product E ⊗ C E ′ is defined to be…”

Section: 41mentioning

confidence: 99%

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Exact orbifold fillings of contact manifolds

Gironella¹,

Zhou²

2021

Preprint

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We study exact orbifold fillings of contact manifolds using Floer theories. Motivated by Chen-Ruan's orbifold Gromov-Witten invariants, we define symplectic cohomology of an exact orbifold filling as a group using classical techniques, i.e. choosing generic almost complex structures. By studying moduli spaces of pseudo-holomorphic curves in orbifolds, we obtain various non-existence, restrictions and uniqueness results for orbifold singularities of exact orbifold fillings of many contact manifolds. For example, we show that exact orbifold fillings of (RP 2n−1 , ξ std ) always have exactly one singularity modeled on C n /(Z/2Z) if n = 2 k . Lastly, we show that in dimension at least 3 there are pairs of contact manifolds without exact cobordisms in either direction, and that the same holds for exact orbifold cobordisms in dimension at least 5.

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Section: 41mentioning

confidence: 99%

Section: Morphisms Between Orbifoldsmentioning

confidence: 99%

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Exact orbifold fillings of contact manifolds

Gironella¹,

Zhou²

2021

Preprint

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“…The identity function is given by id : A → A , such that id(a) = a ∈ A in any arbitrary set A. If G = (X, * ) is a groupoid, then the morphism is given by, α : (x ∈ G) → (y ∈ G) in the groupoid [26]. Moreover, if Ω = {G i : i ∈ I} then the corresponding morphisms can be defined as s : (G ∈ Ω) → Ω and t : (G ∈ Ω) → Ω , such that s(α) = x, t(α) = y.…”

Section: Preliminary Conceptsmentioning

confidence: 99%

Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings

Bagchi

2020

Symmetry

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In general, the group decompositions are formulated by employing automorphisms and semidirect products to determine continuity and compactification properties. This paper proposes a set of constructions of novel topological decompositions of groups and analyzes the behaviour of group actions under the topological decompositions. The proposed topological decompositions arise in two varieties, such as decomposition based on topological fibers without projections and decomposition in the presence of translated projections in topological spaces. The first variety of decomposition introduces the concepts of topological fibers, locality of group operation and the partitioned local homeomorphism resulting in formulation of transitions and symmetric surjection within the topologically decomposed groups. The reformation of kernel under decomposed homeomorphism and the stability of group action with the existence of a fixed point are analyzed. The first variety of decomposition does not require commutativity maintaining generality. The second variety of projective topological decomposition is formulated considering commutative as well as noncommutative projections in spaces. The effects of finite translations of topologically decomposed groups under projections are analyzed. Moreover, the embedding of a decomposed group in normal topological spaces is formulated in this paper. It is shown that Schoenflies homeomorphic embeddings preserve group homeomorphism in the decomposed embeddings within normal topological spaces. This paper illustrates that decomposed group embedding in normal topological spaces is separable. The applications aspects as well as parametric comparison of group decompositions based on topology, direct product and semidirect product are included in the paper.

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“…The identity function is given by id : A → A such that id(a) = a ∈ A in any arbitrary set A. If G = (X, * ) is a groupoid, then the morphism is given by α : (x ∈ G) → (y ∈ G) in the groupoid [22]. The morphisms α 1 , α 2 , α 3 maintain associative composition law given by…”

Section: Preliminary Conceptsmentioning

confidence: 99%

The Sequential and Contractible Topological Embeddings of Functional Groups

Bagchi

2020

Symmetry

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The continuous and injective embeddings of closed curves in Hausdorff topological spaces maintain isometry in subspaces generating components. An embedding of a circle group within a topological space creates isometric subspace with rotational symmetry. This paper introduces the generalized algebraic construction of functional groups and its topological embeddings into normal spaces maintaining homeomorphism of functional groups. The proposed algebraic construction of functional groups maintains homeomorphism to rotationally symmetric circle groups. The embeddings of functional groups are constructed in a sequence in the normal topological spaces. First, the topological decomposition and associated embeddings of a generalized group algebraic structure in the lower dimensional space is presented. It is shown that the one-point compactification property of topological space containing the decomposed group embeddings can be identified. Second, the sequential topological embeddings of functional groups are formulated. The proposed sequential embeddings follow Schoenflies property within the normal topological space. The preservation of homeomorphism between disjoint functional group embeddings under Banach-type contraction is analyzed taking into consideration that the underlying topological space is Hausdorff and the embeddings are in a monotone class. It is shown that components in a monotone class of isometry are not separable, whereas the multiple disjoint monotone class of embeddings are separable. A comparative analysis of the proposed concepts and formulations with respect to the existing structures is included in the paper.

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The groupoid structure of groupoid morphisms

Cited by 7 publications

References 12 publications

Exact orbifold fillings of contact manifolds

Exact orbifold fillings of contact manifolds

Projective and Non-Projective Varieties of Topological Decomposition of Groups with Embeddings

The Sequential and Contractible Topological Embeddings of Functional Groups

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