Dualizability in Low-Dimensional Higher Category Theory

Schommer-Pries, Christopher

doi:10.1090/conm/613/12237

Cited by 7 publications

(6 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Next, we show that the Serre automorphism is actually a pseudo-natural transformation of the identity functor on the maximal subgroupoid of C, as suggested in [Sch13]. To the best of our knowledge, a proof of this statement has not appeared in the literature so far, hence we illustrate the details in the following.…”

Section: Definition 23 An Object X In a Symmetric Monoidal Bicategory...mentioning

confidence: 61%

The Serre automorphism via homotopy actions and the Cobordism Hypothesis for oriented manifolds

Hesse,

Valentino

2017

Preprint

View full text Add to dashboard Cite

We explicitly construct an SO(2)-action on a skeletal version of the 2-dimensional framed bordism bicategory. By the 2-dimensional Cobordism Hypothesis for framed manifolds, we obtain an SO(2)-action on the core of fully-dualizable objects of the target bicategory. This action is shown to coincide with the one given by the Serre automorphism. We give an explicit description of the bicategory of homotopy fixed points of this action, and discuss its relation to the classification of oriented 2d topological quantum field theories.oriented TQFTs. It is relevant to notice that in [Lur09] the homotopy O(n)-action on the framed (∞, n)-category of cobordisms is not explicitly constructed, or even briefly sketched. For an extensive introduction to extended TQFTs and the Cobordism Hypothesis, we refer the reader to [Fre12]. Blurring the distinction between (∞, 2)-categories and bicategories, in [FHLT10] it is argued that in the case where the target is given by the bicategory Alg 2 of algebras, bimodules, and intertwiners, the fully dualizable objects are semisimple finite-dimensional algebras, and that the additional SO(2)-fixed-points structure should correspond to the structure of a symmetric Frobenius algebra. Via a direct construction, in [SP09] it is showed that the bigroupoid Frob of Frobenius algebras, Morita contexts and intertwiners indeed classifies fully extended oriented 2-dimensional TQFTs valued in Alg 2 . In [Dav11], it is observed that the SO(2)-action given by the Serre automorphism on the core of fully-dualizable objects of Alg 2 is trivializable. In a purely bicategorical setting, in [HSV17] the homotopy-fixed-point bigroupoid of the SO(2)-action on Alg 2 is computed, and it is shown that it coincides with Frob.In the present paper we provide an explicit SO(2)-action on the framed bordism bicategory, and show that the SO(2)-action induced on K (C fd ) for any symmetric monoidal bicategory C is given by the Serre automorphism, regarded as a pseudo-natural isomorphism of the identity functor. More precisely, we make use of a presentation of the framed bordism bicategory provided in [Pst14] to construct such an SO(2)-action. By the Cobordism Hypothesis for framed manifolds, which has been proven in the setting of bicategories in [Pst14], there is an equivalence of bicategories (1.1) Fun ⊗ (Cob fr 2,1,0 , C) ∼ = K (C fd ).

show abstract

Section: Definition 23 An Object X In a Symmetric Monoidal Bicategory...mentioning

confidence: 61%

The Serre automorphism via homotopy actions and the Cobordism Hypothesis for oriented manifolds

Hesse,

Valentino

2017

Preprint

View full text Add to dashboard Cite

show abstract

“…When talking about colimits we always mean the appropriate higher categorical concept, for example for d = 2 we mean what is usually called a pseudolimit [JY21, Chapter 5]. The general definition of a fully dualisable will not matter much for us and hence we refer to any of the following references for it [Lur09,GS18,SP14]. The second point implies that the category of framed d-dimensional fully extended topological field theories with values in C is equivalent to the core of the full subcategory on all fully dualisable objects C f.d.…”

Section: The Cobordism Hypothesismentioning

confidence: 99%

Reflection Structures and Spin Statistics in Low Dimensions

Müller¹,

Stehouwer²

2023

Preprint

View full text Add to dashboard Cite

We give a complete classification of topological field theories with reflection structure and spinstatistics in one and two spacetime dimensions. Our answers can be naturally expressed in terms of an internal fermionic symmetry group G which is different from the spacetime structure group. Fermionic groups encode symmetries of systems with fermions and time reversing symmetries. We show that 1-dimensional topological field theories with reflection structure and spin-statistics are classified by finite dimensional hermitian representations of G. In spacetime dimension two we give a classification in terms strongly G-graded stellar Frobenius algebras. Our proofs are based on the cobordism hypothesis. Along the way, we develop some useful tools for the computation of homotopy fixed points of 2-group actions on bicategories.

show abstract

“…Proof Equations (5.2) and (5.3) are the statement that

\mathcal {C}(T,P)

and

\mathcal {C}(\widehat{T},N-P)\lbrace 0,\tfrac{n-m}{2}\rbrace

are dual 1‐morphisms in the bicategory of

\mathbb {Z}

‐algebras, chain complexes of bimodules, and homotopy classes of chain maps [33, Definition 6.1]. Since

\mathcal {C}(T,P)

and

\operatorname{RHom}_{\mathcal {C}(m)}(\mathcal {C}(T,P),\mathcal {C}(m))

are also a dual pair, the result follows from (the proof of) uniqueness of the dual of a dualizable 1‐morphism (essentially [10, Proposition 2.10.5], for instance).…”

Section: Duality Properties Of Khovanov's Tangle Invariants and Their...mentioning

confidence: 99%

Homotopy functoriality for Khovanov spectra

Lawson

Lipshitz

Sarkar

2022

Journal of Topology

View full text Add to dashboard Cite

show abstract

Dualizability in Low-Dimensional Higher Category Theory

Abstract: Abstract. These lecture notes form an expanded account of a course given at the Summer School on Topology and Field

Cited by 7 publications

References 22 publications

The Serre automorphism via homotopy actions and the Cobordism Hypothesis for oriented manifolds

The Serre automorphism via homotopy actions and the Cobordism Hypothesis for oriented manifolds

Reflection Structures and Spin Statistics in Low Dimensions

Homotopy functoriality for Khovanov spectra

Contact Info

Product

Resources

About