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Register a new userTwisted loop transgression and higher Jandl gerbes over finite groupoids
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Given a double cover $pi: mathcal{G} rightarrow hat{mathcal{G}}$ of finite groupoids, we explicitly construct twisted loop transgression maps, $tau_{pi}$ and $tau_{pi}^{ref}$, thereby associating to a Jandl $n$-gerbe $hat{lambda}$ on $hat{mathcal{G}}$ a Jandl $(n-1)$-gerbe $tau_{pi}(hat{lambda})$ on the quotient loop groupoid of $mathcal{G}$ and an ordinary $(n-1)$-gerbe $tau^{ref}_{pi}(hat{lambda})$ on the unoriented quotient loop groupoid of $mathcal{G}$. For $n =1,2$, we interpret the character theory (resp. centre) of the category of Real $hat{lambda}$-twisted $n$-vector bundles over $hat{mathcal{G}}$ in terms of flat sections of the $(n-1)$-vector bundle associated to $tau_{pi}^{ref}(hat{lambda})$ (resp. the Real $(n-1)$-vector bundle associated to $tau_{pi}(hat{lambda})$). We relate our results to Re
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A base $Delta$ generating the topology of a space $M$ becomes a partially ordered set (poset), when ordered under inclusion of open subsets. Given a precosheaf over $Delta$ of fixed-point spaces (typically C*-algebras) under the action of a group $G$, in general one cannot find a precosheaf of $G$-spaces having it as fixed-point precosheaf. Rather one gets a gerbe over $Delta$, that is, a twisted precosheaf whose twisting is encoded by a cocycle with coefficients in a suitable 2-group. We give a notion of holonomy for a gerbe, in terms of a non-abelian cocycle over the fundamental group $pi_1(M)$. At the C*-algebraic level, holonomy leads to a general notion of twisted C*-dynamical system, based on a generic 2-group instead of the usual adjoint action on the underlying C*-algebra. As an application of these notions, we study presheaves of group duals (DR-presheaves) and prove that the dual object of a DR-presheaf is a group gerbe over $Delta$. It is also shown that any section of a DR-presheaf defines a twisted action of $pi_1(M)$ on a Cuntz algebra.
For every $infty$-category $mathscr{C}$, there is a homotopy $n$-category $mathrm{h}_n mathscr{C}$ and a canonical functor $gamma_n colon mathscr{C} to mathrm{h}_n mathscr{C}$. We study these higher homotopy categories, especially in connection with the existence and preservation of (co)limits, by introducing a higher categorical notion of weak colimit. Based on the idea of the homotopy $n$-category, we introduce the notion of an $n$-derivator and study the main examples arising from $infty$-categories. Following the work of Maltsiniotis and Garkusha, we define $K$-theory for $infty$-derivators and prove that the canonical comparison map from the Waldhausen $K$-theory of $mathscr{C}$ to the $K$-theory of the associated $n$-derivator $mathbb{D}_{mathscr{C}}^{(n)}$ is $(n+1)$-connected. We also prove that this comparison map identifies derivator $K$-theory of $infty$-derivators in terms of a universal property. Moreover, using the canonical structure of higher weak pushouts in the homotopy $n$-category, we define also a $K$-theory space $K(mathrm{h}_n mathscr{C}, mathrm{can})$ associated to $mathrm{h}_n mathscr{C}$. We prove that the canonical comparison map from the Waldhausen $K$-theory of $mathscr{C}$ to $K(mathrm{h}_n mathscr{C}, mathrm{can})$ is $n$-connected.
We construct a state-sum type invariant of smooth closed oriented $4$-manifolds out of a $G$-crossed braided spherical fusion category ($G$-BSFC) for $G$ a finite group. The construction can be extended to obtain a $(3+1)$-dimensional topological quantum field theory (TQFT). The invariant of $4$-manifolds generalizes several known invariants in literature such as the Crane-Yetter invariant from a ribbon fusion category and Yetters invariant from homotopy $2$-types. If the $G$-BSFC is concentrated only at the sector indexed by the trivial group element, a cohomology class in $H^4(G,U(1))$ can be introduced to produce a different invariant, which reduces to the twisted Dijkgraaf-Witten theory in a special case. Although not proven, it is believed that our invariants are strictly different from other known invariants. It remains to be seen if the invariants are sensitive to smooth structures. It is expected that the most general input to the state-sum type construction of $(3+1)$-TQFTs is a spherical fusion $2$-category. We show that a $G$-BSFC corresponds to a monoidal $2$-category with certain extra structure, but that structure does not satisfy all the axioms of a spherical fusion $2$-category given by M. Mackaay. Thus the question of what axioms properly define a spherical fusion $2$-category is open.
This paper provides a conceptual study of the twisting procedure, which amounts to create functorially new differential graded Lie algebras, associative algebras or operads (as well as their homoto
This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation--a weak functor--from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail.
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