Related papers: Duals of Higher Vector Spaces

We study the notion of duality in the context of graded manifolds. For graded bundles, somehow like in the case of Gelfand representation and the duality: points vs. functions, we obtain natural dual objects which belongs to a different…

Double vector bundles may be dualized in two distinct ways and these duals are themselves dual. These two dualizations generate a group, denoted $\mathscr{D}\mathscr{F}_2$, which is the symmetric group $S_3$ on three symbols. In the case of…

We compare closed and rigid monoidal categories. Closedness is defined by the tensor product having a right adjoint: the internal hom functor. Rigidity, on the other hand, generalises the duality of finite-dimensional vector spaces. In the…

We introduce higher analogs for cleavages in the context of (Kan) simplicial fibrations. We apply them to obtain geometric models for representations up to homotopy of (higher) Lie groupoids. Concretely, we set an equivalence between…

We show that there is an equivalence in any $n$-topos $\mathcal{X}$ between the pointed and $k$-connective objects of $\mathcal{X}$ and the $\mathbb{E}_k$-group objects of the $(n-k-1)$-truncation of $\mathcal{X}$. This recovers, up to…

For a finite dimensional algebra $A$, we prove that the bounded homotopy category of projective $A$-modules and the bounded derived category of $A$-modules are dual to each other via certain categories of locally-finite cohomological…

This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a…

In this paper we study various convolution-type algebras associated with a locally compact quantum group from cohomological and geometrical points of view. The quantum group duality endows the space of trace class operators over a locally…

The (dual) Dold-Kan correspondence says that there is an equivalence of categories $K:\cha\to \Ab^\Delta$ between nonnegatively graded cochain complexes and cosimplicial abelian groups, which is inverse to the normalization functor. We show…

A natural explicit condition is given ensuring that an action of the multiplicative monoid of non-negative reals on a manifold F comes from homotheties of a vector bundle structure on F, or, equivalently, from an Euler vector field. This is…

We associate to a 2-vector bundle over an essentially finite groupoid a 2-vector space of parallel sections, or, in representation theoretic terms, of higher invariants, which can be described as homotopy fixed points. Our main result is…

In this paper we take some classical ideas from commutative algebra, mostly ideas involving duality, and apply them in algebraic topology. To accomplish this we interpret properties of ordinary commutative rings in such a way that they can…

We study monoidal categories that enjoy a certain weakening of the rigidity property, namely, the existence of a dualizing object in the sense of Grothendieck and Verdier. We call them Grothendieck-Verdier categories. Notable examples…

This paper describes an equivalence of the canonical category of $\mathbb N$-manifolds of degree $2$ with a category of involutive double vector bundles. More precisely, we show how involutive double vector bundles are in duality with…

We solve the differentiation problem for Lie $\infty$-groups. Our approach builds on a classical version of Cartier duality which canonically identifies the Hopf algebra of point distributions supported at the identity of a Lie group with…

We consider discontinuous operations of a group $G$ on a contractible $n$-dimensional manifold $X$. Let $E$ be a finite dimensional representation of $G$ over a field $k$ of characteristics 0. Let $\mathcal{E}$ be the sheaf on the quotient…

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy types of their classifying spaces. Double categories (Ehresmann, 1963) have well-understood geometric realizations, and…

We extend the classical Stone duality between zero dimensional compact Hausdorff spaces and Boolean algebras. Specifically, we simultaneously remove the zero dimensionality restriction and extend to \'etale groupoids, obtaining a duality…

Gelfand duality between unital commutative C*-algebras and Compact Hausdorff spaces is extended to all unital C*-algebras, where the dual objects are what we call compact Hausdorff quantum spaces. We apply this result to obtain, a…

Starting from a generalization of a recent result on self-duality we systematically analyze self-dual models. We find a criterion to judge whether a given model is self-dual or not. With this tool we construct some new self-dual pairs,…