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Authors
Jeffrey C. Morton
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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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We develop the rewriting theory for monoidal supercategories and 2-supercategories. This extends the theory of higher-dimensional rewriting established for (linear) 2-categories to the super setting, providing a suite of tools for constructing bases and normal forms for 2-supercategories given by generators and relations. We then employ this newly developed theory to prove the non-degeneracy conjecture for the odd categorification of quantum sl(2) from arXiv:1307.7816 and arXiv:1701.04133. As a corollary, this gives a classification of dg-structures on the odd 2-category conjectured in arXiv:1808.04924.
We use the notion of isomorphism between two invariant vector fields to shed new light on the issue of linearization of an invariant vector field near a relative equilibrium. We argue that the notion is useful in understanding the passage from the space of invariant vector fields in a tube around a group orbit to the space invariant vector fields on a slice to the orbit. The notion comes from Hepworths study of vector fields on stacks.
In work by Ausoni, Dundas and Rognes a half magnetic monopole is discovered and describes an obstruction to creating a determinant K(ku) to ku*. In fact it is an obstruction to creating a determinant gerbe map from K(ku) to K(Z,3). We describe this obstruction precisely using monoidal categories and define the notion of oriented 2-vector bundles, which removes this obstruction so that we can define a determinant gerbe. We also generalize Brylinskis notion of a connective structure to 2-vector bundles, in a way compatible with the determinant gerbe.
We show that the category of vector fields on a geometric stack has the structure of a Lie 2-algebra. This proves a conjecture of R.~Hepworth. The construction uses a Lie groupoid that presents the geometric stack. We show that the category of vector fields on the Lie groupoid is equivalent to the category of vector fields on the stack. The category of vector fields on the Lie groupoid has a Lie 2-algebra structure built from known (ordinary) Lie brackets on multiplicative vector fields of Mackenzie and Xu and the global sections of the Lie algebroid of the Lie groupoid. After giving a precise formulation of Morita invariance of the construction, we verify that the Lie 2-algebra structure defined in this way is well-defined on the underlying stack.
Given a higher-rank graph $Lambda$, we investigate the relationship between the cohomology of $Lambda$ and the cohomology of the associated groupoid $G_Lambda$. We define an exact functor between the abelian category of right modules over a higher-rank graph $Lambda$ and the category of $G_Lambda$-sheaves, where $G_Lambda$ is the path groupoid of $Lambda$. We use this functor to construct compatible homomorphisms from both the cohomology of $Lambda$ with coefficients in a right $Lambda$-module, and the continuous cocycle cohomology of $G_Lambda$ with values in the corresponding $G_Lambda$-sheaf, into the sheaf cohomology of $G_Lambda$.
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