arXiv subjects

David Michael Roberts

Publications and source records attributed to David Michael Roberts.

At least 19 recordsLinked to original sources

arXiv2024

Homotopy types of topological stacks of categories

This note extends Quillen's Theorem A to a large class of categories internal to topological spaces. This allows us to show that under a mild condition a fully faithful and essentially surjective functor between such topological categories induces a homotopy equivalence of classifying spaces. It follows from this that we can associate a 2-functorial homotopy type to a wide class of topological stacks of categories, taking values in the 2-category of spaces, continuous maps and homotopy classes of homotopies of maps. This generalises work of Noohi and Ebert on the homotopy types of topological stacks of groupoids under the restriction to the site with numerable open covers.

math.AT↗

arXiv2023

Classifying bi-invariant 2-forms on infinite-dimensional Lie groups

A bi-invariant differential 2-form on a Lie group G is a highly constrained object, being determined by purely linear data: an Ad-invariant alternating bilinear form on the Lie algebra of G. On a compact connected Lie group these have an known classification, in terms of de Rham cohomology, which is here generalised to arbitrary finite-dimensional Lie groups, at the cost of losing the connection to cohomology. This expanded classification extends further to all Milnor regular infinite-dimensional Lie groups. I give some examples of (structured) diffeomorphism groups to which the result on bi-invariant forms applies. For symplectomorphism and volume-preserving diffeomorphism groups the spaces of bi-invariant 2-forms are finite-dimensional, and related to the de Rham cohomology of the original compact manifold. In the particular case of the infinite-dimensional projective unitary group PU(H) the classification invalidates an assumption made by Mathai and the author about a certain 2-form on this Banach Lie group.

math.DG↗

arXiv2023

Substructural fixed-point theorems and the diagonal argument: theme and variations

This article re-examines Lawvere's abstract, category-theoretic proof of the fixed-point theorem whose contrapositive is a `universal' diagonal argument. The main result is that the necessary axioms for both the fixed-point theorem and the diagonal argument can be stripped back further, to a semantic analogue of a weak substructural logic lacking weakening or exchange.

math.CT↗

arXiv2022

Rigid models for 2-gerbes I: Chern-Simons geometry

Motivated by the problem of constructing explicit geometric string structures, we give a rigid model for bundle 2-gerbes, and define connective structures thereon. This model is designed to make explicit calculations easier in applications to physics. To compare to the existing definition, we give a functorial construction of a bundle 2-gerbe as in the literature from our rigid model, including with connections. As an example we prove that the Chern--Simons bundle 2-gerbe from the literature, with its connective structure, can be rigidified -- it arises, up to isomorphism in the strongest possible sense, from a rigid bundle 2-gerbe with connective structure via this construction. Further, our rigid version of 2-gerbe trivialisation (with connections) gives rise to trivialisations (with connections) of bundle 2-gerbes in the usual sense, and as such can be used to describe geometric string structures.

math.DG↗

arXiv2022

Explicit String bundles

While higher bundles are of clear relevance to higher gauge theory, examples other than abelian bundle gerbes are hard to come across. One would in particular like to see 2-bundles where the structure 2-group is the String 2-group associated to a compact simple simply-connected Lie group. This talk will outline a method to construct many examples over homogeneous spaces. We shall also consider one example in detail, giving explicit formulas for the crossed-module-valued Cech cocycle arising from a local trivialisation.

math.DG↗

arXiv2021

The elementary construction of formal anafunctors

This article gives an elementary and formal 2-categorical construction of a bicategory of right fractions analogous to anafunctors, starting from a 2-category equipped with a family of covering maps that are fully faithful and co-fully faithful.

math.CT↗

arXiv2021

Many finite-dimensional lifting bundle gerbes are torsion

Many bundle gerbes constructed in practice are either infinite-dimensional, or finite-dimensional but built using submersions that are far from being fibre bundles. Murray and Stevenson proved that gerbes on simply-connected manifolds, built from finite-dimensional fibre bundles with connected fibres, always have a torsion $DD$-class. In this note I prove an analogous result for a wide class of gerbes built from principal bundles, relaxing the requirements on the fundamental group of the base and the connected components of the fibre, allowing both to be nontrivial. This has consequences for possible models for basic gerbes, the classification of crossed modules of finite-dimensional Lie groups, the coefficient Lie-2-algebras for higher gauge theory on principal 2-bundles, and finite-dimensional twists of topological $K$-theory.

math.DG↗

arXiv2020

Extending Whitney's extension theorem: nonlinear function spaces

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restriction map from everywhere-defined functions is a submersion of locally convex manifolds and so admits local linear splittings on charts. This is achieved by considering the corresponding restriction map for locally convex spaces of compactly-supported sections of vector bundles, allowing the even more general case where $C$ only has mild restrictions on inward and outward cusps, and proving the existence of an extension operator.

math.DG↗

arXiv2020

Topological sectors for heterotic M5-brane charges under Hypothesis H

Assuming Fiorenza-Sati-Schreiber's Hypothesis H, on the charge quantization of M-theory's $C$-field, the topological sectors of the resulting $String^{c_2}(4)$-valued higher gauge theory on a heterotic M5-brane are classified by homotopy classes of maps from the worldvolume $Σ_{M5}$ to $BString^{c_2}(4)$. This note calculates the sectors in a number of examples of M5-brane topology, including examples considered in the 3d-3d correspondence, the emergence of skyrmions from higher-dimensional instantons and Witten's analysis of the S-duality of 4d Yang-Mills theory.

hep-th↗

arXiv2020

(Re)constructing Code Loops

The Moufang loop named for Richard Parker is a central extension of the extended binary Golay code. It the prototypical example of a general class of nonassociative structures known today as code loops, which have been studied from a number of different algebraic and combinatorial perspectives. This expository article aims to highlight an experimental approach to computing in code loops, by a combination of a small amount of precomputed information and making use of the rich identities that code loops' twisted cocycles satisfy. As a byproduct we demonstrate that one can reconstruct the multiplication in Parker's loop from a mere fragment of its twisted cocycle. We also give relatively large subspaces of the Golay code over which Parker's loop splits as a direct product.

math.CO↗

arXiv2018

Smooth loop stacks of differentiable stacks and gerbes

Résumé. Nous définissons un groupoïde de Fréchet-Lie Map(S^1,X) d'ana-foncteurs du cercle vers un groupoïde de Lie X. Ceci fournit une présentation du Hom-champ Hom(S^1,\cX), où \cX est le champ différentiable associé à X. Nous appliquons cette construction au groupoïde de Lie sous-jacent au `gerbe fibré' d'une variété différentiable M; le résultat est un gerbe fibré au-dessus de l'espace des lacets LM de M. Abstract. We define a Fréchet--Lie groupoid Map(S^1,X) of anafunctors from the circle into a Lie groupoid X. This provides a presentation of the Hom-stack Hom(S^1,\cX), where \cX is the differentiable stack associated to X. We apply this construction to the Lie groupoid underlying a bundle gerbe on a manifold M; the result is a bundle gerbe on the loop space LM of M.

math.CT↗

arXiv2017

The smooth Hom-stack of an orbifold

For a compact manifold M and a differentiable stack \cX presented by a Lie groupoid X, we show the Hom-stack Hom(M,\cX) is presented by a Fréchet-Lie groupoid Map(M,X) and so is an infinite-dimensional differentiable stack. We further show that if \cX is an orbifold, presented by a proper étale Lie groupoid, then Map(M,X) is proper étale and so presents an infinite-dimensional orbifold.

math.DG↗

arXiv2017

Quasi-periodic paths and a string 2-group model from the free loop group

In this paper we address the question of the existence of a model for the string 2-group as a strict Lie-2-group using the free loop group $LSpin$ (or more generally $LG$ for compact simple simply-connected Lie groups $G$). Baez-Crans-Stevenson-Schreiber constructed a model for the string 2-group using a based loop group. This has the deficiency that it does not admit an action of the circle group $S^1$, which is of crucial importance, for instance in the construction of a (hypothetical) $S^1$-equivariant index of (higher) differential operators. The present paper shows that there are in fact obstructions for constructing a strict model for the string 2-group using $LG$. We show that a certain infinite-dimensional manifold of smooth paths admits no Lie group structure, and that there are no nontrivial Lie crossed modules analogous to the BCSS model using the universal central extension of the free loop group. Afterwards, we construct the next best thing, namely a coherent model for the string 2-group using the free loop group, with explicit formulas for all structure. This is in particular important for the expected representation theory of the string group that we discuss briefly in the end.

math.DG↗

arXiv2016

A bigroupoid's topology (or, Topologising the homotopy bigroupoid of a space)

The fundamental bigroupoid of a topological space is one way of capturing its homotopy 2-type. When the space is semilocally 2-connected, one can lift the construction to a bigroupoid internal to the category of topological spaces, as Brown and Danesh-Naruie lifted the fundamental groupoid to a topological groupoid. For locally relatively contractible spaces the resulting topological bigroupoid is locally trivial in a way analogous to the case of the topologised fundamental groupoid.

math.AT↗

arXiv2016

Equivariant bundle gerbes

We develop the theory of simplicial extensions for bundle gerbes and their characteristic classes with a view towards studying descent problems and equivariance for bundle gerbes. Equivariant bundle gerbes are important in the study of orbifold sigma models. We consider in detail two examples: the basic bundle gerbe on a unitary group and a string structure for a principal bundle. We show that the basic bundle gerbe is equivariant for the conjugation action and calculate its characteristic class; we show also that a string structure gives rise to a bundle gerbe which is equivariant for a natural action of the String 2-group.

math.DG↗

arXiv2016

Simplicial principal bundles in parametrized spaces

In this paper we study the classifying theory of principal bundles in the parametrized setting, motivated by recent interest in higher gauge theory. Using simplicial techniques, we construct a product-preserving classifying space functor for groups in the category of spaces over a fixed space B. Additionally, we prove that the fiberwise geometric realization functor sends a large class of simplicial parametrized principal bundles to ordinary parametrized principal bundles. As an application we show that the fiberwise geometric realization of the universal simplicial principal bundle for a simplicial group G in the category of spaces over B gives rise to a parametrized principal bundle with structure group |G|.

math.AT↗

arXiv2015

On certain 2-categories admitting localisation by bicategories of fractions

Pronk's theorem on bicategories of fractions is applied, in almost all cases in the literature, to 2-categories of geometrically presentable stacks on a 1-site. We give an proof that subsumes all previous such results and which is purely 2-categorical in nature, ignoring the nature of the objects involved. The proof holds for 2-categories that are not (2,1)-categories, and we give conditions for local essential smallness.

math.CT↗

arXiv2015

A topological fibrewise fundamental groupoid

It is well-known that for certain local connectivity assumptions the fundamental groupoid of a topological space can be equipped with a topology making it a topological groupoid. In other words, the fundamental groupoid functor can be lifted through the forgetful functor from topological groupoids to groupoids. This article shows that for a map $Y \to X$ with certain relative local connectivity assumptions, the fibrewise fundamental groupoid can also be lifted to a topological groupoid over the space $X$. This allows the construction of a simply-connected covering space in the setting of fibrewise topology, assuming a local analogue of the definition of an ex-space. When applied to maps which are up-to-homotopy locally trivial fibrations the result is a categorified version of a covering space. The fibrewise fundamental groupoid can also be used to define a topological fundamental bigroupoid of a (suitably locally connected) topological space.

math.AT↗