David Roberts
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This is the private area of David Roberts within the nLab.

I am a visiting research fellow at the University of Adelaide in South Australia, and I work at the National Centre for Vocational Education Research.

Interests

• 2-bundles
• Internal category theory (mostly topological groupoids) and anafunctors
• (2-)Covering spaces
• Homotopy theory, though not of the ${\pi }_{147}(S^{123})$ variety!

Papers

• The inner automorphism 3-group of a strict 2-group, joint with Urs Schreiber, Journal of Homotopy and Related Structures, vol. 3(1), 2008, pp.193–245 (available at the arXiv:07081741)
• Yang-Mills theory for bundle gerbes, joint with Varghese Mathai, Journal of Physics A 39:6039-6044, 2006 (available at the arXiv:0509.3037

Thesis

• This is the final accepted version of my PhD thesis (pdf).

* Abstract: This thesis introduces two main concepts: the fundamental bigroupoid of a topological groupoid and 2-covering spaces, a categorification of covering spaces. The first is applied to the second to show, among other things, that the fundamental 2-group of the 2-covering space is a sub-2-group of the fundamental 2-group of the base. Along the way we derive general results about localisations of the 2-categories of categories and groupoids internal to a site? at classes of weak equivalences, construct a topological fundamental bigroupoid of locally well-behaved spaces, and finish by providing a rich source of examples of 2-covering spaces, including a functorial? 2-connected 2-covering space.

• Internal categories and anafunctors This is the first chapter from my thesis, pretty much as it has been submitted. There were some comments here on an earlier version. It is of independent interest for anyone involved in stacks, even though they aren’t mentioned explicitly.

Talks

• 2-covering spaces - slides from a talk at the 2009 Annual Meeting of the Australian Mathematical Society, at the University of South Australia. 30 September 2009

Notes

• Theorem A for topological categories - this is a version of Quillen’s Theorem A for categories in $\mathrm{Top}$. As a corollary, with a condition on the unit map of the codomain we get that geometric realisation of an $𝒩$-equivalence is a homotopy equivalence.

• There is a monad on $\mathrm{sGrp}(C)$ of simplicial group objects in a category $C$ which sends a simplicial group $K$ to a model of the total space of the universal bundle $\mathrm{EK}\to \overline{W}K$. See groupal model for universal principal infinity-bundles

Ideas

• The (n+1)-ary factorisation system on nCat
• The 2-connected cover of a 2-crossed complex