## On two-dimensional holonomy

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- by João Faria Martins and Roger Picken PDF
- Trans. Amer. Math. Soc.
**362**(2010), 5657-5695 Request permission

## Abstract:

We define the thin fundamental categorical group ${\mathcal P}_2(M,*)$ of a based smooth manifold $(M,*)$ as the categorical group whose objects are rank-1 homotopy classes of based loops on $M$ and whose morphisms are rank-2 homotopy classes of homotopies between based loops on $M$. Here two maps are rank-$n$ homotopic, when the rank of the differential of the homotopy between them equals $n$. Let $\mathcal {C}(\mathcal {G})$ be a Lie categorical group coming from a Lie crossed module ${\mathcal {G}= (\partial \colon E \to G,\triangleright )}$. We construct categorical holonomies, defined to be smooth morphisms ${\mathcal P}_2(M,*) \to \mathcal {C}(\mathcal {G})$, by using a notion of categorical connections, being a pair $(\omega ,m)$, where $\omega$ is a connection 1-form on $P$, a principal $G$ bundle over $M$, and $m$ is a 2-form on $P$ with values in the Lie algebra of $E$, with the pair $(\omega ,m)$ satisfying suitable conditions. As a further result, we are able to define Wilson spheres in this context.## References

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## Additional Information

**João Faria Martins**- Affiliation: Edifício dos Departamentos de Matemática da FCUP, Centro de Matemática da Universidade do Porto, Rua do Campo Alegre, 687, 4169-007 Porto, Portugal
- Address at time of publication: Departamento de Matemática, Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal
- Email: jmartins@math.ist.utl.pt, jn.martins@fct.unl.pt
**Roger Picken**- Affiliation: Departamento de Matemática, Instituto Superior Técnico (Universidade Técnica de Lisboa), Av. Rovisco Pais, 1049-001 Lisboa, Portugal
- Email: rpicken@math.ist.utl.pt
- Received by editor(s): December 4, 2007
- Received by editor(s) in revised form: April 30, 2008
- Published electronically: June 9, 2010
- Additional Notes: The first author was financed by Fundação para a Ciência e Tecnologia (Portugal), post-doctoral grant number SFRH / BPD / 34138 / 2006. This work was supported by the
*Programa Operacional Ciência e Inovação 2010*, financed by the*Fundação para a Ciência e a Tecnologia*(FCT) and cofinanced by the European Community fund FEDER, in part through the research project Quantum Topology POCI / MAT / 60352 / 2004 - © Copyright 2010
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**362**(2010), 5657-5695 - MSC (2010): Primary 53C29; Secondary 18D05
- DOI: https://doi.org/10.1090/S0002-9947-2010-04857-3
- MathSciNet review: 2661492