nLab higher structure

Idea

The term higher structure

(as used in numerous conference titles starting in the 2010s, e.g. CGX 11, BBM 16, SBBM 18, JSSW 18 and by a journal titled Higher Structures)

has come to mean essentially what elsewhere had been and still is called:

• categorified mathematical structure or mathematical structure in higher category theory,

or, more specifically:

• mathematical structure in (∞,1)-category theory,

hence:

• mathematical structure in homotopy theory (higher algebra/homotopical algebra, higher geometry/derived geometry, etc.),

where the identies/laws of ordinary mathematical structures may hold only up to coherent higher homotopies.

For example, L-∞ algebras, which are Lie algebra objects in the (∞,1)-category of chain complexes, would be a core topic in higher structures in Lie theory (elsewhere: ∞-Lie theory). Further basic classes of higher structures in higher algebra (“brave new algebra”) are $A_\infty$ algebras and $E_\infty$-algebras.

Similarly there are higher structures in topology/geometry, for instance principal $\infty$-bundles generalizing principal bundles etc.

Related entries

• higher homotopy

• higher category theory, higher topos theory, (∞,1)-categorification

• homotopy type theory, univalent foundations for mathematics

• higher geometry

• higher differential geometry

• higher Lie theory

• higher prequantum geometry

• higher complex analytic geometry

• higher algebra, homotopical algebra

• higher linear algebra

• higher category theory and physics

• L-∞ algebras in physics

• homotopical algebraic quantum field theory

References

History:

• Johannes Huebschmann: Origins and breadth of the theory of higher homotopies in: Higher Structures in Geometry and Physics – In Honor of Murray Gerstenhaber and Jim Stasheff, Progress in Mathematics 287, Birkhäuser (2001) 25-38 [arXiv:0710.2645, doi:10.1007/978-0-8176-4735-3]

General exposition:

• Urs Schreiber, Higher Structures in Mathematics and Physics, talk at Oberwolfach Workshop 1651a, (Dec 2016)

Discussion in string theory/M-theory:

• Branislav Jurčo, Christian Saemann, Urs Schreiber, Martin Wolf, Higher Structures in M-Theory, Introduction to Higher Structures in M-Theory 2018, Fortsch. d. Phys. 67 (2019) 8-9 [arXiv:1903.02807, doi:10.1002/prop.201910001]

• Domenico Fiorenza, Hisham Sati, Urs Schreiber, The rational higher structure of M-theory

  in Proceedings of the LMS-EPSRC Durham Symposium: Higher Structures in M-Theory 2018, August 2018, Fortschritte der Physik 67 8-9 (2019) [arXiv:1903.02834, doi:10.1002/prop.201910017]

Conferences and proceedings:

• Alberto Cattaneo, Anthony Giaquinto, Ping Xu: Higher Structures in Geometry and Physics – In Honor of Murray Gerstenhaber and Jim Stasheff, Progress in Mathematics 287, Birkhäuser (2011) [doi:10.1007/978-0-8176-4735-3]

• Michael Batanin, Christian Blohmann, Martin Markl, Program on Higher Structures in Geometry and Physics MPI Bonn (2016)

• Vincent Coll, Anthony Giaquinto, Tom Lada, Ping Xu, Higher Structures 2 (2018)

• Stefan Schwede, Clark Barwick, Julie Bergner, Ieke Moerdijk, Higher structures in homotopy theory, Isaac Newton Institute 2018

• Branislav Jurčo, Christian Saemann, Urs Schreiber, Martin Wolf, Higher Structures in M-Theory, LMS EPSRC Durham Symposium 2018

• Higher Structures and Field Theory ESI, Vienna (2022)

Journals:

• Higher Structures, Journal ISSN 2209-0606