Higher category theory

Higher categories are categories that allow morphisms between morphisms, and possibly morphisms between morphisms between morphisms, and so on. “Low-dimensional” higher category theory concerns bicategories and tricategories; beyond that, one turns to general notions of \(n\)-category and \(\infty\)-category.

\(n\)-categories

A strict \(n\)-category is an enriched category in \(\mathbf{Set}\) iterated \(n\) times. What it means to be a weak \(n\)-category is the subject of intense research. The best understood of these are bicategories, treated on a separate page.

Surveys

  • Leinster, 2002: A survey of definitions of n-category (pdf, arxiv)
  • Street, 1995: Higher categories, strings, cubes, and simplex equations (doi, pdf)
    • Sec. 1-3: Monoidal categories and structures in them
    • Sec. 4: Penrose string notation, the precursor to string diagrams
    • Sec. 5: 2-categories and their presentation by computads
    • Sec. 6-10: Extensions to 3-categories and beyond
  • Street, 2010: An Australian conspectus of higher categories (doi), final chapter in Towards higher categories
    • A history of the Australian school’s contributions to higher category theory, written by one of its main workers
  • Cheng, 2011: Comparing operadic theories of \(n\)-category (pdf, arxiv)

Books

  • Leinster, 2004: Higher operads, higher categories (doi, arxiv)
  • Cheng & Lauda, 2004: Higher-dimensional categories: An illustrated guidebook (online )
  • Baez & May, 2010: Towards higher categories (doi, nLab )

\(n\)-fold categories

A strict \(n\)-fold category is an internal category in \(\mathbf{Set}\) iterated \(n\) times. For \(n = 0,1,2,3\), \(n\)-fold categories are sets, categories, double categories, and triple categories.

  • Majard, 2011, talk: N-tuple categories (slides)
    • Nice slides on double, triple, and n-fold categories with lots of pictures
  • Grandis & Paré, 2015: Intercategories (pdf, arxiv)
  • Grandis & Paré, 2017: Intercategories: A framework for three-dimensional category theory (doi, arxiv)
  • Grandis, 2019: Higher dimensional categories: From double to multiple categories (doi), Part II: Multiple categories

Higher-dimensional algebra

John Baez and collaborators have written a series of papers on higher-dimensional algebra , the program of categorifying classical mathematical structures like Hilbert spaces, groups, and Lie algebras.

  1. Baez & Dolan, 1995: Higher-dimensional algebra and topological quantum field theory (doi, arxiv)
  2. Baez & Neuchl, 1996: HDA I: Braided monoidal 2-categories (doi, arxiv, pdf)
  3. Baez, 1997: HDA II: 2-Hilbert spaces (doi, arxiv, pdf)
  4. Baez & Dolan, 1998: HDA III: \(n\)-categories and the algebra of opetopes (doi, arxiv, pdf)
  5. Baez & Langford, 2003: HDA IV: 2-tangles (doi, arxiv, pdf)
  6. Baez & Lauda, 2004: HDA V: 2-groups (tac , arxiv, pdf)
  7. Baez & Crans, 2004: HDA VI: Lie 2-algebras (tac , arxiv, pdf)
  8. Baez, Hoffnung, Walker, 2010: HDA VII: Groupoidification (tac , arxiv, pdf)

A rough draft of “HDA VIII: The Hecke bicategory” by Baez and Hoffnung has been circulated online but is not published.