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
- 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
- Makkai, 2001: On comparing definitions of “weak n-category” (pdf)
- Leinster, 2002: A survey of definitions of n-category (pdf, arxiv)
- 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
\(n\)-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)
- An intercategory is a kind of non-strict triple category
- Slides by Paré on duoidal categories , intercategories , and examples of intercategories
- 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.
- Baez & Dolan, 1995: Higher-dimensional algebra and topological quantum field theory (doi, arxiv)
- Baez & Neuchl, 1996: HDA I: Braided monoidal 2-categories (doi, arxiv, pdf)
- Baez, 1997: HDA II: 2-Hilbert spaces (doi, arxiv, pdf)
- Baez & Dolan, 1998: HDA III: \(n\)-categories and the algebra of opetopes (doi, arxiv, pdf)
- Baez & Langford, 2003: HDA IV: 2-tangles (doi, arxiv, pdf)
- Baez & Lauda, 2004: HDA V: 2-groups (tac , arxiv, pdf)
- Baez & Crans, 2004: HDA VI: Lie 2-algebras (tac , arxiv, pdf)
- Baez, Hoffnung, Walker, 2010: HDA VII: Groupoidification (tac , arxiv, pdf)
A rough draft of “HDA VIII: The Hecke bicategory” by Baez and Hoffnung was circulated online but is not published.