nLab strict (2,1)-category

Idea

A strict (2,1)-category is a 2-category which is both a strict 2-category as well as a (2,1)-category, hence which is a Grpd-enriched category, referred to as a g.e. category in original articles (Fantham & Moore 1983).

Examples

Example

(homotopy 2-category of topological spaces)
There is a strict $(2,1)$-category whose underlying 1-category is that of TopologicalSpaces and whose 2-morphisms are the higher homotopy-classes of homotopies between continuous functions.

If one restricts to topological spaces which admit the structure of CW-complexes, then this is equivalently the homotopy 2-category of the $(\infty,1)$-category of the simplicial localization of TopologicalSpaces at the weak homotopy equivalences (equivalently that of $\infty$-groupoids), hence is the (2,1)-category-enhancement of the classical homotopy category.

Since higher homotopy-classes of homotopies were also known as “tracks”, some authors (following Baues 1991, p. 300) say “track category”, not just for this example, but for $Grpd$-enriched categories generally.

References

Original discussion of $Grpd$-enriched categories motivated from the homotopy 2-category of topological spaces:

• Peter H. H. Fantham, Eric J. Moore, Groupoid Enriched Categories and Homotopy Theory, Canadian Journal of Mathematics 35 3 (1983) 385-416 (doi:10.4153/CJM-1983-022-8)

In this context, some authors use the term “track category” for “$Grpd$-enriched category” (referring to their 2-morphisms, since “track” is a term for relative higher homotopy-classes of homotopies), following:

• Chapter VI of: Hans-Joachim Baues, Combinatorial Homotopy and 4-Dimensional Complexes, Expositions in Mathematics, De Gruyter 1991 (doi:10.1515/9783110854480)

see also

• Klaus Heiner Kamps, Tim Porter, Sec. IV.1 of: Abstract Homotopy and Simple Homotopy Theory, World Scientific 1997 (doi:10.1142/2215, GoogleBooks)

For example, Toda brackets have a neat desciption in the homotopy 2-category of topological spaces regarded (Ex. ) as a strict (2,1)-category (hence: “track category”):

• Keith Hardie, Klaus Heiner Kamps, Rudger Kieboom, Higher homotopy groupoids and Toda brackets, Homology Homotopy Appl. Volume 1, Number 1 (1999), 117-134 (euclid:hha/1139840198)

• Keith Hardie, Howard Marcum, Nobuyuki Oda, Bracket operations in the homotopy theory of a 2-category, Rend. Ist. Mat. Univ. Trieste 33, 19–70 (2001) (rendiconti:33/02)

• Keith Hardie, Klaus Heiner Kamps, Howard Marcum, The Toda bracket in the homotopy category of a track bicategory, Journal of Pure and Applied Algebra Volume 175, Issues 1–3, 8 November 2002, Pages 109-133 (doi:10.1016/S0022-4049(02)00131-7)

• Howard Marcum, Nobuyuki Oda, Long Box Bracket Operations in Homotopy Theory, Appl Categor Struct 19, 137–173 (doi:10.1007/s10485-009-9186-3)

Often strict (2,1)-categories are discussed in the broader context of strict 2-categories, i.e. Cat-enriched categories, and not explicitly identified as the special case that they are, e.g. in:

• Susan B. Niefield, Dorette A. Pronk, Internal groupoids and exponentiability, Cahiers de topologie et géométrie différentielle catégoriques, LX 4 (2019) (pdf)