nLab strict (2,1)-category



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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).



(homotopy 2-category of topological spaces)
There is a strict (2,1)(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 ( , 1 ) (\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 GrpdGrpd-enriched categories generally.


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

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

see also

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”):

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:

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