# nLab strict (2,1)-category

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## 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:

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: