nLab (n,1)-category

Idea

The special case of an (n,r)-category for $r = 1$.

Definition

An $(n,1)$-category, is an $n$-category $C$ that is locally $(n-1)$-groupoidal; that is, for any objects $x$ and $y$, the $(n-1)$-category $C(x,y)$ is an $(n-1)$-groupoid.

Equivalently it is an $(\infty,1)$-category for which the mapping spaces are all $(n-1)$-truncated.

Segal / Rezk completeness

In the higher category theory literature, there is a distinction between $(\infty,1)$-categories and $(n, 1)$-categories which satisfy a Segal completeness or Rezk completeness condition and those which do not, which leads to a distinction between whether $(\infty,1)$-categories and $(n, 1)$-categories by default are Segal / Rezk complete or not:

1. Those authors who start with $(\infty,1)$-categories and define $(n, 1)$-categories from $n$-truncations of the hom-$\infty$-groupoids tend to assume Segal / Rezk completeness by default, where a $(n, 1)$-category which are not Segal complete / Rezk complete is then called a $(n, 1)$-precategory, or when defined internally, a Segal $(n, 1)$-category.

2. Those authors who build $(n, 1)$-categories from components tend not to assume Segal / Rezk completeness by default, where a $(n, 1)$-category which does satisfy Segal completeness or Rezk completeness is called a univalent $(n, 1)$-category, or when defined internally, a complete Segal $(n, 1)$-category or Rezk $(n, 1)$-category.

Special cases:

• A $(1,1)$-category is the same as a $1$-category, which is an ordinary category.
• A $(2,1)$-category is a locally groupoidal $2$-category.
• An $(\infty,1)$-category can be understood as a quasi-category or in many other ways.

Extra stuff, structure, property

• An $(n,1)$-category with the analogous properties of a topos is an (n,1)-topos.

Examples

The canonical example of an $(n+1,1)$-category is nGrpd.

Related concepts

• 0-category, (0,1)-category

• category

• 2-category

• 3-category

• n-category

• (∞,0)-category

• (n,1)-category

• (∞,1)-category

• (∞,2)-category

• (∞,n)-category

• (n,r)-category

References

In Section 11 of

• Charles Rezk, A cartesian presentation of weak n-categories, arXiv:0901.3602

the author describes a presentation of $(n,1)$-categories by a left Bousfield localization of the model structure presenting complete Segal spaces.