Fix a meaning of $\infty$-category, however weak or strict you wish. Then a $1$-category is an $\infty$-category such that every 2-morphism is an equivalence and all parallel pairs of j-morphisms are equivalent for $j \geq 2$. Thus, up to equivalence, there is no point in mentioning anything beyond $1$-morphisms, except whether two given parallel $1$-morphisms are equivalent.

If you rephrase equivalence of $1$-morphisms as equality, which gives the same result up to equivalence, then all that is left in this definition is a category. Thus one may also say that a $1$-category is simply a category.

The point of all this is simply to fill in the general concept of $n$-category; nobody thinks of $1$-categories as a concept in their own right except simply as categories.

Related concepts

The notions of $1$-groupoid and $1$-poset are defined on the same basis.

Revised on March 10, 2012 12:53:11
by Urs Schreiber
(89.204.154.201)