category theory

# Contents

## Idea

By 1-groupoid one means – for emphasis – a groupoid regarded as an ∞-groupoid.

A quick way to make this precise is to says that a 1-groupoid is a Kan complex which is equivalent (homotopy equivalent) to the nerve of a groupoid: a 2-coskeletal Kan complex. More abstractly this is a 1-truncated ∞-groupoid.

More generally and more vaguely: Fix a meaning of $\infty$-groupoid, however weak or strict you wish. Then a $1$-groupoid is an $\infty$-groupoid such that 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 groupoid. Thus one may also say that a $1$-groupoid is simply a groupoid.

The point of all this is simply to fill in the general concept of $n$-groupoid; nobody thinks of $1$-groupoids as a concept in their own right except simply as groupoids. Compare $1$-category and $1$-poset, which are defined on the same basis.

h-level 2 | 0-truncated | discrete space | 0-groupoid/set | sheaf | h-set h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | | h-2-groupoid h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | | h-3-groupoid h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | | h-$n$-groupoid | h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid

Revised on April 29, 2013 21:03:23 by Urs Schreiber (89.204.138.79)