1-groupoid

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

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.

homotopy level | n-truncation | homotopy theory | higher category theory | higher topos theory | homotopy type theory |
---|---|---|---|---|---|

h-level 0 | (-2)-truncated | contractible space | (-2)-groupoid | true/unit type/contractible type | |

h-level 1 | (-1)-truncated | contractible-if-inhabited | (-1)-groupoid/truth value | (0,1)-sheaf/ideal | mere proposition/h-proposition |

h-level 2 | 0-truncated | homotopy 0-type | 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 | (3,1)-sheaf/2-stack | h-2-groupoid |

h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | (4,1)-sheaf/3-stack | h-3-groupoid |

h-level $n+2$ | $n$-truncated | homotopy n-type | n-groupoid | (n+1,1)-sheaf/n-stack | h-$n$-groupoid |

h-level $\infty$ | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h-$\infty$-groupoid |