0-groupoid

A *0-groupoid* or **0-type** a set. This terminology may seem strange at first, but it is very helpful to see sets as the beginning of a sequence of concepts: sets, groupoids, 2-groupoids, 3-groupoids, etc. Doing so reveals patterns such as the periodic table. (It also sheds light on the theory of homotopy groups and n-stuff.)

For example, there should be a $1$-groupoid of $0$-groupoids; this is the underlying groupoid of the category of sets. Then a groupoid enriched over this is a $1$-groupoid (more precisely, a locally small groupoid). Furthermore, an enriched category is a category (or $1$-category), so a $0$-groupoid is the same as a 0-category.

One can continue to define a (−1)-groupoid to be a truth value and a (−2)-groupoid to be a triviality (that is, there is exactly one).

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 | (-1)-groupoid/truth value | (0,1)-sheaf | 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 | h-2-groupoid |

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

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

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

Revised on September 10, 2012 20:24:12
by Urs Schreiber
(131.174.188.17)