## Abstract

We study properties of morphisms of stacks in the context of the homotopy theory of presheaves of groupoids on a small site C. There is a natural method for extending a property P of morphisms of sheaves on C to a property P of morphisms of presheaves of groupoids. We prove that the property P is homotopy invariant in the local model structure on P(C, grpd)L when P is stable under pullback and local on the target. Using the homotopy invariance of the properties of being a representable morphism, representable in algebraic spaces, and of being a cover, we obtain homotopy theoretic characterizations of algebraic and Artin stacks as those which are equivalent to simplicial objects in C satisfying certain analogues of the Kan conditions. The definition of Artin stack can naturally be placed within a hierarchy which roughly measures how far a stack is from being representable. We call the higher analogues of Artin stacks n-algebraic stacks, and provide a characterization of these in terms of simplicial objects. A consequence of this characterization is that, for presheaves of groupoids, n-algebraic is the same as 3-algebraic for all n ≥ 3. As an application of these results we show that a stack is n-algebraic if and only if the homotopy orbits of a group action on it is.

Original language | English |
---|---|

Pages (from-to) | 467-494 |

Number of pages | 28 |

Journal | Mathematische Zeitschrift |

Volume | 269 |

Issue number | 1-2 |

DOIs | |

State | Published - 1 Oct 2011 |

Externally published | Yes |

## Keywords

- 55U10
- Primary 14A20
- Secondary 18G55

## ASJC Scopus subject areas

- Mathematics (all)