# nLab simplicial groupoid

Contents

### Context

#### Higher category theory

higher category theory

# Contents

## Idea

Simplicial groupoids pair the concepts of groupoids and simplicial sets. Via the Dwyer-Kan loop groupoid functor (Dwyer-Kan 84) their homotopy theory is equivalent to the classical homotopy theory of simplicial sets/Kan complexes (both being models for infinity-groupoids).

## Definition

It is probably best to distinguish between the following:

• A simplicial groupoid is a simplicial object in Cat (that is, a functor from $\Delta^{op}$ to $Cat$), in which is all the categories involved are groupoids.

• A simplicially enriched groupoid is a groupoid enriched over the category SimpSet of simplicial sets.

(For a discussion of the terminology of simplicial groupoid and simplicial category, see the entry on the second of these.)

Any simplicially enriched groupoid yields a simplicial groupoid in which the face and degeneracy operators are constant on objects and it is often in this latter form that they are met in homotopy theory.

(Of course, what is ‘best’ is not always done in the literature, so the reader is best advised to check the meaning being used when the term is met in an article or text.)

## References

• W. G. Dwyer and D. M. Kan, Homotopy theory and simplicial groupoids, Nederl. Akad. Wetensch. Indag. Math., 46, (1984), 379 – 385,

• Philip Ehlers, Simplicial groupoids as models for homotopy type Master’s thesis (1991) (pdf)

Last revised on February 20, 2017 at 07:11:52. See the history of this page for a list of all contributions to it.