# nLab simplicial groupoid

### Context

#### Higher category theory

higher category theory

# Contents

## 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

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

Revised on October 12, 2014 16:31:29 by Tim Porter (150.214.205.36)