nLab
simplicial groupoid

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.)

Remarks

• Simplicially enriched groupoids are related to simplicial sets via an adjunction found independently by Dwyer–Kan and Joyal–Tierney; see Dwyer-Kan loop groupoid. This adjunction gives an equivalence of homotopy categories so that simplicially enriched groupoids model all homotopy types.

  • This is a groupoidal version of the equivalence between quasi-categories and simplicially enriched categories via the homotopy coherent nerve.

• A simplicially enriched groupoid having exactly one object is essentially the same as a simplicial group. Notationally however it is often important to distinguish a simplicial group form the corresponding single object simplicially enriched groupoid.

• Many constructions on simplicial groups, such as that of its Moore complex carry over to simplicialy enriched groupoids without difficulty.

Related concepts

• simplicial category

  • simplicially enriched category

  • simplicial object in Cat

• simplicial groupoid

• relation between quasi-categories and simplicial categories

• model structure on presheaves of simplicial groupoids

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)