nLab
simplicial groupoid

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

It is probably best to distinguish between the following:

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

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

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 the 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.