nLab model structure on presheaves of simplicial groupoids



Model category theory

model category, model \infty -category



Universal constructions


Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts


(,1)(\infty,1)-Category theory



The model structure on presheaves of (Dwyer-Kan) simplicial groupoids (sSet-enriched groupoids) is one of the models for ∞-stack (∞,1)-toposes. It is a slight variant on the model structure on simplicial presheaves. (At that link more general information is collected).

For various applications it is useful to

An example is the discussion of principal infinity-bundles in section 3 of (Jardine & Luo)



Write (GW¯):Grpd ΔsSet(G \dashv \bar W) \colon Grpd^\Delta \leftrightarrow sSet for the Quillen adjunction discussed at model structure on simplicial groupoids. This directly prolongs to an adjunction on presheaves

(GW¯):[C op,Grpd Δ]W¯G[C op,sSet] inj. (G \dashv \bar W) \colon [C^{op}, Grpd^\Delta] \stackrel{\overset{G}{\leftarrow}}{\underset{\bar W}{\to}} [C^{op}, sSet]_{inj} \,.

The transferred model structure along W¯\bar W on [C op,Grpd Δ][C^{op}, Grpd^\Delta] of the global injective model structure on simplicial presheaves exists on [C op,sSet,Grpd Δ][C^{op}, sSet, Grpd^\Delta]: fibrations and weak equivalences are those that become global injective fibrations and weak equivalences, respectively, under W¯\bar W.

This appears as (LBK, theorem 3.10).


A model structure on sheaves with values in simplicial groupoids is due to

  • Andre Joyal and Myles Tierney, Strong stacks and classifying spaces, in: Category theory

    (Como, 1990), volume 1488 of Lecture Notes in Math., pages 213–236. Springer, Berlin (1991) (web)

  • Andre Joyal and Myles Tierney, On the homotopy theory of sheaves of simplicial groupoids, Math. Proc. Cambridge Philos. Soc. 120 2 (1996) 263-290

A model structure on presheaves with values in simplicial groupoids:

  • Zhi-Ming Luo, Peter Bubenik, Peter Kim, Closed model categories for presheaves of simplicial groupoids and presheaves of 2-groupoids (arXiv)

A model structure on simplicial sheaves of groupoids is discussed in

Last revised on May 31, 2023 at 17:18:15. See the history of this page for a list of all contributions to it.