nLab
infinity-groupoid

Idea

The notion of $\mathrm{\infty }$-groupoid is the generalization of that of group and groupoids to higher category theory:

an $\mathrm{\infty }$-groupoid – equivalently an (∞,0)-category – is an ∞-category in which all k-morphisms for all $k$ are equivalences.

The collection of all $\mathrm{\infty }$-groupoids forms the (∞,1)-category ∞Grpd.

Special cases of $\mathrm{\infty }$-groupoids include groupoids, 2-groupoids, 3-groupoids, n-groupoids, deloopings of groups, 2-groups, ∞-groups.

Properties

Presentations

There are many ways to present the (∞,1)-category ∞Grpd of all $\mathrm{\infty }$-groupoids, or at least obtain its homotopy category.

A simple and very useful incarnation of $\mathrm{\infty }$-groupoids is available using a geometric definition of higher categories in the form of simplicial sets that are Kan complexes: the $k$-cells of the underlying simplicial set are the k-morphisms of the $\mathrm{\infty }$-groupoid, and the Kan horn-filler conditions encode the fact that adjacent $k$-morphisms have a (non-unique) composite $k$-morphism and that every $k$-morphism is invertible with respect to this composition. See Kan complex for a detailed discussion of how these incarnate $\mathrm{\infty }$-groupoids.

The (∞,1)-category of all $\mathrm{\infty }$-groupoids is presented along these lines by the Quillen model structure on simplicial sets, whose fibrant-cofibrant objects are precisely the Kan complexes:

$$\mathrm{\infty }\mathrm{Grpd}\simeq ({\mathrm{sSet}}_{\mathrm{Quillen}}{)}^{\circ }.$$\infty Grpd \simeq (sSet_{Quillen})^\circ \,.

One may turn this geometric definition into an algebraic definition of ∞-groupoids by choosing horn-fillers . The resulting notion is that of an algebraic Kan complex that has been shown by Thomas Nikolaus to yield an equivalent (∞,1)-category of $\mathrm{\infty }$-groupoids.

There are various model categories which are Quillen equivalent to ${\mathrm{sSet}}_{\mathrm{Quillen}}$. For instance the standard model structure on topological spaces, a model structure on marked simplicial sets and many more. All these therefore present ∞Grpd.

Moreover, the corresponding homotopy category of an (∞,1)-category $\mathrm{Ho}(\mathrm{\infty }\mathrm{Grpd})$, hence a category whose objects are homotopy types of $\mathrm{\infty }$-groupoids, is given by the homotopy category of the category of presheaves over any test category. See there for more details.

Every other algebraic definition of omega-categories is supposed to yield an equivalent notion of $\mathrm{\infty }$-groupoid when restricted to $\omega$-categories all whose k-morphisms are invertible. This is the statement of the homotopy hypothesis, which is for Kan complexes and algebraic Kan complexes a theorem and for other definitions regarded as a consistency condition.

Notably in Pursuing Stacks and the earlier letter to Larry Breen, Alexander Grothendieck initiated the study of $\mathrm{\infty }$-groupoids and the homotopy hypothesis with his original definition of Grothendieck weak infinity-groupoids, that has recently attracted renewed attention.

Strict $\mathrm{\infty }$-groupoids

One may also consider entirely strict $\mathrm{\infty }$-groupoids, usually called $\omega$-groupoids or strict ω-groupoids. These are equivalent to crossed complexes of groups and groupoids.

Relation to $\mathrm{\infty }$-groups

0-connected $\mathrm{\infty }$-groupoids are the delooping $BG$ of ∞-groups (see looping and delooping).

These are presented by simplicial groups. Notably abelian simplicial groups are therefore a model for abelian $\mathrm{\infty }$-groupoids. Under the Dold-Kan correspondence these are equivalent to non-negatively graded chain complexes, which therefore also are a model for abelian $\mathrm{\infty }$-groupoids. This way much of homological algebra is secretly the study of special $\mathrm{\infty }$-groupoids.

Related concepts

• discrete ∞-groupoid, ∞-stack, cohesive ∞-groupoid

• homotopy theory, (∞,1)-category