# nLab infinity-groupoid

## Theorems

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

## Models

#### Higher category theory

higher category theory

# Contents

## Idea

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

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

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

Special cases of $\infty$-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 $\infty$-groupoids, or at least obtain its homotopy category.

A simple and very useful incarnation of $\infty$-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 $\infty$-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 $\infty$-groupoids.

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

$\infty \mathrm{Grpd}\simeq \left({\mathrm{sSet}}_{\mathrm{Quillen}}{\right)}^{\circ }\phantom{\rule{thinmathspace}{0ex}}.$\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 $\infty$-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 $\mathrm{Ho}\left(\infty \mathrm{Grpd}\right)$, hence a category whose objects are homotopy types of $\infty$-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 $\infty$-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 $\infty$-groupoids and the homotopy hypothesis with his original definition of Grothendieck weak infinity-groupoids, that has recently attracted renewed attention.

### Strict $\infty$-groupoids

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

### Relation to $\infty$-groups

0-connected $\infty$-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 $\infty$-groupoids. Under the Dold-Kan correspondence these are equivalent to non-negatively graded chain complexes, which therefore also are a model for abelian $\infty$-groupoids. This way much of homological algebra is secretly the study of special $\infty$-groupoids.

homotopy leveln-truncationhomotopy theoryhigher category theoryhigher topos theoryhomotopy type theory
h-level 0(-2)-truncatedcontractible space(-2)-groupoidtrue/unit type/contractible type
h-level 1(-1)-truncated(-1)-groupoid/truth valueh-proposition
h-level 20-truncateddiscrete space0-groupoid/setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/groupoid(2,1)-sheaf/stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoidh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoidh-3-groupoid
h-level $n+2$$n$-truncatedhomotopy n-typen-groupoidh-$n$-groupoid
h-level $\infty$untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-$\infty$-groupoid

## References

Formulations in homotopy type theory include

• Thorsten Altenkirch, Ondrej Rypácek, A Syntactical Approach to Weak $\omega$-Groupoids (pdf)

See also at category object in an (infinity,1)-category for more along these lines.

category: ∞-groupoid

Revised on December 21, 2012 01:13:48 by Urs Schreiber (82.169.65.155)