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

Higher category theory

higher category theory

Basic concepts

Basic theorems





Universal constructions

Extra properties and structure

1-categorical presentations



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 kk 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.



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 kk-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 kk-morphisms have a (non-unique) composite kk-morphism and that every kk-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:

Grpd(sSet Quillen) . \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 sSet QuillensSet_{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 Ho(Grpd)Ho(\infty Grpd), 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 ω\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 ω\omega-groupoids or strict ∞-groupoids. These are equivalent to crossed complexes of groups and groupoids.

Relation to \infty-groups

Pointed, 0-connected \infty-groupoids are the delooping BG\mathbf{B}G of ∞-groups (see looping and delooping).

These are presented by simplicial groups. Notably, abelian simplicial groups are therefore a model for abelian \infty-groupoids (more precisely, H \mathbb{Z} -modules). 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 (structured) \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)-truncatedcontractible-if-inhabited(-1)-groupoid/​truth value(0,1)-sheaf/​idealmere proposition/​h-proposition
h-level 20-truncatedhomotopy 0-type0-groupoid/​setsheafh-set
h-level 31-truncatedhomotopy 1-type1-groupoid/​groupoid(2,1)-sheaf/​stackh-groupoid
h-level 42-truncatedhomotopy 2-type2-groupoid(3,1)-sheaf/​2-stackh-2-groupoid
h-level 53-truncatedhomotopy 3-type3-groupoid(4,1)-sheaf/​3-stackh-3-groupoid
h-level n+2n+2nn-truncatedhomotopy n-typen-groupoid(n+1,1)-sheaf/​n-stackh-nn-groupoid
h-level \inftyuntruncatedhomotopy type∞-groupoid(∞,1)-sheaf/​∞-stackh-\infty-groupoid


The term ∞-groupoid is sometimes considered to be too unwieldy, and some alternatives have been suggested or used, but none has gained wide acceptance.

Historically, the word “space” is often (ab)used to mean ∞-groupoid, due to the traditional presentation of Gpd\infty Gpd by the model structure on topological spaces. Some have condemned this usage, but others argue that a “homotopy space” is a valid notion of “space”.

The term “homotopy type” is also quite close in meaning to “∞-groupoid”. Historically, it differed in that morphisms of homotopy types were mere homotopy classes of maps of ∞-groupoids, but more recently (especially with the advent of homotopy type theory some have used “homotopy type” synonymously with “\infty-groupoid”.

More radically, Dustin Clausen and Peter Scholze use the term “anima” (plural: anima) as a synonym for \infty-groupoid, in particular in relation to condensed mathematics. Reference: nCafe. In Purity for flat cohomology by Kęstutis Česnavičius and Peter Scholze they write “If the objects of CC are called widgets, then we call those of Ani(C)Ani(C) animated widgets, except that we abbreviate Ani(Set)Ani(Set) to AniAni and the term ‘animated set’ to anima (plural: anima).”


Formulations in homotopy type theory include

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

category: ∞-groupoid

Last revised on August 3, 2020 at 02:56:29. See the history of this page for a list of all contributions to it.