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 ω-groupoids. These are equivalent to crossed complexes of groups and groupoids.

Relation to \infty-groups

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. 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)-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


Formulations in homotopy type theory include

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

category: ∞-groupoid

Revised on June 24, 2017 09:00:10 by Peter Heinig (2003:58:aa24:9a00:cf7:a27a:814:2924)