Higher category theory

higher category theory

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1-categorical presentations



A 22-groupoid is


Fix a meaning/model of ∞-groupoid, however weak or strict you wish. Then a 22-groupoid is an \infty-groupoid such that all parallel pairs of jj-morphisms are equivalent for j3j \geq 3. Thus, up to equivalence, there is no point in mentioning anything beyond 22-morphisms, except whether two given parallel 22-morphisms are equivalent. This definition may give a concept more general than your preferred definition of 22-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of 22-morphisms as equality.

Specific models

There are various objects that model the abstract notion of 22-groupoid.


A bigroupoid is a bicategory in which all morphisms are equivalences.

Bigroupoids may equivalently be thought of in terms of their Duskin nerves. These are precisely the Kan complexes that are 2-hypergroupoids.


A 22-hypergroupoid is a model for a 2-groupoid. This is a simplicial set, whose vertices, edges, and 2-simplices we identify with the objects, morphisms and 2-morphisms of the form

y x z \array{ && y \\ & \nearrow &\Downarrow& \searrow \\ x &&\stackrel{}{\to}&& z }

in the 2-groupoid, respectively.

Moreover, the 3-simplices in the simplicial set encode the composition operation: given three composable 2-simplex faces of a tetrahedron (a 3-horn)

y z x wy z w \array{ y &\to& &\to& z \\ \uparrow &\seArrow& &\nearrow& \downarrow \\ \uparrow &\nearrow& &\Downarrow& \downarrow \\ x &\to&&\to& w } \;\;\; \;\;\; \array{ y &\to& &\to& z \\ &\searrow& &\swArrow& \downarrow \\ && &\searrow& \downarrow \\ &&&& w }

the unique composite of them is is a fourth face κ\kappa and a 3-cell compcomp filling the resulting hollow tetrahedron:

y z x wcompy z κ x w. \array{ y &\to& &\to& z \\ \uparrow &\seArrow& &\nearrow& \downarrow \\ \uparrow &\nearrow& &\Downarrow& \downarrow \\ x &\to&&\to& w } \;\;\; \stackrel{comp}{\to} \;\;\; \array{ y &\to& &\to& z \\ \uparrow &\searrow& &\swArrow& \downarrow \\ \uparrow &{}_\kappa\Downarrow& &\searrow& \downarrow \\ x &\to&&\to& w } \,.

The 3-coskeletal-condition says that every boundary of a 4-simplex made up of five such tetrahedra has a unqiue filler. This is the associativity coherence law on the comoposition operation:

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This says that any of the possible ways to use several of the 3-simpleces to compose a bunch of compsable 2-morphisms are actually equal.

Homotopy 2-types

More generally one may consider a Kan complex that are just homotopy equivalent to a 33-coskeletal one as a 22-groupoid – precisely: as representing the same homotopy type, namely a homotopy 2-type.

Strict 22-groupoids

The general notion of 22-groupoid above is also called weak 22-groupoid to distinguish from the special case of strict 2-groupoids. A strict 22-groupoid is a strict 2-category in which all morphisms are strictly invertible. This is equivalently a certain type of Grpd-enriched category.


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

Last revised on February 17, 2014 at 20:19:32. See the history of this page for a list of all contributions to it.