nLab
3-groupoid

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Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The notion of 3-groupoid is the next higher generalization in higher category theory of groupoid and 2-groupoid.

Definition

A 3-groupoid is an ∞-groupoid such that all parallel pairs of k-morphism are equivalent for k≥4: a 3-truncated ∞-groupoid.

Thus, up to equivalence, there is no point in mentioning anything beyond 3-morphisms, except whether two given parallel 3-morphisms are equivalent. This definition may give a concept more general than your preferred definition of 3-groupoid, but it will be equivalent; basically, you may have to rephrase equivalence of 3-morphisms as equality.

See also n-groupoid.

Models

A general 3-groupoid is geometrically modeled by a 4-coskeletal Kan complex. Equivalently – via the homotopy hypothesis-theorem – by a homotopy 3-type.

A small model of this is a 3-hypergroupoid, where all horn-filelrs in dimension ≥4 are unique .

A 3-groupoid is algebraically modeled by a tricategory in which all morphisms are invertible, and by a 3-truncated algebraic Kan complex.

A semistrict algebraic model for 3-groupoids is provided by the notion of Gray-groupoid. These in turn are encoded by 2-crossed modules.

An entirely strict algebraic model for 3-groupoids (which no longer models all homotopy 3-types) is a 3-truncated strict omega-groupoid.

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+2n-truncatedhomotopy n-typen-groupoidh-n-groupoid
h-level ∞untruncatedhomotopy type∞-groupoid(∞,1)-sheaf/∞-stackh-∞-groupoid

References

  • Simona Paoli, Semistrict models of connected 3-types and Tamsamani’s weak 3-groupoids, Journal of Pure and Applied Algebra 211 (2007), 801-820. (arXiv)

  • Carlos Simpson, Homotopy types of strict 3-groupoids (arXiv)

Revised on September 10, 2012 20:26:30 by Urs Schreiber (131.174.188.17)
Edit | Back in time (9 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: groupoid, vertical categorification, higher category theory, omega-category, oriental, globe, globular set, groupoid cardinality, Simpson's conjecture, stack, sheaf, infinity-groupoid, semi-strict infinity-category, infinity-stack, Cat, set, double category, bicategory, geometric definition of higher categories, geometric shape for higher structures, omega-groupoid, crossed complex, Nonabelian Algebraic Topology, (infinity,1)-category, 0-category, truth value, 0-groupoid, strict 2-category, periodic table, (n,r)-category, infinity-category, n-category, Kan complex, n-groupoid, k-tuply monoidal n-category, Gray-category, strict omega-category, Crans-Gray tensor product, strict omega-groupoid, algebraic definition of higher categories, n-fold category, crossed module, regular epimorphism, tricategory, (-1)-groupoid, generalized universal bundle, k-surjective functor, simplicial T-complex, strict n-category, A-infinity-category, 2-group, homotopy hypothesis, pseudofunctor, (infinity,n)-category, regular monomorphism, (infinity,n)-category of cobordisms, Trimble n-category, simplicial object in Cat, simplicial groupoid, string diagram, homotopy 3-type, homotopy n-type, cat-n-group, 2-vector space, 2-crossed module, crossed square, double groupoid, Dold-Kan correspondence, delooping hypothesis, Pursuing Stacks, k-tuply monoidal (n,r)-category, homotopy 2-type, 3-category, homotopy type, connection on a cubical set, n-fibration, (-2)-groupoid, 1-groupoid, 1-category, 2-groupoid, 2-poset, n-poset, 3-groupoid, T-complex, cubical T-complex, point, contractible space, 2009 March changes, (infinity,1)-sheaf, Godement product, Eckmann-Hilton argument, complicial set, weak complicial set, (infinity,1)-category of (infinity,1)-categories, strict 2-functor, microcosm principle, higher category theory and physics, negative thinking, (2,1)-category, (n,1)-category, replete subcategory, tetracategory, omega-nerve, computad, n-group, transfor, k-morphism, n-functor, (infinity,2)-category, n-fold complete Segal space, opetope, weak inverse, exchange law, essential image, supergravity C-field, path n-groupoid, Gray-groupoid, lax functor, simplicial resolution, Batanin omega-category, differential crossed module, higher category theory - contents, n-truncated object of an (infinity,1)-category, cellular set, Theta-space, Theta category, nPOV, companion pair, coherence theorem for monoidal categories, covering, Grothendieck-Maltsiniotis infinity-category, true, (infinity,n)-module, applications of (higher) category theory, 2Cat, Homotopy Theory of Higher Categories, semicategory, (infinity,1)-category theory, homotopy coherent diagram, hypergroupoid, coherence theorem, hypercrossed complex, vertical composition, globular operad, pasting diagram, associator, unitor, algebraic Kan complex, 2-category theory, automorphism 2-group, embedding, effective monomorphism, Duskin nerve, cosmic cube, symmetric monoidal (infinity,n)-category, Lie 3-group, Gray group, hom-groupoid, horizontal composition, infinity-Chern-Weil theory introduction, differential string structure, holographic principle of higher category theory, principal 3-bundle, infinity-anafunctor, 3-morphism, coherence law, associativity, 1-morphism, (1,1)-category, pentagonator, hexagonator, strict 2-groupoid, bigroupoid, nGrpd, (infinity,n)-presheaf, blob n-category, (infinity,n)Cat, integral transforms on sheaves, integral transform, (infinity,n)-category of spans, fully dualizable object, (2,1)-sheaf, (infinity,2)-sheaf, natural equivalence, T-Duality and Differential K-Theory, (infinity,n)-category with duals, homotopy type theory, looping, sigma-model -- exposition of higher gauge theories as sigma-models, truncated object, unit type, contents of contents, outer automorphism infinity-group, 3-group, homotopy level, bracket type, (infinity,2)-topos, 0-morphism, symmetric 2-group, h-set, h-proposition, model structure on cellular sets, parity complex, higher Segal space, Cartesian fibration of dendroidal sets, (n,r)-congruence, Segal n-category, homotopy 2-category, (infinity,n)-sheaf, coherence theorem for symmetric monoidal categories, (infinity,n)-category with adjoints, homotopy n-types - table, TwoVect, h-groupoid, coherence theorem for braided monoidal categories, coherence theorem for monoidal bicategories, coherence theorem for braided monoidal bicategories, coherence theorem for symmetric monoidal bicategories, (infinity,n)-topos, braided 2-group, symmetric monoidal 2-category, globular theory, double Lie groupoid, Lie algebroid-groupoid