Higher category theory

higher category theory

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Homotopy theory



An nn-groupoid is an


In terms of a known notion of (n,r)-category, we can define an nn-groupoid explicitly as an \infty-category such that:

  • every jj-morphism (at any level) is an equivalence;
  • every parallel pair of jj-morphisms is equivalent, for j>nj > n.

Or we define an nn-groupoid abstractly as an n-truncated object in the (∞,1)-category ∞Grpd.


As Kan complexes

A general model for ∞-groupoids are Kan complexes. In this context an nn-groupoid in the general sense is modeled by a Kan complex all whose homotopy groups vanish in degree k>nk \gt n. In this generality one also speaks of a homotopy n-type.

Every such nn-type is equivalent to a “small” model, an (n+1)(n+1)-coskeletal Kan complex: one in which every kk-sphere Δ k+1\partial \Delta^{k+1} for kn+1k \geq n+1 has a unique filler.

Even a bit smaller than this is a Kan complex that is an nn-hypergroupoid, where in addition to these spheres also the horn fillers in degree n+1n+1 are unique.

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

Revised on April 29, 2013 21:44:32 by Urs Schreiber (