nLab geometric definition of higher categories

• geometric definition of higher categories

• algebraic definition of higher categories

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Idea

In a geometric definition of (n,r)-categories composition of higher morphisms is not an operation with a specified outcome but a relation : the $(n,r)$-category is presented much like a directed space and k-morphisms are $k$-dimensional subspaces in there. When some of these $k$-morphisms are suitably adjacent, there is a guarantee that there exists a $k$-morphism that serves as their composite. But there may be several such. Instead of a rule for picking a specific one, subject to associativity constraints, there is a contractible space of choices of possible composites.

From a geometric presentation of an $(n,r)$-category one can typically obtain an algebraic presentation by choosing composites. The contractibility of the space of choices becomes a coherence law satisfied by the collection of choices.

Conversely, one may typically think of the geometric presentation of an $(n,r)$-category as being the nerve of a corresponding algebraic presentation.

Examples

• a geometric model for (∞,0)-categories = ∞-groupoids are Kan complexes;

  An algebraic Kan complex is an algebraic model that is obtained from a Kan complex by making choices for composition, associator etc.

  • a geometric model for an n-groupoid is a $(n+1)$-coskeletal Kan complex.

• a geometric model for (∞,1)-categories are quasi-categories

  An algebraic quasicategory is an algebraic model that is obtained from a quasi-category by making choices for composition, associator etc.

  • a geometric model for a (1,1)-category = category is a 2-coskeletal quasi-category

• a geometric model for an (n,r)-category is an $(n,r)$-Theta space

• a geometric model for an (∞,n)-category is

  • an n-fold complete Segal space

• a geometric model for a general omega-category is supposed to be a weak complicial set

Properties

When the (∞,1)-category of all (n,r)-categories is presented by a model category, then typically geometric models are cofibrant objects while algebraic models are typically fibrant objects.

For instance all in the standard model structure on simplicial sets, or the standard model structure for quasi-categories all objects are cofibrant.

Related concepts

There are also geometric models for operadic structures: dendroidal sets.