nLab algebraic definition of higher categories

• geometric definition of higher categories

• algebraic definition of higher categories

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Idea

In an algebraic definition of (n,r)-categories composition of higher morphisms is a genuine algebraic operation. As opposed to a geometric model where composites are only guaranteed to exist, in an algebraic model there is prescription for finding these composites.

These choices may be given in an elementary way as binary composition operations as in an ordinary category, or the composition operation may be controled by sophisticated algebraic structures such as operads that, while keeping track of all specified composites, may allow large combinations of possible composition operations.

The composition operation in an algebraic model for an $(n,r)$-category are subject to associativity coherence laws. On the geometric side these reflect the fact that the spaces of possible choices of composites are contractible.

Typically an algebraic model for higher categories admits a nerve operation that turns it into an equivalent geometric model. Conversely, typically one can obtain an algebraic model from a geometric model by making choices of composites.

Examples

• The series of notions

  • category, groupoid

  • bicategory

  • tricategory

  • tetracategory

  are algebraic models for n-categories with $1 \leq n \leq 4$ given in terms of direct explicit operations: no operads or other tools are used but all the possible composition operations are defined elementarily and all the relevant coherence laws are demanded explicitly. This makes these models very concrete and hands-on. But it also has the disadvantage that beginning with tricategories these definitions become quite unwieldy.

• For strict n-categories all subtleties with associators and coherence laws are absent (by definition) and therefore there are straightforward algebraic models for these See strict omega-category, strict omega-groupoid and n-fold category. The drawback is of course that these strict models capture only a very restricted part of higher category theory.

• The Batanin/Leinster approach to higher categories involves algebraic structure imposed all at once (using higher operads) on a globular set. See Batanin omega-category.

• The Trimble/May approach to higher categories involves algebraic structure imposed in stages by a process of iterative enrichment. See also Trimble n-category.

• These models turn out to be closely related to an original idea by Alexandre Grothendieck that was resurrected and formalized by Georges Maltsiniotis: Grothendieck-Maltsiniotis ∞-categories

• Using the tool of model structure on algebraic fibrant objects many geometric models for higher categories may be realized equivalently as algebraic models. This is notably true for ∞-groupoids and (∞,1)-categories. See Algebraic fibrant models for higher categories.

Properties

When an (∞,1)-category of (n,r)-categories is presented by a model category, then algebraic models tend to be fibrant objects (while geometric models tend to be cofibrant objects). For instance in all the folk model structure on algebraic higher categories and groupoids, all objects are fibrant.

In a strict version of the homotopy hypothesis, one may make this ‘algebraicity’ of the model structure (not to be confused with the notion at algebraic model structure) a requirement.