# nLab algebraic definition of higher categories

### Context

#### Higher category theory

higher category theory

# Contents

## 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 form a geometric model by making choices of composites.

## 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.

Revised on September 6, 2013 09:23:53 by Tim Porter (82.123.186.254)