n-category = (n,n)-category
n-groupoid = (n,0)-category
It builds on the classical geometric definition of -groupoids in terms of Kan complexes. A Kan complex is like an algebraic -groupoid in which we have forgotten what precisely the composition operation and what the inverses are, and only know that they do exist. This becomes an algebraic model for -groupoids by adding the specific choices of composites back in.
A morphism of algebraic Kan complexes is a morphism of the underlying Kan complexes that sends chosen fillers to chosen fillers.
This defines the category of algebraic Kan complexes.
A slight variant of this definition is that of a simplicial T-complex.
The category is the category of algebras over a monad
This means that algebraic Kan complexes are formally an algebraic model for higher categories.
See model structure on algebraic fibrant objects for details.
The homotopy hypothesis is true for algebraic Kan complexes:
Moreover, there is a direct Quillen equivalence
to the standard model structure on topological spaces, where the left adjoint is a quotient of the geometric realization of the underlying Kan complexes and is a version of the fundamental ∞-groupoid-functor with values in algebraic Kan complexes.
See homotopy hypothesis – for algebraic Kan complexes for details.
In the absence of AC, one might argue that algebraic Kan complexes are a better model of -groupoids than non-algebraic ones. For instance, an algebraic Kan complex always has the right lifting property with respect to all anodyne morphisms, whereas for a non-algebraic Kan complex this fact requires choice.