homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
The notion of algebraic Kan complex is an algebraic definition of ∞-groupoids.
It builds on the classical geometric definition of $\infty$-groupoids in terms of Kan complexes. A Kan complex is like an algebraic $\infty$-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 $\infty$-groupoids by adding the specific choices of composites back in.
The nontrivial apsect of the definition of algebraic Kan complexes is that they do still present the full (∞,1)-category ∞Grpd. Notably the homotopy hypothesis is true for algebraic Kan complexes.
An algebraic Kan complex is a Kan complex equipped with a choice of horn fillers for all horns.
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 $Alg Kan$ of algebraic Kan complexes.
For more see the section Algebraic fibrant models for higher categories at model structure on algebraic fibrant objects.
A slight variant of this definition is that of a simplicial T-complex.
The category $Alg Kan$ 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:
there is a model category structure on $Alg Kan$ – the model structure on algebraic fibrant objects – and a Quillen equivalence to the standard model structure on simplicial sets.
Moreover, there is a direct Quillen equivalence
to the standard model structure on topological spaces, where the left adjoint $|-|_r$ is a quotient of the geometric realization of the underlying Kan complexes and $\Pi_\infty$ is a version of the fundamental ∞-groupoid-functor with values in algebraic Kan complexes.
See homotopy hypothesis – for algebraic Kan complexes for details.
If we assume the axiom of choice, then any Kan complex can be made into an algebraic Kan complex by making a simultaneous choice of a filler for every horn.
In the absence of AC, one might argue that algebraic Kan complexes are a better model of $\infty$-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.