The notion of a Kan complex is an abstraction of the structure found in the singular simplicial complex of a topological space. There the existence of retractions of a geometric simplex to any of its horns means that all horns in the singular complex can be filled.
The notion of a Kan complex is an abstraction of the structure found in the nerve of a groupoid. If we have a groupoid, , then if we take any inner horn in , it is determined by the sequence of arrows corresponding to , but that also determines a simplex which fills the horn. For the zeroth and last face horns, the outer horns, you can also fill, but that takes a bit more thought, so is left to later. The ‘fillers’ are unique in this case, which suggests some generalisations. In a Kan complex there are no specified ‘composites’ as in a groupoid or its nerve. But one may choose out for all composable -morphisms a composite and thus arrive at an algebraic model for -groupoids: algebraic Kan complexes.
Strict -groupoids correspond to crossed complexes and various other related algebraic models. One of these is the simplicial T-complex. Such an object is the nerve of a crossed complex. There all horns have unique ‘thin’ fillers, so these are Kan complexes corresponding to a strict form of higher dimensional groupoid. Because of these facts, the following are not that surprising as ideas:
A Kan complex is a collection of -simplex-shaped k-morphisms for all , such that for all composable -morphisms a composite does exist (not necessarily uniquely) and such that all -morphisms are invertible under this composition.
Definition
A Kan complex is a simplicial set that satisfies the Kan condition,
which says that all horns of the simplicial set have fillers,
which means equivalently that the unique morphism from to the point is a Kan fibration,
In this last form the Kan condition is useful for defining internal Kan complexes: for instance a smooth Kan complex can be defined as a simplicial object in Diff such that the morphisms are surjective submersions.
Kan complexes are among the most convenient and popular models for ∞-groupoids. The horn filling condition from this point of view is read as guaranteeing that
for all collection of composable -cells (a horn ) there exists an -cell – their composite – and an -cell connecting the original -cells with their composite. Depending on , this interpretation in terms of composition implies that one thinks of all cells as being reversible. Therefore this models an ∞-groupoid.
For illustrations of the horn-filler conditions see Kan fibration.
Whatever other definition of ∞-groupoid one considers, it is expected to map to a Kan complex under the nerve.
We expand now a bit on how a Kan complex may naturally be thought of as a globular∞-groupoid: a higher category in which all k-morphisms for all are invertible. A structure mediating between the simplicial- and the globular model for -groupoids is that of orientals.
For this interpretation, one thinks of a -dimensional cell of a Kan complex as a globulark-morphism whose
source is the -morphism given by the pasting diagram (coproduct) of the -cells that are the faces of ;
target is the -morphism given by the pasting diagram of the -cells that are the faces of
The task of analyzing the combinatorics of k-simplices and their boundaries quickly goes beyond what can be handled in a naive fashion. Luckily, this combinatorial problem has been completely solved by Ross Street in his work on orientals.
The -oriental is precisely the prescription for how exactly to think of a -simplex as being a k-morphism in an omega-category. The first few look like this:
In fact, the omega-nerve of an omega-category is the simplicial set whose collection of -cells is precisely the collection of images of the th oriental in .
This is fully formally the prescription of how to think of a Kan complex as an -groupoid: the Kan complex is the omega-nerve of an omega-category in which all morphism are invertible:
the -cells in are precisely the collection of -morphisms ihn the omega-category of shape the th oriental ;
the horn-filler conditions satisfied by these cells is precisely a reflection of the fact that
there exists a notion of composition of adjacent k-morphisms in the omega-category;
under this composition all -morphisms have an inverse.
This is easy to see in low dimensions:
a 1-cell in the simplicial set has a single source 0-cell and a single target 0-cell and hence may be pictured as a morphism
x \stackrel{\phi}{\to} y
\,.
a 2-cell in the simplicial set has two incoming 1-cells and one outgoing 1-cell , and if we think of the two incoming 1-cells as representing the composite of the corresponding 1-morphisms, we may picture te 2-cell here as a globular 2-morphism
More in detail, one may think of the incoming two adjacent -cells here as not being the composite of these two morphism, but just as a composable pair, and should think of the existence of the 2-morphism here as being a compositor in a bicategory that shows how the composable pair is composed to the morphism .
Notably if is to be regarded as (the nerve of) an ordinary groupoid, every composable pair of morphisms has a unique composite, and hence there should be a unique 2-cell
More generally, in a 2-groupoid there may be non-identity 2-morphisms, and hence for any 1-morphism 2-isomorphic to , there may be many 2-morphisms , hence many 2-cells
All we can say for sure is that at least one such 2-cell exists, and that the 2-cells themselves may be composed in some way. This is precisely what the horn-filler conditions in a Kan complex encode.
We have already seen in low dimension how the existence of composites in an -category is reflected in the fact that in a Kan-complex certain 2-simplices exist, and how the non-uniqueness of these 2-simplices reflects the existence of nontrivial 2-morphisms.
To see in a similar fashion that the Kan condition ensures the existence of inverses consider an outer horn in , a diagram of 1-cells of the form
In general given such a diagram in a category, there is no guarantee that the corresponding triangle as above will exist in its nerve. But if the category is a groupoid, then it is guaranteed that the missing 1-face can be chose to be the inverse of composed with the morphism , and there is at least one 2-morphism
A similar analysis for higher dimensional cells shows that the fact that a Kan complex has all horn fillers encodes precisely the fact that it is the omega-nerve of an omega-category in which allk-morphisms for all are composable if adjacent and have a weak inverse.
(even unique fillers, due to the properties of the nerve of an ordinary category).
This is one way to see and motivate that a simplicial set that is a Kan complex but which does not necessarily have unique fillers makes models an ∞-groupoid.
in the obvious way, the become a simplicial set in the corresponding obvious way. For instance the face maps are induced by restricting maps to along the face inclusions .
That is indeed a Kan complex is intuitively clear. Technically it follows from the fact that the inclusions of topological horns into topological simplices are retracts, in that there are continuous maps given by “squashing” a topological -simplex onto parts of its boundary, such that
({{\Lambda^n}_{Top}}_k \to \Delta^n_{Top} \to
{{\Lambda^n}_{Top}}_k)
=
Id
\,.
Therefore the map is an epimorphism, since it is equal to to which has a right inverse .