The notion of a Kan complex is an abstraction of the combinatorial structure found in the singular simplicial complex of a topological space. There the existence of retractions of any geometric simplex to any of its horns – simplices missing one face and their interior – means that all horns in the singular complex can be filled with genuine simplices, the Kan filler condition.
More in detail, 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.
Specifically for the nerve of a groupoid , a -cell is given by a sequence of morphisms of the form , thought of as a -simplex by taking its -faces to be the the sequences obtained from this by deleting the first or the last morphism or by composing two consecutive morphisms in the sequence.
While this is useful for some purposes, one should beware that this naive generalization, if taken at face value, may break the homotopy theoretic interpretation of (smooth, say) Kan complexes as models for (smooth, say) ∞-groupoids. A homotopy-good theory of Lie Kan complexes is discussed in (NSS, section 4.2). See at internal ∞-groupoid for more.
Some special cases of def. 1 go by their own terminology:
a Kan complex such that for every every -boundary has a unique filler is called -coskeletal;
The horn filling condition from this point of view is read as guaranteeing that:
for all collection of composable k-morphisms (given by a horn ) there exists a k-morphism – their composite – and an -morphism 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.
For illustrations of the horn-filler conditions see also at Kan fibration.
Whatever other definition of ∞-groupoid one considers, it is expected (and in most cases has been shown) to map to a Kan complex under the nerve. See at homotopy hypothesis for more on this.
But in any case it seems clear that there is no “fundamental” conceptual role to prefer Kan complexes over other models for ∞-groupoids. Instead, in view of modern developments it seems right to regard the abstract concept of homotopy type (not as an equivalence class, but as a representative, though) as fundamental, and everything else to be “just a model” for this, which may or may not be useful for a particular computation. This point of view is formalized by the univalent foundations of mathematics in terms of homotopy type theory. Here the theory of homotopy types is given as an abstract foundational notion and then Kan complexes and related structures are shown to be a model. For more on this see at homotopy type theory.
Heuristics on composition and inverses
An ∞-groupoid is first of all supposed to be a structure that consists of k-morphisms for all , which for go between -morphisms.
This means that a simplicial set is a sequence of sets (sets of -simplex shaped -morphisms for all ) equipped with functions that send a -simplex to its -th face, and functions that over a -simplex “erects a flat -simplex” in all possible ways (hence which inserts “identities” called “degeneracies” in this context).
If we write for the category whose objects are abstract cellular simplices and whose morphisms are all cellular maps between these, then such a simplicial set is equivalently a functor of the form
functions that send -morphisms to their boundary -morphisms;
functions that send -morphisms to identity -morphisms on them.
The fact that is supposed to be a functor enforces that these assignments of sets and functions satisfy conditions that make consistent our interpretation of them as sets of -morphisms and source and target maps between these. These are called the simplicial identities.
But apart from this source-target matching, a generic simplicial set does not yet encode a notion of composition of these morphisms.
For instance for the simplicial set consisting of two attached 1-cells
and for an image of this situation in , hence a pair of two composable 1-morphisms in , we want to demand that there exists a third 1-morphisms in that may be thought of as the composition of and . But since we are working in higher category theory (and not to be evil), we want to identify this composite only up to a 2-morphism equivalence
From the picture it is clear that this is equivalent to demanding that for the obvious inclusion of the two abstract composable 1-morphisms into the 2-simplex we have a diagram of morphisms of simplicial sets
A simplicial set where for all such a corresponding such exists may be thought of as a collection of higher morphisms that is equipped with a notion of composition of adjacent 1-morphisms.
For the purpose of describing groupoidal composition, we now want that this composition operation has all inverses. For that purpose, notice that for
the simplicial set consisting of two 1-morphisms that touch at their end, hence for
two such 1-morphisms in , then if had an inverse we could use the above composition operation to compose that with and thereby find a morphism connecting the sources of and . This being the case is evidently equivalent to the existence of diagrams of morphisms of simplicial sets of the form
Demanding that all such diagrams exist is therefore demanding that we have on 1-morphisms a composition operation with inverses in .
In order for this to qualify as an -groupoid, this composition operation needs to satisfy an associativity law up to coherent2-morphisms, which means that we can find the relevant tetrahedrons in . These in turn need to be connected by pentagonators and ever so on. It is a nontrivial but true and powerful fact, that all these coherence conditions are captured by generalizing the above conditions to all dimensions as in the definition of Kan complexes.
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 in 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
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
that is the unique identity 2-morphism
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.
for the set of sequences of composable morphisms of length , for ; schematically:
For each , the two maps and that forget the first and the last morphism in such a sequence and the maps that form the composition of the th morphism in the sequence with the next one, constitute functions denoted
Moreover, the assignments that insert an identity morphism in position constitute functions denoted
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
Therefore the map is an epimorphism, since it is equal to to which has a right inverse .