homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
n-category = (n,n)-category
n-groupoid = (n,0)-category
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.
At the same time, the notion of a Kan complex is an abstraction of the structure found in the nerve of a groupoid, the Duskin nerve of a 2-groupoid and generally (almost by definition, see at cobordism hypothesis) the nerves of n-groupoids for all $n \leq \infty$. In other words, Kan complexes constitute a geometric model for ∞-groupoids/homotopy types which is based on the shape given by the simplex category (as opposed to, say, the globe category, as for what are usually called ω-groupoids). Thus Kan complexes serve to support homotopy theory.
More in detail, a Kan complex is a collection of $k$-simplex-shaped k-morphisms for all $k \in \mathbb{N}$, such that for all composable $k$-morphisms a composite does exist (not necessarily uniquely) and such that all $k$-morphisms are invertible under this composition.
Specifically for the nerve $N(\mathcal{G}_\bullet)$ of a groupoid $\mathcal{G}_\bullet$, a $k$-cell is given by a sequence of morphisms of the form $\{0\to 1\to \ldots \to k\}$, thought of as a $k$-simplex by taking its $(k-1)$-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.
Hence generally, in a Kan complex a $k$-face of an $(k+1)$-simplex may be thought of as the composition of the remaining faces, all regarded as k-morphisms. But unless the Kan complex is the nerve of a groupoid (a 1-groupoid), there is in general not a unique such composite. Indeed, choosing one of the fillers of each horn in a Kan complex to be the composite means passing from Kan complexes to an algebraic model for ∞-groupoids, algebraic Kan complexes.
Among all ∞-groupoids the strict ∞-groupoids correspond to crossed complexes and various other related algebraic models, all or most of of which have a faithful embedding into Kan complexes under a suitable nerve operation. One of these are the simplicial T-complexes, the nerves crossed complexes. There all horns have unique ‘thin’ fillers, so these are Kan complexes corresponding to a strict form of higher dimensional groupoid.
A Kan complex is a simplicial set $S$ that satisfies the Kan condition,
which says that all horns of the simplicial set have fillers/extend to simplices;
which means equivalently that the unique homomorphism $S \to pt$ from $S$ to the point (the terminal simplicial set) is a Kan fibration;
which means equivalently that for all diagrams of the form
there exists a diagonal morphism
completing this to a commuting diagram;
which in turn means equivalently that the map from $n$-simplices to $(n,i)$-horns is an epimorphism
The last characterization in def. 1 is sometimes taken to induce the generalization to internal Kan complexes in ambient geometric contexts. For instance the generalization of Lie groupoids to “Lie Kan complexes” might be defined to be given by simplicial objects in the category SmoothMfd of smooth manifolds such that the morphisms
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 $k \gt n+1$ every $k$-boundary $\partial \Delta$ has a unique filler is called $(n+1)$-coskeletal;
a Kan complex that is $(n+1)$-coskeletal such that in addition the $(n+1)$-horns and $(n+2)$-horns have a unique filler are also called n-hypergroupoids.
These $(n+1)$-coskeletal Kan complexes are models for n-groupoids/homotopy n-types/n-truncated ∞-groupoids.
Here we discuss aspects of how Kan complexes serve as a model (a “geometric model”) for groupoids, 2-groupoids, … n-groupoids and generally ∞-groupoids or homotopy types.
First we survey the general idea in
then we recall 1-groupoids as Kan complexes via their nerves in
Then we discuss basic aspects of the
This is a special case of how weak Kan complexes (quasi-categories) are a model for (∞,1)-categories.
The horn filling condition from this point of view is read as guaranteeing that:
for all collection of $(k-1)$ composable k-morphisms (given by a horn $\Lambda^k[n]$) there exists a k-morphism – their composite – and an $(n+1)$-morphism connecting the original $(n-1)$ $n$-cells with their composite. Depending on $k$, 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.
Of all the models for ∞-groupoids known in the literature, Kan complexes are probably the most widely used, certainly in homotopy theory and related “geometric” approaches to higher category (such as in terms of n-fold complete Segal spaces etc.), less so in “algebraic” approaches to higher category theory. To a large extent this is because the category of Kan complexes – in particular when thought of as the full sub-category of fibrant objects inside the standard model structure on simplicial sets – lends itself usefully to many computations; to some extent it is maybe a historical coincidence that specifically for this model the theory was worked out in such detail. Maybe if Kan – who first tried cubical sets and then rejected them in favor of simplicial sets due to some technical issues – had tried cubical set with connection first, things would have developed differently. See at cubical set for discussion of this issue.
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.
An ∞-groupoid is first of all supposed to be a structure that consists of k-morphisms for all $k \in \mathbb{N}$, which for $k \geq 1$ go between $(k-1)$-morphisms.
In the context of Kan complexes, the tool for organizing such collections of k-morphisms is the notion of a simplicial set, which models $k$-morphisms as being of the shape of $k$-simplices – a vertex for $k = 0$, an edge for $k = 1$, a triangle for $k = 2$, a tetrahedron for $k = 3$, and so on.
This means that a simplicial set $K_\bullet$ is a sequence of sets $\{K_n\}_{n \in \mathbb{N}}$ (sets of $k$-simplex shaped $k$-morphisms for all $k$) equipped with functions $d_i \colon K_{k+1} \to K_{k}$ that send a $(k+1)$-simplex to its $i$-th face, and functions $s_i \colon K_k \to K_{k+1}$ that over a $k$-simplex “erects a flat $(k+1)$-simplex” in all possible ways (hence which inserts “identities” called “degeneracies” in this context).
If we write $\Delta$ 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
Hence we think of this as assigning
a set $[0] \mapsto K_0$ of objects;
a set $[1] \mapsto K_1$ of morphism;
a set $[2] \mapsto K_2$ of 2-morphism;
a set $[3] \mapsto K_3$ of 3-morphism;
and generally
as well as specifying
functions $([n] \hookrightarrow [n+1]) \mapsto (K_{n+1} \to K_n)$ that send $n+1$-morphisms to their boundary $n$-morphisms;
functions $([n+1] \to [n]) \mapsto (K_{n} \to K_{n+1})$ that send $n$-morphisms to identity $(n+1)$-morphisms on them.
The fact that $K$ 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 $k$-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 $\Lambda^1[2]$ the simplicial set consisting of two attached 1-cells
and for $(f,g) : \Lambda^1[2] \to K$ an image of this situation in $K$, hence a pair $x_0 \stackrel{f}{\to} x_1 \stackrel{g}{\to} x_2$ of two composable 1-morphisms in $K$, we want to demand that there exists a third 1-morphisms in $K$ that may be thought of as the composition $x_0 \stackrel{h}{\to} x_2$ of $f$ and $g$. 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 $\Lambda^1[2] \hookrightarrow \Delta[2]$ 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 $(f,g)$ a corresponding such $h$ 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 $K$, then if $g$ had an inverse $g^{-1}$ we could use the above composition operation to compose that with $h$ and thereby find a morphism $f$ connecting the sources of $h$ and $g$. 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 $K$.
In order for this to qualify as an $\infty$-groupoid, this composition operation needs to satisfy an associativity law up to coherent 2-morphisms, which means that we can find the relevant tetrahedrons in $K$. 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.
In order to conceive of the $k$-simplices for higher $k$ as “globular k-morphism” going from a source to a target one needs a bit of combinatorics. This provided by the orientals (due to Ross Street).
The $k$-oriental $O(k)$ is precisely the prescription for how exactly to think of a $k$-simplex as being a k-morphism in an omega-category. The first few look like this:
In fact, the omega-nerve $N(K)$ of an omega-category $K$ is the simplicial set whose collection of $k$-cells $N(K)_k := Hom(O(k),K)$ is precisely the collection of images of the $k$th oriental $O(k)$ in $K$.
This is fully formally the prescription of how to think of a Kan complex as an $\infty$-groupoid: the Kan complex $C$ is the omega-nerve of an omega-category in which all morphism are invertible:
the $k$-cells in $C_k$ are precisely the collection of $k$-morphisms in the omega-category of shape the $k$th oriental $O(k)$;
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 $k$-morphisms have an inverse.
This is easy to see in low dimensions:
a 1-cell $\phi \in C_1$ in the simplicial set $C$ has a single source 0-cell $x := d_1 \phi$ and a single target 0-cell $y := d_0 \phi$ and hence may be pictured as a morphism
a 2-cell $\phi \in C_2$ in the simplicial set $C$ has two incoming 1-cells $d_2 \phi, d_0 \phi \in C_1$ and one outgoing 1-cell $d_1 \phi \in C_1$, 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 $\phi$ here as a globular 2-morphism
More in detail, one may think of the incoming two adjacent $1$-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 $\phi$ here as being a compositor in a bicategory that shows how the composable pair is composed to the morphism $d_1 \phi$.
So if an $\infty$-groupoid is thought of as a globular ω-category in which all k-morphisms are invertible, then the corresponding Kan complex is the nerve or rather the ω-nerve of this ω-category.
Notably if $C$ 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 $k _ x_0 \to x_2$ 2-isomorphic to $h$, there may be many 2-morphisms $g \circ f \Rightarrow k$, 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 $\omega$-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 $C$, 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 $f$ composed with the morphism $h$, 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 all k-morphisms for all $k$ are composable if adjacent and have a weak inverse.
We review how 1-groupoids are incarnated as Kan complexes via their nerve.
A (small) groupoid $\mathcal{G}_\bullet$ is
a pair of sets $\mathcal{G}_0 \in Set$ (the set of objects) and $\mathcal{G}_1 \in Set$ (the set of morphisms)
equipped with functions
where the fiber product on the left is that over $\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1$,
such that
$i$ takes values in endomorphisms;
$\circ$ defines a partial composition operation which is associative and unital for $i(\mathcal{G}_0)$ the identities; in particular
$s (g \circ f) = s(f)$ and $t (g \circ f) = t(g)$;
every morphism has an inverse under this composition.
For $\mathcal{G}_\bullet$ a groupoid, def. 2, we write
for the set of sequences of composable morphisms of length $n$, for $n \in \mathbb{N}$; schematically:
For each $n \geq 1$, the two maps $d_0$ and $d_n$ that forget the first and the last morphism in such a sequence and the $n-1$ maps $d_k$ that form the composition of the $k$th morphism in the sequence with the next one, constitute $(n+1)$ functions denoted
Moreover, the assignments $s_i$ that insert an identity morphism in position $i$ constitute functions denoted
These collections of maps in def. 3 satisfy the simplicial identities, hence make the nerve $\mathcal{G}_\bullet$ into a simplicial set. Moreover, this simplicial set is a Kan complex, where each horn has a unique filler (extension to a simplex).
(A 2-coskeletal Kan complex.)
The nerve operation constitutes a full and faithful functor
Write
for the category of Kan complexes, which is the full subcategory of that of simplicial sets on the Kan complexes.
This means that for $X_\bullet,Y_\bullet \in KanCplx$ two Kan complexes, an element $f_\bullet \colon X_\bullet \to Y_\bullet$ in the hom-set $Hom_{KanCplx}(X_\bullet,Y_\bullet)$ is
such that
For $X_\bullet,Y_\bullet \in KanCplx$ two Kan complexes, their mapping space
is the simplicial set given by
The construction in def. 5 defines an internal hom of Kan complexes.
As such it is also common to write $Y^X$ for $Maps(X,Y)$, as well as $[X,Y]$. Notice that the latter notation is sometimes used instead for just the set of connected components of $Maps(X,Y)$.
Write
for the Kan complex which is 1-groupoid with two objects and one nontrivial morphism and its inverse between them. This comes with two inclusions
of its endpoints.
Then for $X_\bullet \in KanCplx$ any other Kan complex, the mapping space $[I,X]_\bullet$ from def. 5 is the path space object of $X_\bullet$.
A 1-cell in the mapping Kan complex $[X_\bullet, Y_\bullet]_\bullet$ is a homotopy between two morphisms of Kan complexes:
For $f_\bullet, g_\bullet \colon X_\bullet \to Y_\bullet$ two morphisms between two Kan complexes, hence $f_\bullet,g_\bullet \in Hom_{KanCplx}(X,Y)$, a (right-)homotopy $\eta \colon f \Rightarrow g$ is a morphism $\eta_\bullet \colon X_\bullet \to [I_\bullet,X_\bullet]_\bullet$ into the path space object of def. 1 such that we have a commuting diagram
It follows that the category $KanCplx$ is naturally enriched over itself.
We may write ∞Grpd for $KanCplx$ regarded as a $KanCplx$-enriched category, hence as fibrant sSet-enriched category. We write $X$ (without the subscript) for a Kan complex $X_\bullet$ regarded as an object of $\infty Grpd$.
Kan complexes are the fibrant objects in the model structures on simplicial sets for which fibrations are Kan fibrations.
In this context, a weak equivalence between Kan complexes is a morphism of simplicial sets that induces an isomorphism on the simplicial homotopy groups of the two Kan complexes: a weak homotopy equivalence.
The nerve $N(C)$ of a small category is a Kan complex if and only if $C$ is a groupoid.
The existence of inverse morphisms in $D$ corresponds to the fact that in the Kan complex $N(D)$ the “outer” horns
have fillers
(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.
Accordingly
The nerve $N(C)$ of a strict ω-category is a Kan complex if and only if $C$ is a strict ω-groupoid.
The underlying simplicial set of a simplicial group (see there) is a Kan complex.
In particular, via the Dold-Kan correspondence (see there)
every chain complex (of abelian groups, in non-negative degree) is equivalent to a simplicial abelian group and this has an underlying Kan complex. This way homological algebra becomes a special case of the study of homotopy theory (of Kan complexes).
For $X$ a topological space, its singular simplicial complex is the simplicial set $\Pi(X)$ (often denoted $S(X)$) whose set of $n$-simplices is the hom-set
in Top of continuous maps from the standard topological $n$-simplex $\Delta^n_{Top}$ into $X$.
Using the fact that the $\Delta^n_{Top}$ arrange themselves into a cosimplicial space
in the obvious way, the $(\Pi(X)_n)$ become a simplicial set in the corresponding obvious way. For instance the face maps are induced by restricting maps to $X$ along the face inclusions $\delta^i : \Delta^{n-1} \hookrightarrow \Delta^n$.
That $\Pi(X)$ is indeed a Kan complex is intuitively clear. Technically it follows from the fact that the inclusions ${{\Lambda^n}_{Top}}_k \hookrightarrow \Delta^n_{Top}$ of topological horns into topological simplices are retracts, in that there are continuous maps $\Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k$ given by “squashing” a topological $n$-simplex onto parts of its boundary, such that
Therefore the map $[\Delta^n, \Pi(X)] \to [\Lambda^n_k,\Pi(X)]$ is an epimorphism, since it is equal to to $Top(\Delta^n, X) \to Top(\Lambda^n_k, X)$ which has a right inverse $Top(\Lambda^n_k, X) \to Top(\Delta^n, X)$.
The ∞-groupoid represented by the Kan complex $\Pi(X)$ is called the fundamental ∞-groupoid of $X$.
This example is the universal one: up to weak equivalence of Kan complexes every Kan complex is the fundamental $\infty$-groupoid of a (compactly generated, weakly Hausdorff) topological space.
This is the statement of the homotopy hypothesis (which is a theorem for $\infty$-groupoids modeled as Kan complexes.
Other concepts:
Textbook accounts include
section I.3 of
For Kan complexes as such see also the references at simplicial set and at model structure on simplicial sets.
For Kan complexes as $\infty$-groupoids, see for instance section 1.2.5 of
An early mention of this idea was in
For background on the general relation of simplicial- and globular sets see also the references at oriental.
Discussion of the homotopy theory of smooth ∞-groupoids presented by “Lie-Kan complexes” is in section 4.2 of