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Kan complex

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Contents

Idea

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 nn \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 kk-simplex-shaped k-morphisms for all kk \in \mathbb{N}, such that for all composable kk-morphisms a composite does exist (not necessarily uniquely) and such that all kk-morphisms are invertible under this composition.

Specifically for the nerve N(𝒢 )N(\mathcal{G}_\bullet) of a groupoid 𝒢 \mathcal{G}_\bullet, a kk-cell is given by a sequence of morphisms of the form {01k}\{0\to 1\to \ldots \to k\}, thought of as a kk-simplex by taking its (k1)(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 kk-face of an (k+1)(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.

Definition

Definition

A Kan complex is a simplicial set SS 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 SptS \to pt from SS to the point (the terminal simplicial set) is a Kan fibration;

  • which means equivalently that for all diagrams of the form

    Λ i[n] S Δ[n] ptΛ i[n] S Δ[n] \array{ \Lambda^i[n] &\to& S \\ \downarrow && \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow && \\ \Delta[n] }

    there exists a diagonal morphism

    Λ i[n] S Δ[n] ptΛ i[n] S Δ[n] \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \\ \Delta[n] }

    completing this to a commuting diagram;

  • which in turn means equivalently that the map from nn-simplices to (n,i)(n,i)-horns is an epimorphism

    [Δ[n],S][Λ i[n],S]. [\Delta[n], S]\, \twoheadrightarrow \,[\Lambda^i[n],S] \,.
Remark

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

[Δ[n],S][Λ i[n],S] [\Delta[n], S] \to [\Lambda^i[n],S]

are surjective submersions.

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.

Remark

Some special cases of def. 1 go by their own terminology:

  • a Kan complex such that for every k>n+1k \gt n+1 every kk-boundary Δ\partial \Delta has a unique filler is called (n+1)(n+1)-coskeletal;

  • a Kan complex that is (n+1)(n+1)-coskeletal such that in addition the (n+1)(n+1)-horns and (n+2)(n+2)-horns have a unique filler are also called n-hypergroupoids.

These (n+1)(n+1)-coskeletal Kan complexes are models for n-groupoids/homotopy n-types/n-truncated ∞-groupoids.

As models for \infty-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

Remark

This is a special case of how weak Kan complexes (quasi-categories) are a model for (∞,1)-categories.

Remark

The horn filling condition from this point of view is read as guaranteeing that:

for all collection of (k1)(k-1) composable k-morphisms (given by a horn Λ k[n]\Lambda^k[n]) there exists a k-morphism – their composite – and an (n+1)(n+1)-morphism connecting the original (n1)(n-1) nn-cells with their composite. Depending on kk, 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.

Remark

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.

Remark

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.

Heuristics on composition and inverses

An ∞-groupoid is first of all supposed to be a structure that consists of k-morphisms for all kk \in \mathbb{N}, which for k1k \geq 1 go between (k1)(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 kk-morphisms as being of the shape of kk-simplices – a vertex for k=0k = 0, an edge for k=1k = 1, a triangle for k=2k = 2, a tetrahedron for k=3k = 3, and so on.

This means that a simplicial set K K_\bullet is a sequence of sets {K n} n\{K_n\}_{n \in \mathbb{N}} (sets of kk-simplex shaped kk-morphisms for all kk) equipped with functions d i:K k+1K kd_i \colon K_{k+1} \to K_{k} that send a (k+1)(k+1)-simplex to its ii-th face, and functions s i:K kK k+1s_i \colon K_k \to K_{k+1} that over a kk-simplex “erects a flat (k+1)(k+1)-simplex” in all possible ways (hence which inserts “identities” called “degeneracies” in this context).

If we write Δ\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

K:Δ opSet K \colon \Delta^{op} \to Set

Hence we think of this as assigning

  • a set [0]K 0[0] \mapsto K_0 of objects;

  • a set [1]K 1[1] \mapsto K_1 of morphism;

  • a set [2]K 2[2] \mapsto K_2 of 2-morphism;

  • a set [3]K 3[3] \mapsto K_3 of 3-morphism;

and generally

as well as specifying

  • functions ([n][n+1])(K n+1K n)([n] \hookrightarrow [n+1]) \mapsto (K_{n+1} \to K_n) that send n+1n+1-morphisms to their boundary nn-morphisms;

  • functions ([n+1][n])(K nK n+1)([n+1] \to [n]) \mapsto (K_{n} \to K_{n+1}) that send nn-morphisms to identity? (n+1)(n+1)-morphisms on them.

The fact that KK 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 kk-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 Λ 1[2]\Lambda^1[2] the simplicial set consisting of two attached 1-cells

Λ 1[2]={ 1 0 2} \Lambda^1[2] = \left\{ \array{ && 1 \\ & \nearrow && \searrow \\ 0 &&&& 2 } \right\}

and for (f,g):Λ 1[2]K(f,g) : \Lambda^1[2] \to K an image of this situation in KK, hence a pair x 0fx 1gx 2x_0 \stackrel{f}{\to} x_1 \stackrel{g}{\to} x_2 of two composable 1-morphisms in KK, we want to demand that there exists a third 1-morphisms in KK that may be thought of as the composition x 0hx 2x_0 \stackrel{h}{\to} x_2 of ff and gg. 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

x 1 f g x 0 h x 2. \array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{\mathrlap{\simeq}}& \searrow^{\mathrlap{g}} \\ x_0 &&\stackrel{h}{\to}&& x_2 } \,.

From the picture it is clear that this is equivalent to demanding that for Λ 1[2]Δ[2]\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

Λ 1[2] (f,g) K h Δ[2]. \array{ \Lambda^1[2] &\stackrel{(f,g)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists h}} \\ \Delta[2] } \,.

A simplicial set where for all such (f,g)(f,g) a corresponding such hh 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

Λ 2[2]={ 1 0 2} \Lambda^2[2] = \left\{ \array{ && 1 \\ & && \searrow \\ 0 &&\to&& 2 } \right\}

the simplicial set consisting of two 1-morphisms that touch at their end, hence for

(g,h):Λ 2[2]K (g,h) : \Lambda^2[2] \to K

two such 1-morphisms in KK, then if gg had an inverse g 1g^{-1} we could use the above composition operation to compose that with hh and thereby find a morphism ff connecting the sources of hh and gg. This being the case is evidently equivalent to the existence of diagrams of morphisms of simplicial sets of the form

Λ 2[2] (g,h) K f Δ[2]. \array{ \Lambda^2[2] &\stackrel{(g,h)}{\to}& K \\ \downarrow & \nearrow_{\mathrlap{\exists f}} \\ \Delta[2] } \,.

Demanding that all such diagrams exist is therefore demanding that we have on 1-morphisms a composition operation with inverses in KK.

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 KK. 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 kk-simplices for higher kk 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 kk-oriental O(k)O(k) is precisely the prescription for how exactly to think of a kk-simplex as being a k-morphism in an omega-category. The first few look like this:

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In fact, the omega-nerve N(K)N(K) of an omega-category KK is the simplicial set whose collection of kk-cells N(K) k:=Hom(O(k),K)N(K)_k := Hom(O(k),K) is precisely the collection of images of the kkth oriental O(k)O(k) in KK.

This is fully formally the prescription of how to think of a Kan complex as an \infty-groupoid: the Kan complex CC is the omega-nerve of an omega-category in which all morphism are invertible:

  • the kk-cells in C kC_k are precisely the collection of kk-morphisms in the omega-category of shape the kkth oriental O(k)O(k);

  • the horn-filler conditions satisfied by these cells is precisely a reflection of the fact that

    1. there exists a notion of composition of adjacent k-morphisms in the omega-category;

    2. under this composition all kk-morphisms have an inverse.

This is easy to see in low dimensions:

  • a 1-cell ϕC 1\phi \in C_1 in the simplicial set CC has a single source 0-cell x:=d 1ϕx := d_1 \phi and a single target 0-cell y:=d 0ϕy := d_0 \phi and hence may be pictured as a morphism

    xϕy. x \stackrel{\phi}{\to} y \,.
  • a 2-cell ϕC 2\phi \in C_2 in the simplicial set CC has two incoming 1-cells d 2ϕ,d 0ϕC 1d_2 \phi, d_0 \phi \in C_1 and one outgoing 1-cell d 1ϕC 1d_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 ϕ\phi here as a globular 2-morphism

    x 1 d 2ϕ ϕ d 0ϕ x 0 d 1ϕ x 2. \array{ && x_1 \\ & {}^{\mathllap{d_2 \phi}}\nearrow &\Downarrow^\phi& \searrow^{\mathrlap{d_0 \phi}} \\ x_0 &&\underset{d_1 \phi}{\to}&& x_2 } \,.

More in detail, one may think of the incoming two adjacent 11-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 ϕ\phi here as being a compositor in a bicategory that shows how the composable pair is composed to the morphism d 1ϕd_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 CC 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

x 1 f g x 0 h=gf x 2 \array{ && x_1 \\ & {}^{f}\nearrow &\Downarrow& \searrow^{g} \\ x_0 &&\underset{h = g \circ f}{\to}&& x_2 }

that is the unique identity 2-morphism

gf=h. g \circ f \stackrel{=}{\Rightarrow} h \,.

More generally, in a 2-groupoid there may be non-identity 2-morphisms, and hence for any 1-morphism k x0x 2k _ x_0 \to x_2 2-isomorphic to hh, there may be many 2-morphisms gfkg \circ f \Rightarrow k, hence many 2-cells

x 1 f g x 0 k x 2. \array{ && x_1 \\ & {}^{f}\nearrow &\Downarrow^{\simeq}& \searrow^{g} \\ x_0 &&\underset{k }{\to}&& x_2 } \,.

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 ω\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 CC, a diagram of 1-cells of the form

x 1 f x 0 h x 2. \array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow \\ x_0 &&\underset{h}{\to}&& x_2 } \,.

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 ff composed with the morphism hh, and there is at least one 2-morphism

x 1 f hf 1 x 0 h x 2. \array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{\simeq}& \searrow^{\mathrlap{h \circ f^{-1}}} \\ x_0 &&\underset{h}{\to}&& x_2 } \,.

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 kk are composable if adjacent and have a weak inverse.

1-Groupoids as Kan complexes

We review how 1-groupoids are incarnated as Kan complexes via their nerve.

Definition

A (small) groupoid 𝒢 \mathcal{G}_\bullet is

  • a pair of sets 𝒢 0Set\mathcal{G}_0 \in Set (the set of objects) and 𝒢 1Set\mathcal{G}_1 \in Set (the set of morphisms)

  • equipped with functions

    𝒢 1× 𝒢 0𝒢 1 𝒢 1 sit 𝒢 0, \array{ \mathcal{G}_1 \times_{\mathcal{G}_0} \mathcal{G}_1 &\stackrel{\circ}{\to}& \mathcal{G}_1 & \stackrel{\overset{t}{\to}}{\stackrel{\overset{i}{\leftarrow}}{\underset{s}{\to}}}& \mathcal{G}_0 }\,,

    where the fiber product on the left is that over 𝒢 1t𝒢 0s𝒢 1\mathcal{G}_1 \stackrel{t}{\to} \mathcal{G}_0 \stackrel{s}{\leftarrow} \mathcal{G}_1,

such that

  • ii takes values in endomorphisms;

    ti=si=id 𝒢 0, t \circ i = s \circ i = id_{\mathcal{G}_0}, \;\;\;
  • \circ defines a partial composition operation which is associative and unital for i(𝒢 0)i(\mathcal{G}_0) the identities; in particular

    s(gf)=s(f)s (g \circ f) = s(f) and t(gf)=t(g)t (g \circ f) = t(g);

  • every morphism has an inverse under this composition.

Definition

For 𝒢 \mathcal{G}_\bullet a groupoid, def. 2, we write

𝒢 n𝒢 1 × 𝒢 0 n \mathcal{G}_n \coloneqq \mathcal{G}_1^{\times_{\mathcal{G}_0}^n}

for the set of sequences of composable morphisms of length nn, for nn \in \mathbb{N}; schematically:

𝒢 n={x 0f 1x 1f 2x 2f 2f nx n}. \mathcal{G}_n = \left\{ x_0 \stackrel{f_1}{\to} x_1 \stackrel{f_2}{\to} x_2 \stackrel{f_2}{\to} \cdots \stackrel{f_n}{\to} x_n \right\} \,.

For each n1n \geq 1, the two maps d 0d_0 and d nd_n that forget the first and the last morphism in such a sequence and the n1n-1 maps d kd_k that form the composition of the kkth morphism in the sequence with the next one, constitute (n+1)(n+1) functions denoted

d k:𝒢 n𝒢 n1. d_k \colon \mathcal{G}_n \to \mathcal{G}_{n-1} \,.

Moreover, the assignments s is_i that insert an identity? morphism in position ii constitute functions denoted

s i:𝒢 n1𝒢 n. s_i \colon \mathcal{G}_{n-1} \to \mathcal{G}_n \,.
Proposition

These collections of maps in def. 3 satisfy the simplicial identities, hence make the nerve 𝒢 \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.)

Proposition

The nerve operation constitutes a full and faithful functor

N:GrpdKanCplxsSet. N \colon Grpd \to KanCplx \hookrightarrow sSet \,.

Homotopy theory of \infty-groupoids as Kan complexes

Definition

Write

KanCplxsSet KanCplx \hookrightarrow sSet

for the category of Kan complexes, which is the full subcategory of that of simplicial sets on the Kan complexes.

Remark

This means that for X ,Y KanCplxX_\bullet,Y_\bullet \in KanCplx two Kan complexes, an element f :X Y f_\bullet \colon X_\bullet \to Y_\bullet in the hom-set Hom KanCplx(X ,Y )Hom_{KanCplx}(X_\bullet,Y_\bullet) is

  • a sequences of functions f n:X nY nf_n \colon X_n \to Y_n for all nn \in \mathbb{N};

such that

  • these respect all the face maps d kd_k and the degeneracy maps s ks_k.
Definition

For X ,Y KanCplxX_\bullet,Y_\bullet \in KanCplx two Kan complexes, their mapping space

Maps(X ,Y ) KanCplx Maps(X_\bullet,Y_\bullet)_\bullet \in KanCplx

is the simplicial set given by

Maps(X ,Y ):[k]Hom sSet(X ×Δ n,Y ). Maps(X_\bullet,Y_\bullet) \colon [k] \mapsto Hom_{sSet}(X_\bullet \times \Delta^n_\bullet, Y_\bullet) \,.
Proposition

The construction in def. 5 defines an internal hom of Kan complexes.

Remark

As such it is also common to write Y XY^X for Maps(X,Y)Maps(X,Y), as well as [X,Y][X,Y]. Notice that the latter notation is sometimes used instead for just the set of connected components of Maps(X,Y)Maps(X,Y).

Example

Write

I {01} I_\bullet \coloneqq \{0 \stackrel{\simeq}{\to} 1\}

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

i 0,i 1:*I i_0, i_1 \colon \ast \to I

of its endpoints.

Then for X KanCplxX_\bullet \in KanCplx any other Kan complex, the mapping space [I,X] [I,X]_\bullet from def. 5 is the path space object of X X_\bullet.

X [i 0,X ][I ,X ] [i 1,X]X . X_\bullet \stackrel{[i_0,X_\bullet]}{\leftarrow} [I_\bullet,X_\bullet]_\bullet \stackrel{[i_1,X]}{\to} X_\bullet \,.

A 1-cell in the mapping Kan complex [X ,Y ] [X_\bullet, Y_\bullet]_\bullet is a homotopy between two morphisms of Kan complexes:

Definition

For f ,g :X Y f_\bullet, g_\bullet \colon X_\bullet \to Y_\bullet two morphisms between two Kan complexes, hence f ,g Hom KanCplx(X,Y)f_\bullet,g_\bullet \in Hom_{KanCplx}(X,Y), a (right-)homotopy η:fg\eta \colon f \Rightarrow g is a morphism η :X [I ,X ] \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

Y f [i 0,X ] X η [I ,Y ] g [i 1,X ] Y. \array{ && Y_\bullet \\ & {}^{\mathllap{f_\bullet}}\nearrow & \uparrow^{\mathrlap{[i_0, X_\bullet]_\bullet}} \\ X_\bullet &\stackrel{\eta_\bullet}{\to}& [I_\bullet, Y_\bullet] \\ & {}_{\mathllap{g_\bullet}}\searrow & \downarrow_{\mathrlap{[i_1, X_\bullet]_\bullet}} \\ && Y \bullet } \,.

It follows that the category KanCplxKanCplx is naturally enriched over itself.

We may write ∞Grpd for KanCplxKanCplx regarded as a KanCplxKanCplx-enriched category, hence as fibrant sSet-enriched category. We write XX (without the subscript) for a Kan complex X X_\bullet regarded as an object of Grpd\infty Grpd.

Properties

Model category

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.

Examples

Kan complexes from nerves of nn-groupoids

Proposition

The nerve N(C)N(C) of a small category is a Kan complex if and only if CC is a groupoid.

The existence of inverse morphisms in DD corresponds to the fact that in the Kan complex N(D)N(D) the “outer” horns

d 0 f d 1 Id d 1 d 1and d 1 f d 0 Id d 0 d 0 \array{ && d_0 \\ & && \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 }

have fillers

d 0 f 1 f d 1 Id d 1 d 1and d 1 f f 1 d 0 Id d 0 d 0 \array{ && d_0 \\ & {}^{f^{-1}}\nearrow&& \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \searrow^{f^{-1}} \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 }

(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

Proposition

The nerve N(C)N(C) of a strict ω-category is a Kan complex if and only if CC is a strict ω-groupoid.

Kan complexes from simplicial groups

The underlying simplicial set of a simplicial group (see there) is a Kan complex.

In particular, via the Dold-Kan correspondence (see there)

Ch 0DKsAbforgetKanCplxsSet Ch_{\bullet \geq 0} \underoverset{\simeq}{DK}{\to} sAb \stackrel{forget}{\to} KanCplx \hookrightarrow sSet

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).

Singular simplicial complexes / fundamental \infty-groupoids

For XX a topological space, its singular simplicial complex is the simplicial set Π(X)\Pi(X) (often denoted S(X)S(X)) whose set of nn-simplices is the hom-set

Π(X) n:=Top(Δ Top n,X) \Pi(X)_n := Top(\Delta^n_{Top}, X)

in Top of continuous maps from the standard topological nn-simplex Δ Top n\Delta^n_{Top} into XX.

Using the fact that the Δ Top n\Delta^n_{Top} arrange themselves into a cosimplicial space

Δ Top:ΔTop \Delta_{Top} : \Delta \to Top

in the obvious way, the (Π(X) n)(\Pi(X)_n) become a simplicial set in the corresponding obvious way. For instance the face maps are induced by restricting maps to XX along the face inclusions δ i:Δ n1Δ n\delta^i : \Delta^{n-1} \hookrightarrow \Delta^n.

That Π(X)\Pi(X) is indeed a Kan complex is intuitively clear. Technically it follows from the fact that the inclusions Λ n Top kΔ Top n{{\Lambda^n}_{Top}}_k \hookrightarrow \Delta^n_{Top} of topological horns into topological simplices are retracts, in that there are continuous maps Δ Top nΛ n Top k\Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k given by “squashing” a topological nn-simplex onto parts of its boundary, such that

(Λ n Top kΔ Top nΛ n Top k)=Id. ({{\Lambda^n}_{Top}}_k \to \Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k) = Id \,.

Therefore the map [Δ n,Π(X)][Λ k n,Π(X)][\Delta^n, \Pi(X)] \to [\Lambda^n_k,\Pi(X)] is an epimorphism, since it is equal to to Top(Δ n,X)Top(Λ k n,X)Top(\Delta^n, X) \to Top(\Lambda^n_k, X) which has a right inverse Top(Λ k n,X)Top(Δ n,X)Top(\Lambda^n_k, X) \to Top(\Delta^n, X).

The ∞-groupoid represented by the Kan complex Π(X)\Pi(X) is called the fundamental ∞-groupoid of XX.

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.

  • A slight weakening of the Kan condition, the weak Kan condition leads to the definition of quasi-category.

Other concepts:

References

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

Revised on September 5, 2013 21:42:58 by Tim_Porter (83.202.4.5)