Definition of the standard n-simplex

The striped blue simplex in the domain has to exist in order for this map to be a Kan fibration

For each n ≥ 0, recall that the standard ${\displaystyle n}$-simplex, ${\displaystyle \Delta ^{n}}$, is the representable simplicial set

${\displaystyle \Delta ^{n}(i)=\mathrm {Hom} _{\mathbf {\Delta } }([i],[n])}$

Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard ${\displaystyle n}$-simplex: the convex subspace of ℝⁿ⁺¹ consisting of all points ${\displaystyle (t_{0},\dots ,t_{n})}$ such that the coordinates are non-negative and sum to 1.

Definition of a horn

For each k ≤ n, this has a subcomplex ${\displaystyle \Lambda _{k}^{n}}$, the k-th horn inside ${\displaystyle \Delta ^{n}}$, corresponding to the boundary of the n-simplex, with the k-th face removed. This may be formally defined in various ways, as for instance the union of the images of the n maps ${\displaystyle \Delta ^{n-1}\rightarrow \Delta ^{n}}$ corresponding to all the other faces of ${\displaystyle \Delta ^{n}}$.^[1] Horns of the form ${\displaystyle \Lambda _{k}^{2}}$ sitting inside ${\displaystyle \Delta ^{2}}$ look like the black V at the top of the adjacent image. If ${\displaystyle X}$ is a simplicial set, then maps

${\displaystyle s:\Lambda _{k}^{n}\to X}$

correspond to collections of ${\displaystyle n}$ ${\displaystyle (n-1)}$-simplices satisfying a compatibility condition, one for each ${\displaystyle 0\leq k\leq n-1}$. Explicitly, this condition can be written as follows. Write the ${\displaystyle (n-1)}$-simplices as a list ${\displaystyle (s_{0},\dots ,s_{k-1},s_{k+1},\dots ,s_{n})}$ and require that

${\displaystyle d_{i}s_{j}=d_{j-1}s_{i}\,}$ for all ${\displaystyle i<j}$ with ${\displaystyle i,j\neq k}$.^[2]

These conditions are satisfied for the ${\displaystyle (n-1)}$-simplices of ${\displaystyle \Lambda _{k}^{n}}$ sitting inside ${\displaystyle \Delta ^{n}}$.

Definition of a Kan fibration

Lifting diagram for a Kan fibration

A map of simplicial sets ${\displaystyle f:X\rightarrow Y}$ is a Kan fibration if, for any ${\displaystyle n\geq 1}$ and ${\displaystyle 0\leq k\leq n}$, and for any maps ${\displaystyle s:\Lambda _{k}^{n}\rightarrow X}$ and ${\displaystyle y:\Delta ^{n}\rightarrow Y\,}$ such that ${\displaystyle f\circ s=y\circ i}$ (where ${\displaystyle i}$ is the inclusion of ${\displaystyle \Lambda _{k}^{n}}$ in ${\displaystyle \Delta ^{n}}$), there exists a map ${\displaystyle x:\Delta ^{n}\rightarrow X}$ such that ${\displaystyle s=x\circ i}$ and ${\displaystyle y=f\circ x}$. Stated this way, the definition is very similar to that of fibrations in topology (see also homotopy lifting property), whence the name "fibration".

Simplicial sets from singular homology

An important example comes from the construction of singular simplices used to define singular homology, called the singular functor^{[4]pg 7}

${\displaystyle S:{\text{Top}}\to s{\text{Sets}}}$.

Given a space ${\displaystyle X}$, define a singular ${\displaystyle n}$-simplex of X to be a continuous map from the standard topological ${\displaystyle n}$-simplex (as described above) to ${\displaystyle X}$,

${\displaystyle f:\Delta _{n}\to X}$

Taking the set of these maps for all non-negative ${\displaystyle n}$ gives a graded set,

${\displaystyle S(X)=\coprod _{n}S_{n}(X)}$.

To make this into a simplicial set, define face maps ${\displaystyle d_{i}:S_{n}(X)\to S_{n-1}(X)}$ by

${\displaystyle (d_{i}f)(t_{0},\dots ,t_{n-1})=f(t_{0},\dots ,t_{i-1},0,t_{i},\dots ,t_{n-1})\,}$

and degeneracy maps ${\displaystyle s_{i}:S_{n}(X)\to S_{n+1}(X)}$ by

${\displaystyle (s_{i}f)(t_{0},\dots ,t_{n+1})=f(t_{0},\dots ,t_{i-1},t_{i}+t_{i+1},t_{i+2},\dots ,t_{n+1})\,}$.

Since the union of any ${\displaystyle n+1}$ faces of ${\displaystyle \Delta _{n+1}}$ is a strong deformation retract of ${\displaystyle \Delta _{n+1}}$, any continuous function defined on these faces can be extended to ${\displaystyle \Delta _{n+1}}$, which shows that ${\displaystyle S(X)}$ is a Kan complex.^[5]

Simplicial sets underlying simplicial groups

It can be shown that the simplicial set underlying a simplicial group is always fibrant^{[4]pg 12}. In particular, for a simplicial abelian group, its geometric realization is homotopy equivalent to a product of Eilenberg-Maclane spaces

${\displaystyle \prod _{i\in I}K(A_{i},n_{i})}$

In particular, this includes classifying spaces. So the spaces ${\displaystyle S^{1}\simeq K(\mathbb {Z} ,1)}$, ${\displaystyle \mathbb {CP} ^{\infty }\simeq K(\mathbb {Z} ,2)}$, and the infinite lens spaces ${\displaystyle L_{q}^{\infty }\simeq K(\mathbb {Z} /q,2)}$ are correspond to Kan complexes of some simplicial set. In fact, this set can be constructed explicitly using the Dold–Kan correspondence of a chain complex and taking the underlying simplicial set of the simplicial abelian group.

Geometric realizations of small groupoids

Another important source of examples are the simplicial sets associated to a small groupoid ${\displaystyle {\mathcal {G}}}$. This is defined as the geometric realization of the simplicial set ${\displaystyle [\Delta ^{op},{\mathcal {G}}]}$ and is typically denoted ${\displaystyle B{\mathcal {G}}}$. We could have also replaced ${\displaystyle {\mathcal {G}}}$ with an infinity groupoid. It is conjectured that the homotopy category of geometric realizations of infinity groupoids is equivalent to the homotopy category of homotopy types. This is called the homotopy hypothesis.

Non-example: standard n-simplex

It turns out the standard ${\displaystyle n}$-simplex ${\displaystyle \Delta ^{n}}$ is not a Kan complex^{[6]pg 38}. The construction of a counter example in general can be found by looking at a low dimensional example, say ${\displaystyle \Delta ^{1}}$. Taking the map ${\displaystyle \Lambda _{0}^{2}\to \Delta ^{1}}$ sending

${\displaystyle {\begin{matrix}[0,2]\mapsto [0,0]&[0,1]\mapsto [0,1]\end{matrix}}}$

gives a counter example since it cannot be extended to a map ${\displaystyle \Delta ^{2}\to \Delta ^{1}}$ because the maps have to be order preserving. If there was a map, it would have to send

${\displaystyle {\begin{aligned}0\mapsto 0\\1\mapsto 1\\2\mapsto 0\end{aligned}}}$

but this isn't a map of simplicial sets.

Simplicial enrichment and function complexes

For simplicial sets ${\displaystyle X,Y}$ there is an associated simplicial set called the function complex ${\displaystyle {\textbf {Hom}}(X,Y)}$, where the simplices are defined as

${\displaystyle {\textbf {Hom}}_{n}(X,Y)={\text{Hom}}_{s{\text{Sets}}}(X\times \Delta ^{n},Y)}$

and for an ordinal map ${\displaystyle \theta$ :[m]\to [n]} there is an induced map

${\displaystyle \theta ^{*}:{\textbf {Hom}}(X,Y)_{n}\to {\textbf {Hom}}(X,Y)_{m}}$

(since the first factor of Hom is contravariant) defined by sending a map ${\displaystyle f:X\times \Delta ^{n}\to Y}$ to the composition

${\displaystyle X\times \Delta ^{m}\xrightarrow {1\times \theta } X\times \Delta ^{n}\xrightarrow {f} Y}$

Kan fibrations and pull-backs

Given a (Kan) fibration ${\displaystyle p:X\to Y}$ and an inclusion of simplicial sets ${\displaystyle i:K\hookrightarrow L}$, there is a fibration^{[4] pg 21}

${\displaystyle {\textbf {Hom}}(L,X)\xrightarrow {(i^{*},p_{*})} {\textbf {Hom}}(K,X)\times _{{\textbf {Hom}}(K,Y)}{\textbf {Hom}}(L,Y)}$

(where ${\displaystyle {\textbf {Hom}}}$ is in the function complex in the category of simplicial sets) induced from the commutative diagram

${\displaystyle {\begin{matrix}{\textbf {Hom}}(L,X)&\xrightarrow {p_{*}} &{\textbf {Hom}}(L,Y)\\i^{*}\downarrow &&\downarrow i^{*}\\{\textbf {Hom}}(K,X)&\xrightarrow {p_{*}} &{\textbf {Hom}}(K,Y)\end{matrix}}}$

where ${\displaystyle i^{*}}$ is the pull-back map given by pre-composiiton and ${\displaystyle p_{*}}$ is the pushforward map given by post-composition. In particular, the previous fibration implies ${\displaystyle p_{*}:{\textbf {Hom}}(L,X)\to {\textbf {Hom}}(L,Y)}$ and ${\displaystyle i^{*}:{\textbf {Hom}}(L,Y)\to {\textbf {Hom}}(K,Y)}$ are fibrations.

Homotopy groups of Kan complexes

The homotopy groups of a fibrant simplicial set may be defined combinatorially, using horns, in a way that agrees with the homotopy groups of the topological space which realizes it. For a Kan complex ${\displaystyle X}$ and a vertex ${\displaystyle x:\Delta ^{0}\to X}$, as a set ${\displaystyle \pi _{n}(X,x)}$ is defined as the set of maps ${\displaystyle \alpha$ :\Delta ^{n}\to X} of simplicial sets fitting into a certain commutative diagram:

${\displaystyle \pi _{n}(X,x)=\left\{\alpha$ :\Delta ^{n}\to X:{\begin{matrix}\Delta ^{n}&{\overset {\alpha }{\to }}&X\\\uparrow &&\uparrow x\\\partial \Delta ^{n}&\to &\Delta ^{0}\end{matrix}}\right\}}

Notice the fact ${\displaystyle \partial \Delta ^{n}}$ is mapped to a point is equivalent to the definition of the sphere ${\displaystyle S^{n}}$ as the quotient ${\displaystyle B^{n}/\partial B^{n}}$ for the standard unit ball

${\displaystyle B^{n}=\{x\in \mathbb {R} ^{n}:||x||_{eu}\leq 1\}}$

Defining the group structure requires a little more work. Essentially, given two maps ${\displaystyle \alpha ,\beta$ :\Delta ^{n}\to X} there is an associated ${\displaystyle (n+1)}$-simplice ${\displaystyle \omega$ :\Delta ^{n+1}\to X} such that ${\displaystyle d_{n}\omega$ :\Delta ^{n}\to X} gives their addition. This map is well-defined up to simplicial homotopy classes of maps, giving the group structure. Moreover, the groups ${\displaystyle \pi _{n}(X,x)}$ are Abelian for ${\displaystyle n\geq 2}$. For ${\displaystyle \pi _{0}(X)}$, it is defined as the homotopy classes ${\displaystyle [x]}$ of vertex maps ${\displaystyle x:\Delta ^{0}\to X}$.