References:
Simplicial Homotopy Theory by Paul G. Goerss and John F. Jardine
Simplicial Objects in Algebraic Topology by J. Peter May
Higher Topos Theory by Jacob Lurie
Simplicial Sets
Consider the category \(\mathbf{LO}\) of linearly ordered sets and order-preserving maps. Let \(\Delta\subset \mathbf{LO}\) be the subcategory called ordinal number category consisting of finite linearly ordered sets whose objects we denote by \(\Delta = \{[0], [1], [2], \cdots\}\), where the combinatorial simplices \([n]\) is generated by \(0\le 1\le \cdots\le n\) and depicted by arrows as
\[[n]=\{0\to 1\to \cdots \to n\}.\]
A simplicial object in any category \(\mathcal{C}\) is a contravariant functor \(S:\Delta^{op}\to \mathcal{C}\).
A cosimplicial object in \(\mathcal{C}\) is a covariant functor \(C:\Delta\to \mathcal{C}\).
A simplicial set is a simplicial object in the category of sets, i.e. a contravariant functor \(S:\Delta^{op}\to {\bf Set}\).
Concretely speaking it consists of
For each \(n\ge 0\) a set \(S_n = S([n])\), the set of \(n\)-simplices.
For each order-preserving map \(f:[m]\to [n]\) a map \(f^* = S(f): S_n\to S_m\) compatible with composition.
The category of simplicial sets is denoted by \({\bf Set}_\Delta\).
Simplicial Identities
Face and Degeneracy Maps
On combinatorial simplices \(\Delta\) all maps can be generated by the face maps and degeneracy maps
\(p_{i,n}: [n-1]\to [n]\) is the order-preserving map that jumps over \(i\), i.e. \(p_{i,n}(i) = i+1\). Here \(0\le i\le n\).
\(q_{i,n}: [n+1]\to [n]\) is the order-preserving map that duplicates \(i\), i.e. \(q_{i,n}(i) = q_{i,n}(i+1) = i\), \(0\le i\le n\).
The face maps \(d_i=p_{i,n}^*: S_n \to S_{n-1}\) are the functorial images of \(p_{i,n}\).
It obtains the \(i\)-th face of any simplex by omitting the \(i\)-th vertex.
The degeneracy maps \(s_i=q_{i,n}^*: S_n \to S_{n+1}\) are the functorial images of \(q_{i,n}\).
It duplicates the \(i\)-th vertex of any simplex.
Simplicial Identities
Axiomatically simplicial sets can be described as \(\{S_n,d_n,s_n\}\) satisfying the simplicial identities
\(d_i d_j = d_{j-1} d_i\) for \(i<j\).
This will jump over \(i,j\) vertices and obtain an \(n-2\) face. The \(j-1\) is due to the fact that \(d_i\) jumps over \(i\) and affects the indexing for \(j>i\).
\(d_i s_j = s_{j-1} d_i\) for \(i<j\).
This will jump over \(i\) and duplicate \(j\) vertices, which is equivalent to duplicating the \(j-1\)-th vertice after jumping over \(i\).
\(d_i s_i = d_{i+1} s_i = \mathrm{id}\).
First duplicate the \(i\)-th vertex and then jump over one of them, which is equivalent to doing nothing.
\(s_i s_j = s_{j+1} s_i\) for \(i<j\).
Duplicate \(i\) and \(j\).
Important Examples
for any linearly ordered set \(L\) we can write \(\Delta^L = h_L = \mathrm{Hom}_\mathbf{LO}(,L)\) for the representable functor which is a simplicial set (by restricting to the subcategory \(\Delta\)).
Write \(\Delta^n = h_{[n]} = \mathrm{Hom}_\Delta(,[n])\) for the standard \(n\)-simplex, the most important simplicial set. Yoneda’s Lemma gives
\[\mathrm{Hom}_{{\bf Set}_\Delta}(\Delta^n, S) = S([n]) = S_n.\]
Horns
The \(j\)-th horn \(\Lambda^n_j\subset \Delta^n\) is a simplicial set, a subfunctor of \(\Delta^n\) generated by all the faces except \(j\), i.e. \(\{d_i(\Delta^n): i\neq j\}\). It consists of
\[\mathrm{Hom}(\Delta^m, \Lambda^n_j) = \left\{f\in \mathrm{Hom}([m],[n]): f([m])\cup \{j\}\neq [n]\right\},\]
here \(f([m])\cup \{j\}\neq [n]\) exactly rules out the \(j\)-th face and its subfaces, note that the interior is also removed. This construction can be generalized to \(\Lambda^J_j\subset\Delta^J\) for any \(J\in \mathbf{LO}\).
Realization Functor
The realization functor \(|\cdot|: {\bf Set}_\Delta\to \mathbf{Top}\) sends a simplicial set \(S\) to its geometric realization \(|S|\), which is the real topological space obtained by gluing together the standard simplices \(\Delta^n\) according to the simplicial structure of \(S\).
For standard simplex, define
\[|\Delta^n| = \left\{ (t_0,\cdots,t_n)\in \mathbb{R}^{n+1}: \sum t_i = 1, t_i\ge 0\right\}.\]
For any map \(f:\Delta^n\to \Delta^m\in {\bf Set}_\Delta\) or equivalently any map \(f:[n]\to [m]\) in \(\Delta\), we have an induced map \(|f|:|\Delta^n|\to |\Delta^m|\) defined by
\[|f|(t_0,\dots,t_n) := \left(i\mapsto \sum_{j\in f^{-1}(i)} t_j\right)_i.\]
If we think of \(|\Delta^n|\) as the space of probability distributions on \([n]\), then \(|f|\) is the pushforward of probability measures >w<.
In the general case,
\[|X| := \mathrm{colim}_{\{\Delta^n\to X\}} |\Delta^n|,\]
where \(\{\Delta^n\to X\}\) is the category of all simplicial maps from \(\Delta^n\) to \(X\) with morphisms all over \(X\). Clearly the assignment \(S\mapsto \{\Delta^n\to S\}\) is functorial, so the functor \(|\cdot|\) is functorial.
\((|\cdot|, S)\) is an adjoint pair with \(S:\mathbf{Top}\to {\bf Set}_\Delta\) the singular functor \(S(X)([n]) := \mathrm{Hom}(|\Delta^n|, X)\). We have
\[\mathrm{Hom}_{\mathbf{Top}}(|X|, Y) = \mathrm{Hom}_{{\bf Set}_\Delta}(X, S(Y)).\]
Proof. \[\begin{align*} \mathrm{Hom}_{\mathbf{Top}}(|X|, Y) &= \mathrm{Hom}_{\mathbf{Top}}\left(\mathrm{colim}_{\{\Delta^n\to X\}} |\Delta^n|, Y\right) \\ &= \lim_{\{\Delta^n\to X\}} \mathrm{Hom}_{\mathbf{Top}}(|\Delta^n|, Y) \\ &= \lim_{\{\Delta^n\to X\}} \mathrm{Hom}_{{\bf Set}_\Delta}(\Delta^n, S(Y)) \\ &= \mathrm{Hom}_{{\bf Set}_\Delta}\left(\mathrm{colim}_{\{\Delta^n\to X\}} \Delta^n, S(Y)\right) \\ &= \mathrm{Hom}_{{\bf Set}_\Delta}(X, S(Y)). \end{align*}\]
Kan Model Structure
\({\bf Set}_\Delta\) has a model structure with weak equivalences, fibrations, and cofibrations defined as follows,
\(f:X\to Y\in {\bf Set}_\Delta\) is called a fibration if it is a Kan fibration, i.e. for any diagram
of solid commutative arrows, there exists a dashed arrow making the diagram commute.
One can understand it as higher homotopy lifting property, any image of \(\Delta^n\) whose horn can be lifted to \(X\) can be completed to \(X\).
Another way to understand is, \(\Delta^n\) can be seen as the higher composition of \(\Lambda^n_k\) (it actually includes higher inversions as well so behaves more like a groupoid than an ordinary category). So it requires if certain arrows in \(X\to Y\) has a composition in \(Y\), then it must have a corresponding composition in \(X\) lying over it.
\(f:X\to Y\in {\bf Set}_\Delta\) is called a cofibration if it is a subfunctor, i.e. \(f\) is injective on \(n\)-simplices for all \(n\).
\(f:X\to Y\in {\bf Set}_\Delta\) is called a weak equivalence if the induced map of geometric realization \(|f|:|X|\to |Y|\) is a homotopy equivalence.
These model structures need to satisfy five axioms,
limit axiom: \({\bf Set}_\Delta\) has all finite limits and colimits.
2-out-of-3 property: In a composition, if two out of three maps \(f,g,h\) are weak equivalences, then so is the third.