\( \newcommand{\Sing}{\mathrm{Sing}} \)

Turning simplicial complexes into simplicial sets

Table of Contents

The statement

One way to produce a simplicial set from a simplicial complex \(K\) is to pick a partial order \(\le\) on the vertices such that each simplex is a chain and form the simplicial set \(\Sing_\le(K)\) with \(n\)-simplices given by: \[\Sing_\le(K)_n := \{ (x_0, \ldots, x_n) \in V(K)^{n+1} : x_0 \le \cdots \le x_n, \{x_0, \ldots, x_n\} \in K\}.\]

This has the advantage that its geometric realization is homeomorphic to the geometric realziation of \(K\), \(|\Sing_\le(K)| \cong |K|\), but, since it involves a choice of order, it is not a functor from the category of simplicial complexes and simplicial maps to the category of simplicial sets.

There are also functorial constructions. One such construction is to take the nerve of the poset of simplices in \(K\), say \(N(K, \subseteq)\). This also has geometric realization homeomorphic to the geometric realization of (the barycentric subdivision of) \(K\), but this note is about another functorial construction: the simplicial set \(\Sing(K)\) with \(n\)-simplices given by: \[\Sing(K)_n := \{ (x_0, \ldots, x_n) \in V(K)^{n+1} : \{x_0, \ldots, x_n\} \in K\}.\]

Notice that \(\Sing(K)_n\) can also be described as the set of morphisms in the category of simplicial complexes from an \(n\)-simplex to \(K\). This is analogous to the definition of the singular simplicial set of a topological space, which explains the choice of notation.

The geometric realization of this simplicial set is much larger than \(|K|\), but it is homotopy equivalent to it as we shall now prove.

Proof with Reedy model structures

Our first step will be to express \(\Sing(K)\) in terms of \(\Sing_\le(K)\), (for some arbitrarily chosen \(\le\)) which already has the homotopy type of \(K\). To this end, let \(E : \mathbf{\Delta} \to \mathsf{sSet}\) be the cosimplicial simplicial set given by \(E[n] = \mathrm{cosk}_0(\coprod^{n+1} \Delta^0)\). Another way of describing \(E[n]\) is as the nerve of the indiscrete category with \(n+1\) objects (indiscrete means there is a unique morphism \(x \to y\) for any pair of objects).

Intuitively, to obtain \(\Sing(K)\) from \(\Sing_\le(K)\) we replace each simplex \(\Delta^n\) with \(E[n]\). Indeed, \(m\)-simplices of \(\Delta^n\) are sequences \((i_0, \ldots, i_m)\) with \(0 \le i_0 \le \cdots i_m \le n\) while \(m\)-simplices of \(E[n]\) are all sequences \((i_0, \ldots, i_m)\) with \(0 \le i_0, \ldots, i_m \le n\); which is exactly the difference between \(\Sing_\le(K)\) and \(\Sing(K)\).

If \(\Delta : \mathbf{\Delta} \to \mathsf{sSet}\) denotes the canonical cosimplicial simplicial set, namely, the Yoneda embedding for \(\mathbf{\Delta}\), we have that for any simplicial set \(X\), \(X \cong \Delta \otimes_{\mathbf{\Delta}} X\). So this operation of replacing \(\Delta^n\)’s with \(E[n]\)’s sends \(X\) to \(E \otimes_{\mathbf{\Delta}} X\).

In summary, we are claiming that the relation between \(\Sing(K)\) and \(\Sing_\le(K)\) is given by \(\Sing(K) \cong E \otimes_{\mathbf{\Delta}} \Sing_\le(K)\), which can be checked using the description above of simplices in \(\Delta^n\) and \(E[n]\).

There is a canonical inclusion \(\Delta \to E\) of cosimplicial simplicial sets, and we claim that for any simplicial set \(X\), this inclusion induces a homotopy equivalence \(X \cong \Delta \otimes_{\mathbf{\Delta}} X \to E \otimes_{\mathbf{\Delta}} X\). This will follow from existing machinery for Reedy model structures.

By proposition A.2.9.26 from Lurie’s Higher Topos Theory, the functor tensor product \[\otimes_{\Delta} : \mathop{\mathrm{Fun}}(\mathbf{\Delta}, \mathsf{sSet}) \times \mathop{\mathrm{Fun}}(\mathbf{\Delta}^{\mathrm{op}}, \mathsf{sSet}) \to \mathsf{sSet}\] is a left Quillen bifunctor when we equip \(\mathsf{sSet}\) with the Quillen model structure and both functor categories with the corresponding Reedy model structure.

Any simplicial set \(X\) can be regarded as a functor \(\mathbf{\Delta}^{\mathrm{op}} \to \mathsf{sSet}\), \([n] \mapsto X_n\) by regarding the set \(X_n\) of \(n\)-simplices as a discrete or constant simplicial set. When regarded as such, \(X\) is Reedy cofibrant (in fact, all bisimplicial sets are Reedy cofibrant). Therefore, \(X \otimes_{\Delta} -\) is a left Quillen functor.

Now, both \(\Delta^n\) and \(E[n]\) are contractible simplicial sets, so the inclusion \(\Delta \to E\) is an objectwise weak equivalence. Thus, we need only check that both \(\Delta\) and \(E\) are Reedy cofibrant to conclude from Ken Brown’s lemma and the proposition that the functor tensor product will send the inclusion to a weak equivalence, as desired.

For \(\Delta\) this is well known: the latching map \(L_n \Delta \to \Delta^n\) is readily seen to be the inclusion \(\partial \Delta^n \to \Delta^n\), a monomorphism and thus a cofibration in \(\mathsf{sSet}\). The case of \(E\) is very similar. Indeed, the latching object is given by \(L_n(E) = \mathop{\mathrm{colim}}_{[k] \hookrightarrow [n]} E[k]\), which one can check consists of all simplices of \(E[n]\) that do not involve all \(n+1\) vertices, and the canonical map \(L_n(E) \to E\) is then a monomorphism.

Proof using symmetric simplicial sets

Andrea Gagna pointed out on MathOverflow that one can also use results about symmetric simplicial sets to prove this result. Let \(\mathbf{\Upsilon}\) be the category of finite non-empty sets and all functions between them. The category of symmetric simplicial sets is defined to be \(\mathsf{\Sigma Set} := \mathop{\mathrm{Fun}}(\mathbf{\Upsilon}^{\mathrm{op}}, \mathsf{Set})\). There is an obvious functor \(v : \mathbf{\Delta} \to \mathbf{\Upsilon}\), including monotone functions into all functions. That functor (thought of as a functor \(\mathbf{\Delta}^{\mathrm{op}} \to \mathbf{\Upsilon}^{\mathrm{op}}\)) induces adjunctions \(v_{!} \dashv v^{\ast} \dashv v_{\ast}\), where \(v^{\ast} : \mathsf{\Sigma Set} \to \mathsf{sSet}\) is precomposition with \(v\) and \(v_{!}\) and \(v_{\ast}\) are left and right Kan extension along \(v\).

We can use these functors to relate \(\Sing(K)\) with \(\Sing_{\le}(K)\), namely, we have \(v^{\ast} v_{!} \Sing_\le(K) \cong \Sing(K)\). More generally, for any simplicial set \(X\) we have \(v^{\ast} v_{!} X = E \otimes_{\mathbf{\Delta}} X\). This is not hard to do directly for \(\Sing_\le(K)\). A slicker way is to notice that \(E[n] \cong v^{\ast}(\Upsilon_n)\) where \(\Upsilon_n\) is the representable symmetric simplicial set corresponding to \([n]\). Thus \(E[n] \cong v^{\ast}v_{!}\Delta^n\) and since both \(v^{\ast}\) and \(v_{!}\) are left adjoints we have \[E \otimes_{\mathbf{\Delta}} X \cong (v^{\ast} \circ v_{!} \circ \Delta) \otimes_{\mathbf{\Delta}} X \cong v^{\ast} v_{!} (\Delta \otimes_{\mathbf{\Delta}} X) \cong v^{\ast} v_{!} X.\]

Moreover, it is straightforward to check that the canonical map \(X \to E \otimes_{\mathbf{\Delta}} X\) used in the first proof corresponds to the unit \(X \to v^{\ast} v_{!} X\) under the above isomorphism. It only remains to prove the unit is a weak equivalence.

In section 8.3 of Les Préfaisceaux comme Modèles des Types d’Homotopie, Cisinski proves there is a model structure on \(\mathsf{\Sigma Set}\) for which \((v_{!}, v^{\ast})\) is a Quillen equivalence in which \(v^{\ast}\) creates the weak equivalences (that is, a map \(f\) of symmetric simplicial sets is a weak equivalence if and only if \(v^{\ast}(f)\) if a weak equivalence of simplicial sets). When a right Quillen functor creates weak equivalences, it is a right Quillen equivalence if and only if the components of the unit of the adjunction at cofibrant objects are weak equivalences. Since all simplicial sets are cofibrant, we obtain that \(X \to v^{\ast} v_{!} X\) is a weak equivalence, as desired.

Omar Antolín Camarena