Simplicial objects and quasicategories

Definition 1.6.1. The simplex category \(\simp \) is the 1-category of non-empty finite linearly ordered sets and order-preserving maps between them.

Observation 1.6.2. Every morphism in \(\simp \) can be written as a finite composite of face maps \(d_i\colon [n-1] \to [n]\) and degeneracy maps \(s_i\colon [n+1] \to [n]\) (see Notation 1.2.10). The relations these maps satisfy are known as the simplicial identities.

Definition 1.6.3. Let \(C\) be an \(\infty \)-category. A simplicial object in \(C\) is defined to be a functor \(\simp \catop \to C\). We will denote the \(\infty \)-category of simplicial objects by \[ \s C := \Fun (\simp \catop ,C). \]

Definition 1.6.4 (Simplices). For every natural number \(n\), we define the \(n\)-simplex \(\Delta ^n\) as the simplicial set represented by \([n]\): \[ \Delta ^n := \Hom _{\simp }(-, [n]) \colon \simp \catop \to \Set . \] By the Yoneda lemma this defines a fully faithful functor \[ \Delta ^{\bullet }\colon \simp \hookrightarrow \sSet . \] Again by the Yoneda lemma there is a natural bijection \(X_n \cong \Hom _{\sSet }(\Delta ^n, X)\), so that we may think of an \(n\)-simplex in \(X\) as a morphism \(\sigma \colon \Delta ^n \to X\) of simplicial sets.

Definition 1.6.5 (Horns). For \(n \geq 1\) and \(0 \leq k \leq n\), we define the \(k\)-horn \(\Lambda ^n_k\) as the subsimplicial set \[ \Lambda ^n_k \subseteq \Delta ^n \] whose \(m\)-simplices are those maps \(\phi \colon [m] \to [n]\) in \(\simp \) such that there is some element \(i \in [n] \setminus \{k\}\) which is not in the image of \(\phi \). Equivalently, it is the union inside \(\Delta ^n\) of all of its faces except for the \(k\)-th face.

Example 1.6.6. For \(n = 2\) the three horns \(\Lambda ^2_0\), \(\Lambda ^2_1\) and \(\Lambda ^2_2\) may be displayed as follows:

Definition 1.6.7. Let \(X\) be a simplicial set.

(1)

We say that \(X\) is a Kan complex if for every \(n \geq 1\) and \(0 \leq k \leq n\), every morphism of simplicial sets \(\Lambda ^n_k \to X\) extends to a morphism \(\Delta ^n \to X\):

(2)

We say that \(X\) is a quasicategory if the above condition holds for all \(n \geq 1\) but only for \(0 < k < n\).

We denote by \[ \Kan \quad \subseteq \quad \qCat \quad \subseteq \quad \sSet \] the full subcategories on the Kan complexes and the quasicategories, respectively.

Example 1.6.8 (Singular simplicial complex). Let \(X\) be a topological space. We will construct a simplicial set \(\Sing (X) \in \sSet \) called the singular simplicial complex of \(X\).

• For every \(n \geq 0\), let \(\abs {\Delta ^n}\) denote the topological \(n\)-simplex: \[ \abs {\Delta ^n} := \{ (x_0, \dots , x_n)\in \R ^{n+1} \mid x_{0} + \dots + x_n=1, \,\, x_{i}\geq 0{\text { for }}i=0,\dots ,n\}. \]
• For every order-preserving map \(\phi \colon [n] \to [m]\) we get a continuous map \[ \abs {\Delta ^{\phi }}\colon \abs {\Delta ^n} \to \abs {\Delta ^m}, \quad (x_0, \dots , x_n) \mapsto (y_0, \dots , y_m), \qquad y_i := \sum _{j \in \phi ^{-1}(i)} x_j. \] This defines a functor \[ \abs {\Delta ^{\bullet }} \colon \simp \to \Top , \qquad [n] \mapsto \abs {\Delta ^n}, \qquad \phi \mapsto \abs {\Delta ^{\phi }}. \]
• We now define the simplicial set \(\Sing (X)\) as \[ \Sing (X)_n := \Hom _{\Top }(\abs {\Delta ^n}, X), \] i.e. it is the composite \(\simp \catop \xrightarrow {\abs {\Delta ^{\bullet }}} \Top \catop \xrightarrow {\Hom _{\Top }(-,X)} \Set \).

The simplicial set \(\Sing (X)\) is a familiar object from algebraic topology, and is used for example in the definition of the singular (co)homology of \(X\). It is not difficult to see that \(\Sing (X)\) is functorial in \(X\), so that it defines a functor \[ \Sing \colon \Top \to \sSet . \]

Definition 1.6.9 (Geometric realization). Let \(K\colon \simp \catop \to \Set \) be a simplicial set. Its geometric realization is the quotient space \[ \abs {K}:=\left (\bigsqcup _{n\geq 0}K_n\times \abs {\Delta ^n}\right )/\sim , \] where each set \(K_n\) carries the discrete topology and the equivalence relation is generated by \[ (\phi ^*(x),y)\sim (x,\abs {\Delta ^\phi }(y)), \] for all maps \(\phi \colon [n]\to [m]\) in \(\simp \), elements \(x\in K_m\), and points \(y\in \abs {\Delta ^n}\). This construction defines a functor \[ \abs {-}\colon \sSet \to \Top \] which is left adjoint to the singular complex functor \(\Sing \colon \Top \to \sSet \).

Exercise 1.6.10. Show that the singular simplicial complex \(\Sing (X)\) of a topological space \(X\) is a Kan complex.

Example 1.6.11 (Nerve of a 1-category). Let \(C\) be a (small, classical) 1-category. We define the nerve of \(C\) as the simplicial set \(N(C) \in \sSet \) given by \[ N(C)_n := \Hom _{\Cat ^{(1)}}([n],C), \] i.e. it is the composite \(\simp \catop \hookrightarrow (\Cat ^{(1)}_1)\catop \xrightarrow {\Hom _{\Cat ^{(1)}}(-,C)} \Set \). Here \(\Cat ^{(1)}_1\) denotes the 1-category of small 1-categories.¹ Again this construction is functorial in \(C\), resulting in a functor \[ N\colon \Cat ^{(1)}_1 \to \sSet . \]

Exercise 1.6.12. Show that the nerve \(N(C)\) of a classical 1-category \(C\) is a quasicategory. Provide an example for which it is not a Kan complex. Give a complete characterization of those 1-categories \(C\) for which \(N(C)\) is a Kan complex.

Exercise 1.6.13. Show that the functor \(N\colon \Cat ^{(1)}_1 \to \sSet \) is fully faithful, and that its essential image consists precisely of the simplicial sets in which every inner horn has a unique filler.

Example 1.6.14 (Homotopy coherent nerve). A further family of quasicategories is given by the homotopy coherent nerve \(\Ntop (D)\) of a topologically enriched category \(D\) (see Definition 24.3.1). Since this construction will not be used in the present book, we do not discuss it here in detail; see [Lurie (2009), Definition 1.1.5.5]. In Section 24.3, we will instead construct the homotopy coherent nerve directly as an \(\infty \)-category, without passing through quasicategories.

Proposition 1.6.15. Quasicategories satisfy axioms A–K.

Warning 1.6.16. Readers who prefer a concrete model may henceforth think of \(\infty \)-categories as quasicategories. This should only be used as a conceptual aid: statements that refer to the underlying simplicial set, such as equality of vertices or horn fillers, are statements about the model rather than about the abstract \(\infty \)-category. In the rest of the book we therefore continue to formulate everything model-independently.