The simplex category has objects which are linearly ordered sets \([n] = \{ 0 < 1 < \dots < n \},\) and it has morphisms which are functions $ f \colon [m] \to [n] $ that respect the ordering, i.e., $ i \leq j $ implies $ f(i) \leq f(j) $. We denote the simplex category by the letter $ \Delta $.
A presheaf of sets on $ \Delta $, i.e., a functor $ X \colon \Delta^{\mathrm{op}} \to \text{Set} $, is called a simplicial set; we denote the image $ X([n]) $ by $ X_n $. A morphism of simplicial sets is a natural transformation of functors. Let us write $ \text{Set}_{\Delta} $ for the category of simplicial sets.
The simplicial set
\[\Delta^n([m]) = \text{Hom}_{\Delta}([m], [n])\]is called the standard $ n $-simplex. The Yoneda lemma tells us
\[X_n = \text{Hom}_{\text{Set}_{\Delta}}(\Delta^n, X).\]Likewise, for $n\geq1$ and $0\leq i\leq n$, the simplicial subset of $ \Delta^n $
\[\Lambda_i^n([m]) = \{f \in \Delta^n([m]) \mid f([m]) \cup \{ i \} \neq [n] \}\]is called the $ i $-th horn; it is referred to as being inner if $ 0 < i < n $ and outer if $ i = 0, n $. Lurie visualizes $ \Lambda_0^2 $, $ \Lambda_1^2 $, and $ \Lambda_2^2 $, respectively, as follows:
\[\xymatrix{ \{1\} \ar@{.>}[dr] \\ \{0\} \ar[u] \ar[r] & \{2\} } \quad \xymatrix{ \{1\} \ar[dr] \\ \{0\} \ar[u] \ar@{.>}[r] & \{2\} } \quad \xymatrix{ \{1\} \ar[dr] \\ \{0\} \ar@{.>}[u] \ar[r] & \{2\} }\]The condition says that, even after adding the vertex $ i $ to the image of $ f $, at least one vertex of $ [n] $ is still missing. Equivalently, $ \Lambda_i^n $ is the union of all codimension-one faces of $ \Delta^n $ except the face opposite the vertex $ i $.
Let $ X $ be a simplicial set and $ \iota \colon \Lambda_i^n \hookrightarrow \Delta^n $ the inclusion. We say $ X $ is a Kan complex if, for any horn $ \Lambda_i^n $ and morphism $ f_0 \colon \Lambda_i^n \to X $, there exists a morphism $ f \colon \Delta^n \to X $ such that $ f \circ \iota = f_0 $; pictorially, the following diagram must commute:
\[\xymatrix{ \Lambda_i^n \ar[r]^{f_0} \ar[d]_{\iota} & X \\ \Delta^n \ar@{.>}[ur]_{f} }\]Example 1. Let $ A $ be a compactly generated topological space. We define a simplicial set $ \text{Sing}(A) $ as follows. Write $ \vert \Delta^n \vert $ for the geometric realization of $ \Delta^n $, i.e., the standard topological $ n $-simplex in $ \mathbb{R}^{n+1} $. We put
\[\text{Sing}_n(A) = \text{Hom}_{\text{Top}}(\vert \Delta^n \vert, A)\]to be the set of singular $ n $-simplices. Each $ f \colon [m] \to [n] $ determines a morphism $ \text{Sing}_n(A) \to \text{Sing}_m(A) $ by precomposing with the map
\[\vert \Delta^m \vert \to \vert \Delta^n \vert, \qquad (t_0, \dots, t_m) \longmapsto \left( \sum_{\substack{0 \leq i \leq m \\ f(i)= 0}} t_i, \dots, \sum_{\substack{0 \leq i \leq m \\ f(i)= n}} t_i \right).\]$ \text{Sing} $ is a functor from topological spaces to simplicial sets, whose left adjoint is the geometric realization functor $ \vert \cdot \vert $.
Proposition 2. $ \text{Sing}(A) $ is a Kan complex.
Proof. The adjunction $ \vert \cdot \vert \dashv \text{Sing}(\cdot) $ implies the following diagram:
\[\xymatrix{ \text{Hom}_{Top}(\vert \Delta^n \vert, A) \ar[r]^{\cong} \ar[d]_{\vert \iota \vert^*} & \text{Hom}_{Set_{\Delta}}(\Delta^n, \text{Sing}(A)) \ar[d]^{\iota^*} \\ \text{Hom}_{Top}(\vert \Lambda_i^n \vert, A) \ar[r]^{\cong} & \text{Hom}_{Set_{\Delta}}(\Lambda_i^n, \text{Sing}(A)) }\]This reduces the problem of lifting $ f_0 \colon \Lambda_i^n \to \text{Sing}(A) $ to extending the associated map $ \vert f_0 \vert \colon \vert \Lambda_i^n \vert \to A $. There is a continuous retraction $ r \colon \vert \Delta^n \vert \to \vert \Lambda_i^n \vert $. We conclude that the continuous map $ \vert f \vert \colon \vert \Delta^n \vert \to A $ given by $ \vert f \vert = \vert f_0 \vert \circ r $ corresponds under the adjunction to our desired filler. Q.E.D.
Example 3. Let $ \mathcal{C} $ be a small category. Define a simplicial set $ N(\mathcal{C}) $, the nerve of $ \mathcal{C} $, by considering functors
\[N_n(\mathcal{C}) = \text{Fun}([n], \mathcal{C}).\]Here, we are considering $ [n] $ as the category with objects $ {0, 1, \dots, n} $ and arrows $ i \to j $ if $ i \leq j $. Explicitly, objects of $ N_n(\mathcal{C}) $ are composable sequences of morphisms
\[\xymatrix{ C_0 \ar[r]^{f_1} & C_1 \ar[r]^{f_2} & \cdots \ar[r]^{f_n} & C_n. }\]The following proposition tells us we can consider the nerve as a weak Kan complex, meaning it satisfies the Kan lifting condition for all inner horns.
Proposition 4. Let $ X $ be a simplicial set. The following are equivalent:
(1) There exists a small category and an isomorphism $ X \cong N(\mathcal{C}) $.
(2) For each inner horn, $ 0 < i < n $, and diagram
\[\xymatrix{ \Lambda_i^n \ar[r]^{f_0} \ar[d]_{\iota} & X \\ \Delta^n \ar@{.>}[ur]_{f} }\]there exists a unique dotted arrow making it commute.
Proof. This is 1.1.2.2 of [3]. We only sketch a couple of main ideas without providing a complete proof.
$ (1) \implies (2) $ Write $ C_k $ for the vertex of $ \mathcal C $ corresponding to the $ k $th vertex of the horn, and let $ g_k \colon C_{k-1} \to C_k $ be the morphism corresponding to the restriction $ f_0 \mid \Delta^{{ k-1, k }} $. The composable sequence
\[\xymatrix{ C_0 \ar[r]^{g_1} & C_1 \ar[r]^{g_2} & \cdots \ar[r]^{g_n} & C_n, }\]determines a unique functor $ [n] \to \mathcal C $, hence a unique $ n $-simplex $ f \colon \Delta^n \to X $ extending the horn.
$ (2) \implies (1) $ We mention the proof of associativity law of the composition operator. Consider a sequence of morphisms
\[\xymatrix{ w \ar[r]^{f} & x \ar[r]^{g} & y \ar[r]^{h} & z. }\]We have the following 3 faces of the 4-sided 3-simplex:
\[\xymatrix{ x \ar[dr]^{g} \\ w \ar[u]^{f} \ar[r]_{g \circ f} & y } \quad \xymatrix{ y \ar[dr]^{h} \\ x \ar[u]^{g} \ar[r]_{h \circ g} & z } \quad \xymatrix{ y \ar[dr]^{h} \\ w \ar[u]^{g \circ f} \ar[r]_{h \circ (g \circ f)} & z }\]By (2), we get a unique fourth face:
\[\xymatrix{ x \ar[dr]^{h \circ g} \\ w \ar[u]^{f} \ar[r]_{h \circ (g \circ f)} & z }\]Thus, the associativity law $ h \circ (g \circ f) = (h \circ g) \circ f $. Q.E.D.
We define a simplicial set $ X $ to be an $ \infty $-category if it is a weak Kan complex; i.e., for each inner horn, $ 0 < i < n $, and diagram
\[\xymatrix{ \Lambda_i^n \ar[r]^{f_0} \ar[d]_{\iota} & X \\ \Delta^n \ar@{.>}[ur]_{f} }\]there exists a dotted arrow making it commute. Note that the dotted arrow is not required to be unique, contrasting the case of the nerve of a category, and it is not required to exist on outer horns, unlike $ \text{Sing}(A) $. Thus, $ \infty $-categories can be thought of as a generalized framework for small category theory and algebraic topology.
The weak Kan complexes defined above are also called quasicategories. They form one model for $ (\infty, 1) $-categories: higher categories with $ n $-morphisms for each $ n \in \mathbb{N} $, where the $ n $-morphisms for $ n > 1 $ are invertible. Section 1.1 of [3] compares this model with categories enriched in spaces.
In particular, a topological category $ T $ is a category enriched over compactly generated Hausdorff spaces. A simplicial category $ C $ is a category enriched over simplicial sets; we denote this category \(\text{Cat}_{\Delta}\). The simplicial nerve \(N \colon \text{Cat}_{\Delta} \to \text{Set}_{\Delta}\) is characterized by
\[\text{Hom}_{\text{Set}_{\Delta}}(\Delta^n, N(C)) \cong \text{Hom}_{\text{Cat}_{\Delta}}(\mathfrak{C}[\Delta^n], C)\]where $ \mathfrak{C}[\Delta^n] $ is a simplicial category which we will not define here. We set $ N(T) $ to be $ N(\text{Sing}(T)) $. Theorem 1.1.5.13 in [3] asserts that the counit
\[\vert \text{Hom}_{\mathfrak{C}[N(T)]}(x, y) \vert \to \text{Hom}_T(x, y)\]is a weak homotopy equivalence of topological spaces. Thus, after passing to the appropriate notions of categorical and homotopy equivalence, quasicategories and topological categories model the same homotopy theory.
5. References
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