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 contravariant functor $ X \colon \Delta \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, 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:

\[\xymatrix{ \{1\} \ar@{.>}[dr] \\ \{0\} \ar[u] \ar[r] & \{2\} } \quad \xymatrix{ \{1\} \ar[dr] \\ \{0\} \ar[u] \ar@{.>}[r] & \{2\} } \quad \xymatrix{ \{1\} \ar[dr] \\ \{0\} \ar@{.>}[u] \ar[r] & \{2\} }\]

Here, when performing the operation "\(\cup \{ i \}\)", we add any arrows touching the vertex \(\{ i \}\) to the diagram.

  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 topological $ n $-simplex in $ \mathbb{R}^n $. 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

\[\vert \Delta^m \vert \to \vert \Delta^n \vert, \qquad (t_0, \dots, t_n) \to \left( \sum_{f(i)= 0} t_i, \dots, \sum_{f(i)= n} t_i \right).\]

$ \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:

\[\xymatrix{ \text{Hom}_{Top}(\vert \Delta^n \vert, A) \ar[r]^{\cong} \ar[d]_{\vert \iota \vert^*} & \text{Hom}_{Set_{\Delta}}(\Delta^n, \text{Sing}(A)) \ar[d]^{\iota^*} \\ \text{Hom}_{Top}(\vert \Gamma_i^n \vert, A) \ar[r]^{\cong} & \text{Hom}_{Set_{\Delta}}(\Lambda_i^n, \text{Sing}(A)) }\]

This reduces the problem of lifting $ f_0 \colon \Lambda_i^n \to \text{Sing}(A) $ to lifting the associated $ \vert f_0 \vert \colon \vert \Lambda_i^n \vert \to A $. Let $ r \colon \Delta^n \to \Lambda_i^n $ be a continuous retract. We conclude that $ \vert f \vert \colon \vert \Delta^n \vert \to A $ given by $ \vert f \vert = \vert f_0 \vert \circ r $ is our desired map. 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

\[\xymatrix{ C_1 \ar[r]^{f_1} & C_2 \ar[r]^{f_2} & \cdots \ar[r]^{f_n} & C_n. }\]

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) $   Let $ g_k \colon X_{k-1} \to X_k $ denote the restriction $ f_0 \mid \Delta^{{ k-1, k }} $. Composing our $ g_k $,

\[\xymatrix{ X_1 \ar[r]^{g_1} & X_2 \ar[r]^{g_2} & \cdots \ar[r]^{g_n} & X_n, }\]

determines an $ n $-simplex $ f \colon \Delta^n \to X $.

$ (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 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.

  Up to a notion of homotopy equivalence, $ \infty $-categories are equivalent to $ (\infty, 1) $-categories. That is categories with $ n $-morphisms for each $ n \in \mathbb{N} $, where the $ n $-morphisms for $ n > 1 $ are invertible. Lurie proves this in 1.1. of [3].

  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

\[\text{Hom}_{\text{Set}_{\Delta}}(\Delta, N(C)) \cong \text{Hom}_{\text{Cat}_{\Delta}}(\mathfrak{C}[\Delta^n], C)\]

for some simplicial category $ \mathfrak{C}[\Delta^n] $ 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 conunit

\[\vert \text{Hom}_{\mathfrak{C}[N(T)]}(x, y) \vert \to \text{Hom}_T(x, y)\]

is a weak homotopy equivalence of topological spaces. Since we are interested in objects up to homotopy equivalence, we consider $ \infty $-categories and topological categories to be the same.

References

  1. D. Gaitsgory and N. Rozenblyum, A study in derived algebraic geometry, volume I: Correspondences and duality, American Mathematical Society, Mathematical Surveys and Monographs, Volume 221, 2017.
  2. A. Krause and T. Nikolaus, Lectures on topological Hochschild homology and cyclotomic spectra, lecture notes.
  3. J. Lurie, Higher Topos Theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, Princeton, NJ, 2009.
  4. ————, Higher Algebra, 2017.
  5. ————, Kerodon. Link
  6. D. Quillen, Homotopical algbera, Springer lecture notes in mathematics, 1965.