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3.1 The Homotopy Theory of Kan Complexes

Let $X_{}$ and $Y_{}$ be simplicial sets, and suppose we are given a pair of maps $f_0, f_1: X_{} \rightarrow Y_{}$. A homotopy from $f_0$ to $f_1$ is a morphism of simplicial sets $h: \Delta ^1 \times X_{} \rightarrow Y_{}$ satisfying $f_0 = h|_{ \{ 0\} \times X_{} }$ and $f_1 = h|_{ \{ 1\} \times Y_{} }$ (Definition 3.1.4.2). Beware that, for general simplicial sets, this terminology can be misleading: for example, the existence of a homotopy from $f_0$ to $f_1$ need not imply the existence of a homotopy from $f_1$ to $f_0$. However, the situation is better in the case if we assume that $Y_{\bullet }$ is a Kan complex. In general, we can identify morphisms from $X_{}$ to $Y_{}$ as vertices of the simplicial set $\operatorname{Fun}( X_{}, Y_{} )$ of Construction 1.4.3.1, and homotopies with edges of the simplicial set $\operatorname{Fun}( X_{}, Y_{} )$. In §3.1.3, we will show that when $Y_{}$ is a Kan complex, then $\operatorname{Fun}( X_{}, Y_{} )$ is also a Kan complex (Corollary 3.1.3.4). In §3.1.4, we exploit this fact to construct an $\infty $-category $\operatorname{Kan}$ (Construction 3.1.4.8) whose objects are Kan complexes, which will play an essential role throughout this book.

Our approach to Corollary 3.1.3.4 is somewhat indirect. We begin in §3.1.1 by introducing the notion of a Kan fibration between simplicial sets. Roughly speaking, a Kan fibration $f: X_{} \rightarrow S_{}$ can be viewed as a family of Kan complexes parametrized by $S_{}$: in particular, if $f$ is a Kan fibration, then each fiber $X_{s} = \{ s\} \times _{ S_{} } X_{}$ is a Kan complex (Remark 3.1.1.6). In §3.1.3, we will deduce Corollary 3.1.3.4 as a consequence of a more general stability result for Kan fibrations under exponentiation (Theorem 3.1.3.1). Our proof of this result uses a characterization of Kan fibrations in terms of homotopy lifting properties (Proposition 3.1.2.4), which we formulate and prove in §3.1.2.

We say that a morphism of Kan complexes $f: X_{} \rightarrow Y_{}$ is a homotopy equivalence if its image in the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ is an isomorphism: that is, if $f$ admits a homotopy inverse $g: Y_{} \rightarrow X_{}$. This definition makes sense for more general simplicial sets (Definition 3.1.5.1), but is of somewhat limited utility. When working with simplicial sets which are not Kan complexes, it is often better to consider the more liberal notion of weak homotopy equivalence (Definition 3.1.5.10), which we introduce and study in §3.1.5. In §3.1.6, we introduce the Gabriel-Zisman calculus of anodyne morphisms, which can be used to construct a large supply of weak homotopy equivalences between simplicial sets (see Proposition 3.1.6.7 and Corollary ). For example, every simplicial set $X_{\bullet }$ admits an anodyne morphism $f: X_{\bullet } \rightarrow Q_{\bullet }$ (Corollary 3.1.7.2): we prove this in §3.1.7, using a simple incarnation of Quillen's “small object argument.”

Structure

  • Subsection 3.1.1: Kan Fibrations
  • Subsection 3.1.2: Left and Right Fibrations
  • Subsection 3.1.3: Exponentiation of Kan Fibrations
  • Subsection 3.1.4: The $\infty $-Category of Kan Complexes
  • Subsection 3.1.5: Homotopy Equivalences and Weak Homotopy Equivalences
  • Subsection 3.1.6: Anodyne Morphisms
  • Subsection 3.1.7: Fibrant Replacement