Warning 1.2.0.1. Let $f_0, f_1: S \rightarrow T$ be morphisms of simplicial sets. We define a homotopy from $f_0$ to $f_1$ to be a morphism of simplicial sets $h: \Delta ^1 \times S \rightarrow T$ satisfying $h|_{ \{ 0\} \times S} = f_0$ and $h|_{ \{ 1\} \times S} = f_1$ (Definition 3.1.5.2). In the special case where $T = \operatorname{Sing}_{\bullet }(X)$ is the singular simplicial set of a topological space $X$, this recovers the usual definition of homotopy between the associated continuous functions $F_0, F_1: |S| \rightarrow X$ (Example 3.1.5.5). Beware that, if $T$ is a general simplicial set, then the definition of homotopy is not symmetric: the existence of a homotopy from $f_0$ to $f_1$ does not imply the existence of a homotopy from $f_1$ to $f_0$ (for example, take $T = \Delta ^1$ to be the standard simplex, and $f_ i: \{ i\} \hookrightarrow \Delta ^1$ to be the inclusion maps).
1.2 From Topological Spaces to Simplicial Sets
Simplicial sets are connected to algebraic topology by two closely related constructions:
To every topological space $X$, one can associate a simplicial set $\operatorname{Sing}_{\bullet }(X)$, whose $n$-simplices are given by continuous functions from the topological $n$-simplex
\[ | \Delta ^{n} | = \{ (t_0, t_1, \ldots , t_ n) \in [0,1]^{n+1} : t_0 + t_1 + \cdots + t_ n = 1 \} \]to $X$. We will refer to $\operatorname{Sing}_{\bullet }(X)$ as the singular simplicial set of $X$ (Construction 1.2.2.2). These simplicial sets tend to be quite large: in any nontrivial example, the sets $\operatorname{Sing}_{n}(X)$ will be uncountable for every nonnegative integer $n$.
Any simplicial set $S_{\bullet }$ can be regarded as a “blueprint” for constructing a topological space $| S_{\bullet } |$ called the geometric realization of $S_{\bullet }$, which can be obtained as a quotient of the disjoint union $\coprod _{n \geq 0} S_ n \times | \Delta ^ n |$ by an equivalence relation determined by the face and degeneracy operators of $S_{\bullet }$. Many topological spaces of interest (for example, any space which admits a finite triangulation) can be realized as a geometric realization of a simplicial set $S_{\bullet }$ having only finitely many nondegenerate simplices.
These constructions determine adjoint functors
relating the category $\operatorname{Set_{\Delta }}$ of simplicial sets to the category $\operatorname{Top}$ of topological spaces. We review the constructions of these functors in §1.2.2 and §1.2.3, viewing them as instances of a general paradigm (Variant 1.2.2.8 and Proposition 1.2.3.15) which will appear repeatedly in Chapter 2.
Under mild assumptions, the entire homotopy type of $X$ can be recovered from the simplicial set $\operatorname{Sing}_{\bullet }(X)$. More precisely, there is a canonical map $| \operatorname{Sing}_{\bullet }(X) | \rightarrow X$ (given by the counit of the preceding adjunction), and Giever showed that it is always a weak homotopy equivalence (hence a homotopy equivalence when $X$ has the homotopy type of a CW complex; see Proposition 3.6.3.8). Consequently, for the purpose of studying homotopy theory, nothing is lost by replacing $X$ by $\operatorname{Sing}_{\bullet }(X)$ and working in the setting of simplicial sets, rather than topological spaces. In fact, it is possible to develop the theory of algebraic topology in entirely combinatorial terms, using simplicial sets as surrogates for topological spaces. In §1.2.1, we consider a simple example of this idea. We say that a simplicial set is connected if it is nonempty and cannot be decomposed as a disjoint union of nonempty simplicial subsets (Definition 1.2.1.6). Every simplicial set $S$ decomposes uniquely as disjoint union of connected simplicial subsets (Proposition 1.2.1.13), indexed by a set which we denote by $\pi _0(S)$. In the special case where $S = \operatorname{Sing}_{\bullet }(X)$ is the singular simplicial set of a topological space $X$, this construction recovers the set $\pi _0(X)$ of path components of $X$ (Remark 1.2.2.5).
The discussion of connectedness in §1.2.1 illustrates a general phenomenon: many useful concepts from topology have combinatorial counterparts in the setting of simplicial sets. However, one must take some care when applying those concepts to simplicial sets which are not of the form $\operatorname{Sing}_{\bullet }(X)$.
In §1.2.5, we introduce a class of simplicial sets called Kan complexes, for which the bad behavior described in Warning 1.2.0.1 cannot occur: if $T$ is a Kan complex and $S$ is any simplicial set, then homotopy determines an equivalence relation on the collection of morphisms $f: S \rightarrow T$ (see Proposition 3.1.5.4). By definition, $T$ is a Kan complex if it satisfies an extension condition with respect to certain maps of simplicial sets $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ called horn inclusions, which we introduce in §1.2.4. For every topological space $X$, the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is a Kan complex (Proposition 1.2.5.8). Moreover, a classical theorem of Milnor ([MR0084138]) guarantees that the functor $X \mapsto \operatorname{Sing}_{\bullet }(X)$ induces an equivalence from the homotopy category of CW complexes to the homotopy category of Kan complexes. In particular, every Kan complex $T$ is homotopy equivalent to a Kan complex of the form $\operatorname{Sing}_{\bullet }(X)$, where $X$ is a topological space (in fact, we can take $X$ to be the geometric realization $|T|$; see Theorem 3.6.4.1). Heuristically, one can think that Kan complexes are simplicial sets which “behave like” the singular simplicial sets of topological spaces. However, there are many other examples having a more combinatorial flavor: for example, any simplicial set which admits a group structure is automatically a Kan complex (Proposition 1.2.5.9).