A homotopy from the continuous map \(f : X \to Y\) to the continuous map \(g : X \to Y\) is a continuous map

\[ F : X \times [0,1] \to Y \]

such that

\[ F(x, 0) = f(x) \text{ for all } x \in X \]

\[ F(x, 1) = g(x) \text{ for all } x \in X. \]

If there exists a homotopy from \(f\) to \(g\), we say that \(f\) is homotopic to \(g\) and write \(f \sim g\). This is an equivalence relation:

\[ f \sim g \]

\[ f \sim g \Rightarrow g \sim f \]

\[ f \sim g \text{ and } g \sim h \Rightarrow f \sim h \]

The map \(f : X \to Y\) is called a homotopy equivalence, if there exists a map \(g : Y \to X\) such that

\[ f \circ g \sim \mathrm{id}_Y \]

\[ g \circ f \sim \mathrm{id}_X \]

If there exists a homotopy equivalence \(f : X \to Y\) we say that \(X\) and \(Y\) are homotopy equivalent and write \(X \simeq Y\).

Let \(*\) a space with one point. If \(X \simeq *\), we say that X is a contractible space.

Ex. \(\mathbb{R}^n\) is contractible.

\(\mathbb{R}^n \setminus \{0\} \simeq S^{n-1} = \{ x \in \mathbb{R}^n \mid \| x \| = 1 \}\).

We will associate a space \(B \C\) to every category \(\C\). First, we introduce simplicial sets.

Let \(\Delta\) be the categoy whose objects are the ordered sets

\[[n] = \{ 0 < 1 < 2 < \cdots < n \} \quad (n \geq 0)\]

and whose morphisms are all non-decreasing maps. We let \(\mathbb{R}[n]\) be the \(\mathbb{R}\)-vector space with basis \([n]\) and define

\[ \Delta [n] \subset \mathbb{R}[n] \]

to be the convex hull of the set of basis elements \([n] \subset \mathbb{R}[n]\):

[image]

So every \(x \in \Delta [n]\) can be written uniquely as

\[ x = \sum_{i \in [n]} a_i \cdot i \]

where \(a_i \in [0, 1]\) and

\[ \sum_{i \in [n]} a_i = 1. \]

The map \(\theta : [m] \to [n]\) gives rise to continuous map

\[ \theta_* : \Delta [m] \to \Delta [n] \]

defined by

\[ \begin{align*} & \theta_* \left( \sum_{i \in [m]} a_i \cdot i \right) \\[1em] &= \sum_{i \in [m]} a_i \cdot \theta(i) \\ &= \sum_{j \in [n]} \left( \sum_{i \in \theta^{-1} (j)} a_i \right) \cdot j \end{align*} \]

This defines a functor

\[ \Delta[-] : \Delta \to \J \]

from \(\Delta\) to the category \(\J\) of topological spaces and continuous maps.

\[ \begin{array}{ccc} [m] & \mapsto & \Delta[m] \\ \downarrow \theta & \mapsto & \downarrow \theta_* \\ [n] & \mapsto & \Delta[n] \end{array} \]

A simplicial set is a functor

\[ \begin{array}{lccc} X[-] : & \Delta^{\op} & \to & \Sets \\[0.5em] & [m] & \mapsto & X[m] \\ & \downarrow \theta & \mapsto & \uparrow \theta^* \\ & [n] & \mapsto & X[n] \end{array} \]

This is a “recipe” for building a space out of simplices: We define the geometric realization of \(X[-]\) to be the space

\[ | X[-] | := \left( \coprod_{n \geq 0} X[n] \times \Delta[n] \right) / \sim \]

where \(\sim\) is the equivalence relation generated by the relation that, for all \(\theta : [m] \to [n]\) in \(\Delta\), all \(x \in X[n]\), and all \(z \in \Delta[m]\), identifies

\[ \begin{array}{ccc} (x, \theta_*(z)) & \in & X[n] \times \Delta[n] \\ \sim & ~ & ~ \\ (\theta^*(x), z) & \in & X[m] \times \Delta[m]. \end{array} \]

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