nLab infinite projective space

Definition

Given a (topological) ground field $\mathbb{K}$ (or more generally a topological ground ring), its projective spaces $\mathbb{K}P^n$ canonical form a sequence of subspaces

(1)$$\begin{array}{ccc} \mathbb{K}P^n &\xhookrightarrow{\phantom{--}}& \mathbb{K}P^{n+1} \\ [x_1 \colon \cdots \colon x_{n+1}] &\mapsto& [x_1 \colon \cdots \colon x_{n+1} \colon 0] \mathrlap{\,.} \end{array}$$

Definition

The infinite projective space $\mathbb{K}P^\infty$ over $\mathbb{K}$ is the union of the $\mathbb{K}P^n$, hence the colimit over the above sequence (1):

$$\mathbb{K}P^\infty \;\coloneqq\; \underset{\underset{n \to \infty}{\longrightarrow}}{\lim} \, \mathbb{K}P^n \,.$$

Examples

Example

The infinite projective space over the real numbers, $\mathbb{K} \coloneqq \mathbb{R}$, is a model for (the homotopy type of) the classifying space of the cyclic group of order 2:

$$\mathbb{R}P^\infty \;\simeq\; B C_2 \,.$$

Example

The infinite projective space over the complex numbers, $\mathbb{K} \coloneqq \mathbb{C}$, is a model for (the homotopy type of) the classifying space of the circle group U(1):

$$\mathbb{C}P^\infty \;\simeq\; B \mathrm{U}(1) \,.$$

Example

The infinite projective space over the quaternions, $\mathbb{K} \coloneqq \mathbb{H}$, is a model for (the homotopy type of) the classifying space of the quaternion unitary group Sp(1):

$$\mathbb{H}P^\infty \;\simeq\; B \mathrm{Sp}(1) \,.$$

Related entries

• projective line

• projective plane