nLab orthogonal space

Definition

The category of orthogonal spaces is the category of topological functors from the topological category $\mathbf{L}$ of finite-dimensional real inner product spaces (with linear isometric embeddings) to the topological category of topological spaces (with continuous maps).

Here a topological space, or simply a space, refers to an object in a convenient category of spaces, namely, compactly generated weakly Hausdorff space.

Orthogonal spaces are turned into a relative category using the notion of a global equivalence, which are introduced via an equivariant version of the Whitehead theorem. A morphism $f\colon X\to Y$ is a global equivalence if for any compact Lie group $G$, any continuous orthogonal representation $V$ of $G$ on a finite-dimensional real inner product space, any $k\ge0$, and any continuous maps $\alpha\colon\partial D^k\to X(V)^G$ and $\beta\colon D^k\to Y(V)^G$ such that $\beta|_{\partial D^k}=f(V)^G\circ\alpha$, we can find a continuous orthogonal representation $W$ of $G$, a $G$-equivariant linear isometric embedding $\phi\colon V\to W$ and a continuous map $\lambda\colon D^k\to X(W)^G$ such that $\lambda|_{\partial D^k}=X(\phi)^G\circ\alpha$ and $f(W)^G\circ\lambda$ is homotopic to $Y(\phi)^G\circ\beta$ relative $\partial D^k$.

In particular, if $X$ and $Y$ are constant functors, then $f\colon X\to Y$ is a global equivalence if and only if $f(0)\colon X(0)\to Y(0)$ is a weak homotopy equivalence of topological spaces.

Global equivalences can also be characterized as morphisms $f\colon X\to Y$ such that for any compact Lie group $G$ and for some (hence any) complete $G$-universe $U$, the canonical map of underlying $G$-spaces

$$hocolim_V f(V)\colon hocolim_V X(V)\to hocolim_V Y(V)$$

is a $G$-equivariant weak equivalence? (i.e., a weak homotopy equivalence on $H$-fixed points for any closed subgroup $H$ of $G$), where $V$ runs over all finite-dimensional subrepresentations of $U$.

Example: the global classifying space of a compact Lie group

The global classifying space of a compact Lie group is defined as

$$\mathrm{B}_{gl} G = \mathbf{L}_{G,V} = \mathbf{L}(V,-)/G,$$

where $V$ is any faithful representation of $G$. Here $\mathbf{L}(V,-)$ denotes the enriched hom in the topological category $\mathbf{L}$.

If $V'$ is another such faithful representation, then the canonical maps

$$\mathbf{L}_{G,V} \gets \mathbf{L}_{G,V\oplus V'} \to \mathbf{L}_{G,V'}$$

are global equivalences.

For example, if $G=Z/2Z$, then the underlying nonequivariant space of $\mathrm{B}_{gl} G$ is $\mathbf{RP}^\infty$.

References

• Stefan Schwede, Global homotopy theory.