nLab
orthogonal space

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Definition

The category of orthogonal spaces is the category of topological functors from the topological category L\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:XYf\colon X\to Y is a global equivalence if for any compact Lie group GG, any continuous orthogonal representation VV of GG on a finite-dimensional real inner product space, any k0k\ge0, and any continuous maps α:D kX(V) G\alpha\colon\partial D^k\to X(V)^G and β:D kY(V) G\beta\colon D^k\to Y(V)^G such that β| D k=f(V) Gα\beta|_{\partial D^k}=f(V)^G\circ\alpha, we can find a continuous orthogonal representation WW of GG, a GG-equivariant linear isometric embedding ϕ:VW\phi\colon V\to W and a continuous map λ:D kX(W) G\lambda\colon D^k\to X(W)^G such that λ| D k=X(ϕ) Gα\lambda|_{\partial D^k}=X(\phi)^G\circ\alpha and f(W) Gλf(W)^G\circ\lambda is homotopic to Y(ϕ) GβY(\phi)^G\circ\beta relative D k\partial D^k.

In particular, if XX and YY are constant functors, then f:XYf\colon X\to Y is a global equivalence if and only if f(0):X(0)Y(0)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:XYf\colon X\to Y such that for any compact Lie group GG and for some (hence any) complete GG-universe UU, the canonical map of underlying GG-spaces

hocolim Vf(V):hocolim VX(V)hocolim VY(V)hocolim_V f(V)\colon hocolim_V X(V)\to hocolim_V Y(V)

is a GG-equivariant weak equivalence? (i.e., a weak homotopy equivalence on HH-fixed points for any closed subgroup HH of GG), where VV runs over all finite-dimensional subrepresentations of UU.

Example: the global classifying space of a compact Lie group

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

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

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

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

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

are global equivalences.

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

References

Last revised on November 21, 2020 at 23:07:09. See the history of this page for a list of all contributions to it.