nLab G-universe

Idea

In equivariant stable homotopy theory over a compact Lie group $G$, a $G$-universe is a $G$-representation that contains “all” representations of $G$ of sorts.

This is used in one definition of G-spectra via looping and delooping by representation spheres.

Definition

A G-universe in this context is (e.g. Greenlees-May, p. 10) an infinite dimensional real inner product space equipped with a linear $G$-action that is the direct sum of countably many copies of a given set of (finite dimensional) representations of $G$, at least containing the trivial representation on $\mathbb{R}$ (so that $U$ contains at least a copy of $\mathbb{R}^\infty$).

Applications

Infinite complex projective $G$-space

Definition

(infinite complex projective G-space)

For $G$ an abelian compact Lie group, let

(1)$$\mathcal{U}_G \;\coloneqq\; \underset{k \in \mathbb{N}}{\bigoplus} \underset{\mathbf{1}_V \in R(G)}{\bigoplus} \mathbf{1}_V$$

be the G-universe being the infinite direct sum of all complex 1-dimensional linear representations of $G$, regarded as a topological G-space with toplogy the colimit of its finite-dimensional linear subspaces.

Then the infinite complex projective G-space is the colimit

$$P\big( \mathcal{U}_G \big) \;\coloneqq\; \underset{ \underset{ { V \subset \mathcal{U}_G } \atop { dim(V) \lt \infty } }{\longrightarrow} }{\lim} P\big( V \big)$$

of the projective G-spaces for all the finite-dimensional $G$-linear representations inside the G-universe (1).

(e.g. Greenlees 01, Sec. 9.2)

This concept is at the heart of equivariant complex oriented cohomology theory.

Related concepts

• RO(G)-grading

• infinite projective G-space

References

• John Greenlees, Peter May, Equivariant stable homotopy theory (pdf)