nLab extended power

Definition

Write $\mathbb{F}_2 \coloneqq \mathbb{Z}/2\mathbb{Z}$ for the field with two elements.

For $V$ an $\mathbb{F}_2$-module, hence an $\mathbb{F}_2$-vector space, and for $n \in \mathbb{N}$, write

$$V^{\otimes n}_{h \Sigma_n} \in \mathbb{F}_2 Mod$$

for the homotopy quotient of the $n$-fold tensor product of $V$ with itself by the action of the symmetric group. Explicitly this is presented, up to quasi-isomorphism by the ordinary coinvariants $D_n(V)$ of the tensor product of $V^{\otimes n}$ with a free resolution $E \Sigma_n^\bullet$ of $\mathbb{F}_2$:

$$V^{\otimes n}_{h \Sigma_n} \simeq D_n(V) \coloneqq (V^{\otimes n} \otimes E\Sigma_n)_{\Sigma_n} \,.$$

This is called the $n$th extended power of $V$.

Application

• Steenrod square

Related concepts

• tensor power

• H-infinity ring spectrum

References

• Jacob Lurie, 18.917 Topics in Algebraic Topology: The Sullivan Conjecture, Fall 2007. (MIT OpenCourseWare: Massachusetts Institute of Technology), Lecture notes

  Lecture 2 Steenrod operations (pdf)