Background - Strictification as Trivialization

Credit: This was explained by Carmelli in a beautiful talk at Minnesoda

Let $X$ be a spectrum. We can perform (0-)semi-additive integration along the discrete anima with $m$ components, which is a fun way to say we’re using the fact that product and coproduct coincide to compute the diagonal and codiagonal/fold maps on an object $X\stackrel{\Delta }{\to }{X}^{\oplus m}\stackrel{\nabla }{\to }X$ In terms of the underlying $E_{\infty }$ group of elements, this operation is adding an element to itself $m$ times: $x\mapsto x+x+\cdots +x$ In particular, we see that the image of the map has a permutation symmetry that we can exploit. This has the following precise incarnation, which is that the above map refines to a map $X\to X^{B{\Sigma }_m}:=ma{p}_{Sp}({\Sigma }_{+}^{\infty }B{\Sigma }_m,X)$ this comes from induction/transfer of local systems along the map $pt\to B{\Sigma }_m$.

Remark: By adjunction this is of course equivalent to data ${\Sigma }_{+}^{\infty }B{\Sigma }_m\otimes X\to X$. This coresponds to a “$X$-homological” (rather than $X$-cohomological) version of this discussion.