Shift and suspension of sequential prespectra

For a sequential prespectrum $E$, its shift is

$(\mathrm{sh}E{)}_n=E_{n+1},$

with structure map ${\sigma }_{n+1}:S^1\wedge E_{n+1}\to E_{n+2}$. Its sequential suspension (the right-shifted object) is

$(sE{)}_0=\ast ,(sE{)}_{n+1}=E_n,$

with the unique structure map out of $S^1\wedge \ast$ at level zero and ${\sigma }_{n-1}:S^1\wedge E_{n-1}\to E_n$ at level $n\ge 1$.

With the grading convention ${\pi }_k(E)={\mathrm{colim}}_n{\pi }_{n+k}(E_n)$, there are canonical isomorphisms

$\boxed{{\displaystyle {\pi }_k(\mathrm{sh}E)\cong {\pi }_{k-1}(E),{\pi }_k(sE)\cong {\pi }_{k+1}(E)\mathrm{.}}}$

For the shift, put $m=n+1$ and delete the missing finite initial term:

${\mathrm{colim}}_n{\pi }_{n+k}(E_{n+1})={\mathrm{colim}}_m{\pi }_{m+(k-1)}(E_m)={\pi }_{k-1}(E)\mathrm{.}$

For $sE$, discard its zero level and put $m=n-1$:

${\mathrm{colim}}_{n\ge 1}{\pi }_{n+k}(E_{n-1})={\mathrm{colim}}_{m\ge 0}{\pi }_{m+(k+1)}(E_m)={\pi }_{k+1}(E)\mathrm{.}$

The cofinal-tail lemma makes both reindexings canonical. These signs correct the reversed formulas in the Step-1 scaffold.