Tate diagonal

While the symmetric monoidal (∞,1)-category of spectra is not a cartesian monoidal (∞,1)-category, hence while the smash product of spectra does not admit diagonal morphisms, it turns out that for each prime number $p$ there is a analogue of the diagonal into the $p$-fold smash product. This is the *Tate diagonal* (def. ). One way that this really does behave like a diagonal map is the construction of the Frobenius morphism on E-∞ rings (see this def. and this example)

For $n \in \mathbb{N}$ a natural number, write $C_p \coloneqq \mathbb{Z}/n\mathbb{Z}$ for the cyclic group of order $n$.

For $X \in Spectra$ a spectrum and $n$ a natural number, consider the Tate spectrum

$(X \wedge \cdots \wedge X)^{t C_n} \in Spectra$

where the $n$-fold smash product of spectra is regarded as equipped with the ∞-action by the cyclic group given by cyclic permutation of smash factors. This construction canonically extends to an (∞,1)-functor

$T_n
\;\colon\;
Spectra
\longrightarrow
Spectra$

on the (∞,1)-category of spectra.

For $p$ a prime number, the ∞-groupoid of natural transformations

$id_{spectra} \longrightarrow T_p$

from the identity functor on the (∞,1)-category of spectra to the (∞,1)-functor $T_p$ from def. is equivalent to the underlying space of $T_p$ applied to the sphere spectrum:

$Hom( id_{Specta}, T_p )
\;\simeq\;
\Omega^\infty T_p(\mathbb{S})
\simeq
Hom_{Specta}(\mathbb{S}, T_p(\mathbb{S}))
\,.$

For $p$ a prime number, the *Tate diagonal* is the natural transformation

$\array{
id_{Specta} &\overset{\Delta_p}{\longrightarrow}& T_p
\\
X &\mapsto& (X \wedge \cdots X)^{t C_p}
}$

is the one which under the equivalence of prop. corresponds to the composite morphism

$\mathbb{S}
\overset{}{\longrightarrow}
\mathbb{S}^{C_p}
\overset{}{\longrightarrow}
\mathbb{S}^{t C_p}$

where the first is the unit map into the homotopy fixed point spectrum, and the second is the defining one of the Tate spectrum.

(Nikolaus-Scholze 17, def. III.1.4)

- Thomas Nikolaus, Peter Scholze,
*On topological cyclic homology*(arXiv:1707.01799)