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My question is both for the topological and for the algebraic/motivic version of the stable homotopy category $\mathbf{SH}$.

It is well known that most cohomologies are represented in $\mathbf{SH}$ by certain spectra which are monoids (also called ring spectra in this setting). This means that $K$-theory, motivic and singular cohomology, cobordism cohomologies, etc... are represented by certain spectra $\mathbb{E}$ that are endowed with maps $$ \mu \colon \mathbb{E}\otimes \mathbb{E} \to \mathbb{E} \ \ \mbox{ and }\ \ e\colon \mathbb{1}\to \mathbb{E}\ , $$ where $\mathbb{1}$ is the unit object in $\mathbf{SH}$ for the tensor product, satisfying certain axioms (check here). The $\mu$ is called multiplication, and induces the cup and cap products, and $e$ is called the unit and correspond to the unit of the cohomology.

A comonoid (or coalgebra) in $\mathbf{SH}$ is an object $\mathbb{F}$ which is a monoid in the opposite category $\mathbf{SH}^\mathrm{op}$. In other words, $\mathbb{F}$ is endowed with maps $$ \Delta \colon \mathbb{F} \to \mathbb{F} \otimes \mathbb{F} \ \ \mbox{ and } \ \ \xi \colon \mathbb{F} \to \mathbb{1} \ , $$ satisfying certain axioms.

My question is:

Are there relevant examples of comonoids (coalgebras) in either the algebraic/motivic or the topological version of $\mathbf{SH}$?

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    $\begingroup$ By "algebraic" SH, do you mean the stable motivic category? $\endgroup$
    – Denis Nardin
    Commented Jul 7, 2022 at 11:28
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    $\begingroup$ Yes, I mean Voevodsky's stable homotopy category in the $\mathbb{A}^1$-homotopy setting $\endgroup$
    – Tintin
    Commented Jul 7, 2022 at 11:52
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    $\begingroup$ They’re fairly common: every suspension spectrum gives a comonoid, and a lesser known theorem of Bousfield says that in the K(n)-local category every spectrum admits such a structure. The “Thom diagonal” is an adjacent concept, presenting a Thom spectrum as a cotorsor for the comonoid associated to the suspension spectrum of the base space. I don’t know whether there’s an out-and-out classification available. $\endgroup$
    – Eric Peterson
    Commented Jul 7, 2022 at 14:15
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    $\begingroup$ Also the Koszul duals of augmented monoids and descent corings. $\endgroup$
    – Jonathan Beardsley
    Commented Jul 7, 2022 at 15:03

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