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I have a basic question on homotopy theory, and I would welcome answers or references both from the classic and the motivic context of homotopy theory.

Let $\mathbb{E}=(E_n)_{n\in \mathbb{N}}$ be an $\Omega$-spectrum which is also a ring spectrum. That is to say, there is a map $\mu\colon \mathbb{E} \wedge \mathbb{E}\to \mathbb{E}$ satisfying the usual properties so that the cohomology defined by $\mathbb{E}$ has a cup-product $$ \mathbb{E}^{p}(\underline{\phantom{a}})\otimes \mathbb{E}^q (\underline{\phantom{a}})\to \mathbb{E} ^{p+q}(\underline{\phantom{a}}) \ $$ A fact from classic homotopy theory is that $E_0=\Omega^\infty \mathbb{E}$ is also a ring object (since the functor $\mathbb{E}^{0}(\underline{\phantom{a}})$ maps spaces to rings). Hence, there is a map $u\colon E_0\times E_0\to E_0$ with usual properties representing the cup-product $\mathbb{E}^{0}(\underline{\phantom{a}})\otimes \mathbb{E}^0 (\underline{\phantom{a}})\to \mathbb{E} ^{0}(\underline{\phantom{a}})$. Since $E_n=\Omega^\infty \mathbb{E}[n]$ represents the groups $\mathbb{E}^n(\underline{\phantom{a}})$, my question is the following:

Question:$$\mbox{Are there maps } u_{p,q}\colon E_p\times E_q\to E_{p+q} \,, \, p,q\in \mathbb{N}\, ,\, \mbox{ inducing the cup product}? $$ $$\mbox{Maybe maps from } E_p\wedge E_q \ \mbox{ to } \ E_{p+q}?$$

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    $\begingroup$ Yes, the Yoneda lemma guarantees the existence of the map $E_p\times E_q\to E_{p+q}$, and an easy auxiliary argument shows that it factors through $E_p\wedge E_q$. $\endgroup$
    – Neil Strickland
    Nov 26, 2019 at 18:48
  • $\begingroup$ Thanks Neil, what a lapse not to have noticed it. I write it as an answer just to mark the question as solved. $\endgroup$
    – Tintin
    Nov 26, 2019 at 19:21
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    $\begingroup$ Can't you just use the fact that $\Omega^\infty$ is lax symmetric monoidal for the smash product in the source and the target (the "fact from classical homotopy theory" I assume you are alluding to) applied to the maps $Σ^pE∧Σ^qE→Σ^{p+q}E$? $\endgroup$
    – Denis Nardin
    Nov 27, 2019 at 10:04

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As Neil Strickland points out at the comments section: the existence of $u_{p,q}$ follows from Yoneda lemma, as in the case of $u_{0,0}$, since $E_p\times E_q$ represents $\mathbb{E}^p(\underline{\phantom{a}})\otimes \mathbb{E}^q(\underline{\phantom{a}})$.

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