# Infinite loop space of ring spectra: the cup product

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

• 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 $E_p\wedge E_q$. Nov 26, 2019 at 18:48
• Thanks Neil, what a lapse not to have noticed it. I write it as an answer just to mark the question as solved. Nov 26, 2019 at 19:21
• 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 $Σ^pE∧Σ^qE→Σ^{p+q}E$? Nov 27, 2019 at 10:04

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 $$\mathbb{E}^p(\underline{\phantom{a}})\otimes \mathbb{E}^q(\underline{\phantom{a}})$$.