Suppose I have a coalgebra $\mathcal{C}$ in the $\infty$-category of presentable stable $\infty$-categories (with continuous functors). If the structure functors for $\mathcal{C}$ all have continuous right adjoints, then passing to right adjoints I get an algebra structure on $\mathcal{C}$. However, in the general situation, I only have a full stable subcategory $\mathcal{D}$ of $\mathcal{C}$ such that the right adjoints preserve $\mathcal{D}$, i.e. they induce $\mathcal{D}\otimes \mathcal{D}\rightarrow\mathcal{D}$, and restricting to $\mathcal{D}$ they are continuous. Note that $\mathcal{D}$ is not necessarily a sub-coalgebra of $\mathcal{C}$.

Is there a clean way to show that in the above situation, $\mathcal{D}$ naturally induces an algebra structure? All I can say is that all the homotopical data needed are contained in the coalgebra structure of $\mathcal{C}$, but this seems not so clean.

  • $\begingroup$ So if I read correctly, you have a monoidal structure on $\mathcal{C}$ (obtained by the right adjoint of your coalgebra structure) and a full subcategory $\mathcal{D} \subset \mathcal{C}$ stable under the tensor product and you want to say it is a monoidal category ? I assume that either you are working without unit or you also assume that the unit is in $\mathbb{D}$ right ? $\endgroup$ – Simon Henry May 14 at 18:32
  • $\begingroup$ Let's say the unit is in $\mathcal{D}$. The thing that I'm confused about is that it seems the tensor product on the category of presentable stable $\infty$-categories with continuous functors don't extend to the category with all functors. I'm not sure whether that causes any issue on a monoidal structure on $\mathcal{C}$. $\endgroup$ – Wonderfield May 14 at 20:34

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