n-categories and Day convolution

Glasman’s theorem about $\infty$-categorical Day convolutions shows: Let $X^{\otimes },Y^{\otimes }$ be a symmetric monoidal $\infty$-categories, we have an equivalence

$Fu{n}^{lax-\otimes }(X^{\otimes },Y^{\otimes })\simeq CAlg(Fun(X,Y{)}^{Day})$ Example: dualizable objects $Fu{n}^{\otimes }(nBor{d}^{\bigsqcup },(n+1)P{r}^{L,\otimes })\subset CAlg(Fun(nBord,(n+1)P{r}^L{)}^{Day})$ where the inclusion is from strong symmetric monoidal functors to lax ones. Hence, by cobordism hypothesis, the data of a fully dualizable object has a description as an $E_{\infty }$-algebra object in a (non-symmetric monoidal!) functor category, with respect to Day convolution.

Example: filtered rings: $CAlg(Fil=Fun({\mathbb{Z}}^{\downarrow },Sp{)}^{Day})\simeq Fu{n}^{lax-\otimes }({\mathbb{Z}}^{\downarrow ,+},S{p}^{\otimes })$ A filtered $E_{\infty }$ ring spectrum is the data of a symmetric monoidal functor from the symmetric monoidal poset $\mathbb{Z}$ to $Sp$.

Example: categorified presheaves Let $C^{\otimes }$ be a symmetric monoidal $\infty$-category $CAlg(nP(C)=Fun(C^{op},nP{r}^L{)}^{Day})\simeq Fu{n}^{lax-\otimes }(C^{op,\otimes },nP{r}^{L,\otimes })$ Example: 6-functor formalisms In the above example, take $C=Span(Z)$, where $Z$ is some category of geometric objects, like schemes - in particular $Z$ has all finite products, and hence a Cartesian monoidal structure, and this induces a monoidal structure on $Span(Z)$. Lukas Mann defines a 6-functor formalism as an object of the LHS $Fu{n}^{lax-\otimes }(Span(C{)}^{\times },P{r}^{L,\otimes })\simeq CAlg(Fun(Span(C{)}^{op},P{r}^L))$