Given symmetric monoidal $\infty$-categories $A, B, C$ and lax symmetric monoidal maps $F:A\to C$, $G:B\to C$, I am curious if the pullback (when I say pullback here I will really mean homotopy pullback) of $F$ along $G$ is necessarily lax symmetric monoidal in a natural way? The pullback is necessarily an $\infty$-operad still, but it doesn't seem immediately clear that one can still get cocartesian lifts of all morphisms, not just inert ones.


I am sort of curious if there is an $\infty$-cosmos (in the sense of Riehl-Verity) of symmetric monoidal $\infty$-categories with lax symmetric monoidal maps. If there were, one could just define it as cosmologically embedded in $\infty$-operads, full on the underlying simplicial categories. This would work if symmetric monoidal $\infty$-categories were stable under pullback along isofibrations in $\infty$-operads (which should be equivalent to the above question). I suspect this would be false, but I can't think of a counterexample.



Can there be a cospan of symmetric monoidal $\infty$-categories whose maps are lax symmetric monoidal but the pullback is not symmetric monoidal?
Oops, this is actually not hard, just using 1-categories. Explicitly, take two maps from the terminal category to Ab, one landing in $\mathbb{Z}$, one landing in 0, both are lax symmetric monoidal. The pullback (after replacing the first map by an equivalent isofibration but we can mostly ignore this technicality) is empty so not a symmetric monoidal $\infty$-category (where it would be an ordinary category anyways). So indeed, symmetric monoidal $\infty$-categories with lax symmetric monoidal functors do not form an $\infty$-cosmos.


Though, if we are pulling back $(G,F):C\to A\times B$ along a two-sided fibration $(p\_1,p\_0):D\to A\times B$ from $A$ to $B$ (which is to say, $p\_1$ is cocartesian, $p\_0$ is cartesian and lifts are compatible), where $p\_1,p\_0,$ and $F$ are symmetric monoidal, while $G$ is lax symmetric monoidal, this should work. This case can be used to show that cyclotomic spectra are symmetric monoidal, for example.


