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This question may be very naive, or the answer may be well-known. In any case, a good amount of googling did not bring up anything useful (maybe I'm using the wrong words?).

If $f:A\to B$ is a morphism between certain objects (classically rings or schemes), then one can make sense of the unit, Picard and brauer groups of $A,B$, and there are certain five-term exact sequences relating those (in good cases, these are just excerpts of long exact sequences from etale cohomology, but as far as I understand the five-term sequences exist more generally).

In more genererality, in [1] it is shown that one may take for $A,B$ symmetric monoidal categories with coequalisers stable under the tensor product, and that the two five-term sequences arise as $\pi_0$ and $\pi_1$ of a five-term sequence of "cat-groups".

Since i work with tensor triangulated categories, I am wondering if it is possible to relax the requirements regarding coequalisers (which typically do not exist in my context). The coequalisers come from considering module categories over algebras in the symmetric monoidal category. This leads me to believe that the right way to formulate this problem is on the level of monoidal model or infinity-categories.

So: have Picard-brauer type sequences been worked out for monoidal infinity categories (or perhaps tensor triangulated categories)?

[1] Enrico M. Vitale A Picard-Brauer exact sequence of categorical groups

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