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May 31, 2021 at 7:58 comment added Maxime Ramzi Can't you use the monadic bar construction for this ? Namely $A$ should be the colimit of its bar construction, $B$ too, so that $A\coprod B$ should be the colimit of the coproducts of their bar construction; now the bar constructions are made up of free $M$-algebras, so that their coproduct is readily computable. Would that be a suitable model for what you want ? This should also work for general colimits, because the monadic bar construction is natural
May 31, 2021 at 3:38 history edited Dmitry Vaintrob
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May 31, 2021 at 0:36 history asked Dmitry Vaintrob CC BY-SA 4.0