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We study weak commutative algebras in a symmetric monoidal model category$\mathscr{M}$. We provide a model structure on these algebras for any symmetricmonoidal model category that is combinatorial and left proper. Our motivationwas to have a homotopy theory of weak commutative dg-algebras in characteristic$p>0$, since there is no such theory for strict commutative dg-algebras. For ageneral $\mathscr{M}$, we show that if the projective model structure on strictcommutative algebras exists, then the inclusion from strict to weak algebras isa Quillen equivalence. The results of this paper can be generalized tosymmetric co-Segal $\mathcal{P}$-algebras for any operad $\mathcal{P}$. Andsurprisingly, the axioms of a monoidal model category are not necessary to getthe model structure on co-Segal commutative algebras

Model structure on co-Segal commutative dg-algebras in characteristic p>0