Let \(\mathcal{S}\) be a small category, and suppose that we are given two (non-full) subcategories \(\mathcal{S}^{sm}\) and \(\mathcal{S}^{cl}\) that generate all morphisms of \(\mathcal{S}\) under composition in the same way as morphisms of quasi-projective algebraic varieties are generated by smooth morphisms and closed immersions. We show that...

Constructing monoidal structures on fibered categories via factorizations