Let D be a free monoidal category containing a monoid, in the sense that for any monoidal category (E, ⊗, I) there is an equivalence
MonCat wk ((D,+,0),(E,⊗, I)) ≅ Mon(E,⊗,I)
between the category of weak monoidal functors D → E and the category of monoids in E.
Now suppose E is a category with finite products. Then there is an isomorphism of categories
MonCatcolax ((D,+,0), (E,x,1)) ≅ [Δop ,E]
between the category of colax functors D → E and the category of simplicial objects in E.
It could argued that if (E,⊗, I) is general monoidal category, it would be better to define a simplicial object in E not as a functor Δop → E, but rather as a colax monoidal functor D → E. For example, it was colax monoidal version that made possible the definition of homotopy differential graded algebra (здесь он ссылается на два своих препринта math.QA/9912084 и math.QA/0002180 ).