Comonoidal properties of I-chains

In joint work with Steffen Sagave we showed that there is an integral version of the Sullivan cochains of a space in I-chain complexes where I is the category of finite sets and injections. One could hope that there are further integral models that correspond to the Lie- and coalgebra models of spaces in rational homotopy theory. I show that the homotopy colimit of a cocommutative coalgebra in I-chains is an $E_\infty$-coalgebra, but I also prove that the norm map is not an isomorphism in general and give an example for an acyclic I-chain complex whose connected cocommutative cofree coalgebra has homology in positive degrees and whose homotopy colimit is not acyclic. This shows that certain left induced model structures on comonoids cannot exist.