We can use the comultiplication of a Hopf-algebra to get that modules over a noncommutative Hopf-algebra are still a monoidal category. Can we do the same thing for a one-fold loop space, where "comultiplication" is replaced by the diagonal of the space?

It is presumably true that Omega X-modules are the same as fibrations over X. In that case, you certainly can: there is a product of two fibrations as a fibration over X x X, and you can then pull back along the diagonal.

Simplicial commutative rings can be interpreted as functors $\text{Poly}^{op} \rightarrow \text{Spc}$ which send finite coproducts to finite products. Is there an analogous way to rephrase simplicial modules over a simplicial commutative ring?

@LPK idk, but a common trick is to trade a module for a square-zero extension of a ring. maybe you'll find that has traction here

If X is a sufficiently nice connected topological space, then H^1(X,Z/n) should correspond to morphisms \pi_1(X) -> Z/n. This should just be connected Z/n-covering spaces, right? . On the other hand, we can interpret H^1(X,Z/n) as being given by principal homogeneous spaces. Do these also have to be connected if X is? What is stopping me from say taking a disjoint union of two connected Z/5-covering spaces and letting Z/10 \cong Z/2 x Z/5

act by (1,0) switching the two components and (0,1) in the original way?

I think the connected Z/n-covering spaces correspond to the surjective morphisms \pi_1(X) -> Z/n

thanks!

@SaulGlasman Will it be the case that if it the map is not surjective, say that f: \pi_1(X) -> Z/n has image mZ/n, where m|n. Will it then be a disjoint union of n/m identical covering spaces?