Monoid cohomology

Question

It is well-known that the group cohomology groups of a group $G$ are isomorphic to the cohomology groups of the classifying space $|G|$ of (the nerve of the delooping) of $G$.

More generally, we have also classifying spaces for monoids, and hence we may analogously define the monoid cohomology groups of a monoid $A$ to be the cohomology groups of $|A|$.

Questions:

1. Has this notion been studied before?
2. How does it relate to group cohomology and group completion? More precisely, given a monoid $A$, how are the the co/homology groups of $|A|$ and $|A^{\mathrm{grp}}|$ related?

Answer 1

1. Every connected CW complex is the classifying space of a monoid. This is proven by McDuff in On the classifying spaces of discrete monoids. This does not discount studying the classifying spaces of specific monoids, but it does discount a general theory, since well, it would be the same as all of algebraic topology.

2. This Master's thesis by Taelman studies the group-completion map on classifying spaces. In particular, it is an isomorphism on fundamental groups and since the classifying space of a group is a $K(G,1)$ this means it is the first Postnikov truncation. The author shows that if the monoid is free or commutative, then this map is a homotopy equivalence (i.e. cohomology of commutative and free monoids is the same as group cohomology of their groupification). In general, there will be no relation aside from the fact that their first (co)homology will be isomorphic.