Morita semi-equivalences

Question

Recall that algebras (or linear 1-categories) $A$ and $B$ are Morita equivalent if there exist bimodules $_AM_B$ and $_BN_A$ and isomorphisms $u: {}_A(M \otimes_BN)_A \to {}_AA_A$ and $v: {}_B(N \otimes_AM)_B \to {}_BB_B$ such that $u$ and $v$ satisfy the zig-zag relations.

I'm interested in what can said under the weaker hypothesis that ${}_B(N \otimes_AM)_B \cong {}_BB_B$ but ${}_A(M \otimes_BN)_A$ is not necessarily isomorphic to ${}_AA_A$.

One easy consequence is that the isomorphism classes of $B$-modules inject into the isomorphism classes of $A$-modules. Can the $A$-modules which do not come from $B$-modules be understood in terms of the bimodule ${}_A(M \otimes_BN)_A$?

In the situation I'm mainly interested in, $A$ and $B$ are *-algebras (actually *-categories) and $_AM_B$ is the same as $_BN_A$ under the usual identification of left and right modules.

Answer 1

This was slightly too long to be a comment, but it certainly is not a complete answer.

Since module categories behave very much like abelian groups, I expect you can say a lot by thinking in analogy with the lower-categorical-level situation: you have two abelian groups $A$ and $B$, and homomorphisms $n : A \to B$ and $m: B \to A$, such that $nm = \mathrm{id}_B$. Then you certainly can say that $mn : A\to A$ is a projection onto a direct summand isomorphic to $B$, and $\mathrm{id}_A - mn$ is a projection onto the other direct summand.

Back to module categories, you may not always be able to make sense of the difference ${_A A_A} \ominus ({_A {M\otimes_B N}_A})$. Two best cases are: perhaps you have a natural inclusion ${_A {M\otimes_B N}_A}\hookrightarrow {_A A_A}$, in which case the difference is the quotient; or perhaps you have a natural projection ${_A A_A} \twoheadrightarrow {_A {M\otimes_B N}_A}$, in which case the difference is the kernel. More generally, you may be in the situation where the functors $M\otimes_B$ and $N\otimes_A$ are one-sided adjoints, in which case you can use whichever of the unit or counit of the adjunction is appropriate. Recall also that the cleanest way to take differences is to work in the derived category, in which case the difference is the cone of the (co)unit. (In the derived category, there are many available differences, one of which is the cone of the zero map, but that almost certainly will not be useful to you.) There probably will be some conditions on the map between $A$ and $M\otimes_B N$ coming from the request that ${_A A_A} \ominus ({_A {M\otimes_B N}_A})$ is a projection.