I should say that I'm not a category theorist or an abstract algebraist, so maybe this will be very pedestrian. I have the following, somewhat vague question:

I have categories C and D, a forgetful functor $U:C\rightarrow D$. This has a left adjoint, but does not have a right adjoint. Are there other situations where this occurs, and what "workarounds" are there.

Let's be more precise with an example which is close to my actual situation. Let $A$ be a $k$-algebra ($k=\mathbb C$ if you like) and let $U:A{\sf -mod} \rightarrow k{\sf -vect}$ the forgetful functor from left $A$-modules to $k$-vector spaces. This has a left adjoint $A \otimes \underline{\ \ }$; you have a natural bijection, for a vector space $V$ and a module $M$,

`\[ \text{Hom}_{k{\sf -vect}}(V,U(M)) \cong \text{Hom}_{A{\sf-mod}}(A\otimes V,M), \]`

which sends $T:V\rightarrow U(M)$ to $a\otimes x\mapsto a\cdot T(x)$. Then a module of the form $A\otimes V$ is "free", and this leads to consideration of projective modules etc. In my situation (dealing with topological algebras etc.) I can parallel all of this.

Similarly, we have a right adjoint $\text{Hom}_{k{\sf -vect}}(A,\underline{\ \ })$. This gives a natural bijection

$$ \text{Hom}_{A{\sf -mod}}(M,\text{Hom}_{k{\sf -vect}}(A,V)) \cong \text{Hom}_{k{\sf -vect}}(U(M),V), $$ which identifies $T:U(M)\rightarrow V$ with $x \mapsto (a\mapsto T(a\cdot x))$. Then one might call a module of the form $\text{Hom}_{k{\sf -vect}}(A,V)$ "cofree", and this leads to injective modules, etc.

In my situation, my category of modules is not a "closed category"; if cofree modules existed, the naturality of the above bijection would force them to $\text{Hom}(A,V)$ (for a suitable meaning of $\text{Hom}$), but this object is not in my category. I'm hence missing a good notion of "cofree"; but really I'm interested in injectives. Are there situations (say, in algebra) where something similar occurs?

**Edit:** People demand a more explicit example. Suppose $A$ is a von Neumann algebra, and my category is those modules which are dual Banach spaces $E$, with a bounded left action of $A$, and such that for each $x\in E$, the orbit map $A\rightarrow E; x\mapsto a\cdot x$ is weak$^*$-continuous. The forgetful functor is to the category of dual Banach spaces with weak$^*$-continuous bounded linear maps as morphisms. The right adjoint "should be" $B^\sigma(A,\underline{\ \ })$, the weak$^*$-cts bounded linear maps. But this is rarely itself even a dual Banach space.

muchmore precise about this. Does this happen? Yes, all the time! The forgetful functor from groups to sets has a left adjoint but not a right adjoint, and this is extremely common. Is there a workaround? I don't know: workaround for what exactly? Instead of the highly elliptical "dealing with topological algebras, etc.", please tell us exactly what you are dealing with, and then maybe one can tell if there are injectives or whatever, and maybe you can tell us what you want to do if there aren't. $\endgroup$7more comments