If (C,tensor,1) is a symmetric monoidal category and f:A-->B is a morphism of PROPs (or monoidal cats = colored PROPs), one gets a forgetful functor f^*:B-Alg(C)-->A-Alg(C) (where B-Alg(C)=tensor-preserving functors from B to C) defined by precomposing with f.
Does anyone conditions on A,B,C under which this functor has a left or a right adjoint? (e.g. if C has the monoidal structure coming from products, it has a left adjoint, is there more to say?)
Paul-André Melliès has quite an interesting paper on this topic:
http://hal.archives-ouvertes.fr/docs/00/33/93/31/PDF/free-models.pdf
...but phrased in the more general terms of T-algebras of a pseudomonad. The idea is that a pseudomonad on a 2-category (especially Cat), let you put algebraic structures on categories the same way monads let you put them on objects of a category, like sets. This is motivated by the need to put PROPs, PROBs, PROs, Lawvere theories, etc. all under one roof.
He begins by talking about how a T-algebra homomorphism (a monoidal functor in the case where the T-algebras are monoidal categories) j : A -> B induces a forgetful functor U_j from Models(B,C) to Models(A,C) in the way you mentioned. Looking for left adjoint to U_j amounts to looking for a way to push some functor backwards along j in a suitably natural way. As Tom already mentioned, this is the left Kan extension. This process is functorial, and usually written Lan_j : [A,C] -> [B,C]. Furthermore, Lan_j -| U_j.
But if we were done there, all PROPs would have free algebras, which we know is not true in general (cf. bialgebras). The hard part is proving the Lan_j is a T-algebraic left Kan-extension. In the case of Lawvere theories, this is easy, because the product structure guarantees all natural transformations of cartesian functors are cartesian, but in the monoidal case, this stuff all needs to be checked.
This is where the story starts to get more complicated. It seems quite tricky to come up with suitably weak conditions under which Lan_j is T-algebraic. Mellies phrases these in terms of distributers (aka profunctors, modules, depending on who you ask and what country you are in :-P). If functors are like functions, this are a bit like relations. The nice thing about them is they always come in adjoint pairs f_* and f^* for any functor f.
So, thm 1 in the paper is (roughly) this. If j and j^* are T-algebraic in the suitable 2-categories, C is (T-algebraically) complete and co-complete, and for any model f : A -> C, f_* o j^* factors through the up-star of the Yoneda embedding y : C -> Psh(C), then U_j has a left adjoint computed as Lan_j that is indeed the free functor.
This is quite heavy-duty (pro-arrow equipment, ends, etc.), but it seems to get the job done. It would be nice to see more concrete/specific examples of this.
I don't know the answer to that question, but I know the answer to some similar questions.
The simplest (which you probably know about) is that if f: A --> B is a map of (small) categories and C has all small colimits (respectively, limits) then f^* has a left (respectively, right) adjoint. These adjoints are called left and right Kan extension along f.
A more complex setting is finite product categories. If f: A --> B is a finite-product preserving functor between categories with finite products, then it's a fact that f^*: FP(B, Set) --> FP(A, Set) has a left adjoint. (Here FP(A, C) means the category of finite-product preserving functors from A to C, and all natural transformations between them.) Looking at the proof, it seems to work if we replace Set by any category with finite products and small colimits such that the former distribute over the latter.
Perhaps it's true that for monoidal categories, f^* has a left adjoint as long as A and B are small, C has small colimits, and tensor in C distributes over small colimits. (And a dual statement should hold for right adjoints.) But that's only a guess.
