>>>>>>>>>>>>>>>>>>>>>>>>>>>>>> let F:C -> D be a functor between two small categories. Its induced functor F^*:Fun(D,Set) -> Fun(C,Set) has both a left adjoint L and a right adjoint R.
1. Under which precise conditions on F is L F^* = Id. 2. Under which precise conditions on F is R F^* = Id. 3. Under which precise conditions on F is F^* R = Id. 4. Under which precise conditions on F is F^* L = Id. Does anybody know any answers to one of these queries? What happens if equality is replaced by natural equivalence? Does anybody know of good references to these or similar problems? Markus Pfenniger Andy Tonks <<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<<< I don't know the answers. However, I can describe an analogous context (ring theory) where there is a surprising answer to 2: the appropriate analogue holds iff F is an epimorphism. Perhaps someone already knows whether the corresponding conjecture is true in the category (or monoid) context, and anyway perhaps the different perspective will cast light on the problem. First, as Mike Barr points out, natural equivalence is the appropriate property to look for, and in fact I don't know how you might go beyond the question of when the units or counits of the adjunctions are natural isos. If R is a ring, then the category of (right, say) modules over R is very like a functor category. R _is_ a category (one object, lots of morphisms), and a module is a functor from R to Abelian Groups. Of course, it's more special than that, because we also require that the functor should preserve the additive structure. However, the idea that a functor F from a category C to Sets is a kind of module is quite a natural one. The "elements of the module" are the elements of the sets F(i) where i ranges over objects of C, and a morphism f: i -> j of C "acts on" the elements (at least, those in F(i)) by xf = F(f)(x). You'll see this more clearly in the case where C is a monoid (only one object). This perspective is explained in Popescu, "Abelian categories with applications to rings and modules" (LMS Monographs 3, Academic Press, 1973). It extends readily to "ringoids", i.e. categories enriched over Abelian groups (Barry Mitchell "Rings with several objects", Advances in Mathematics 37 (1972) 1-161). If F: R -> S is a ring homomorphism, then you get a functor F^*: Mod-S -> Mod-R ("restriction of scalars") with left adjoint L, M |-> M tensor_R S, and right adjoint R, M |-> S hom_R M. It is known that the counit of the adjunction L -| F^* is a natural iso if and only if F is an epimorphism in the category of rings. (See, e.g., Stenstro"m "Rings of Quotients", Springer 1975.) (Note that epimorphisms are not always surjective - e.g. the embedding of the integers in the rationals is epi.) This is also true for ringoids (I believe it's in Mitchell's paper). It's conceivable that the direct analogue holds for categories, i.e. (2) (in suitable form) iff F is an epimorphism in the category of (small) categories, though at first glance the proof in Stenstro"m doesn't seem to generalize. If you look into this, I'd suggest you study the monoid case first. Steve Vickers ==============================================================================