A primary example of such a situation is homotopy (or coherent) Kan extension of a topological functor along another topological functor. A theory of such extensions was considering in Cordier J.M., Extension de Kan simplicialement coherent, preprint, Amiens,1985. Cordier J.M., Porter T., Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54. This is all done simplicially, but it is not hard to translate it to topological language. Formally your general situation can be reduced (up to weak equivalence) to such a coherent Kan extension. Of coarse it is not a practical method to work with adjunctions. More specific examples of this situation were studied in my paper Categorical strong shape theory, Cahiers de Topologie et Geom, Diff., v XXXVIII-1, 1997, 3-67. Roughly speaking, I call strong shape theory of a simplicial functor F:A ---> B its right extension along itself in a bicategory of coherent distributors. This bicategory has simplicial categories as objects, simplicial bimodules as 1-morphisms and coherent natural transformations as 2-morphisms. You also should take a "derived" composition instead of usual compositions of bimodules. I call strong shape category of a functor the Kleisli category of its strong shape theory. If F has a left adjoint G in the homotopy sense then this extension can be calculated as the profunctor A(G(-),-). We also can consider the dual situation and define strong coshape theories. Examples include : 1. If Q is a simplicial Quillen model category all objects of which are cofibrant. Let Q_f be the full subcategory of fibrant objects, then the inclusion Q_f ---> Q has homotopy left adjoint (which is in general not a functor but a coherent functor) given by fibrant replacement. The corresponding strong shape category is ho(Q). 2. Let B be the category of algebras of some simplicial monad T, and let A be its full subcategory of free T-algebras. Then the inclusion A ----> B often has a homotopy right adjoint given by classical bar construction B(T,T,X). The corresponding coshape category is the category of T-algebras and their coherent homomorphisms. 3. Homotopy category of simplicial diagrams is another example. In general we do not need this left adjoint G to exists. What we need is some sort of coherent left adjoint (functor up to all higher homotopies). I call this resolution of F, so strong shape theory is actually a theory of resolutions in a very general sense. Moreover, this resolution is not necessary exists in B but can exists in some extension of B, for example in pro(B). This is enough to construct strong shape category. For example, this is the case of strong shape theory considered by topologists (and where the name came from). They study the inclusion ANR ----> Top where ANR is the category of absolute neighbourhood retracts. The so called ANR-expension of a topological space is an example of a resolution in our sense in pro(Top). We also can replace ANR by the category of CW-complexes. There is also a number of other interesting and important examples, which show that strong shape and coshape theories are what homotopy theorists actually study. I hope this will be helpful. Michael. on 24/7/02 9:16 PM, Philippe Gaucher at gaucher@math.u-strasbg.fr wrote:
Hello,
Here is the situation :
C and D are two categories enriched over the category of Hausdorff compactly generated topological spaces. So for any object X,Y, C(X,Y) and D(X,Y) are topological spaces and everything is continuous.
F:C-->D and G:D-->C are two functors satisfying :
there exists a natural continuous map C(FX,Y)-->D(X,GY) which is always a weak homotopy equivalence.
pg.
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