Street and Walters (1973) called a category A total when its Yoneda embedding has a left adjoint and proved that such a category is compact in the sense of Isbell, that is: every functor F: A --> B preserving all colimits has a right adjoint (some provisions on the size of hom sets need to be made). There are many papers on the fairly rich supply of total categories, by Wood (1982) and myself with B"orger (1990) and Adamek (1990), and by others. There is also paper dealing specifically with compact categories, by B"orger, Tholen, Wischnewsky and Wolff (JPAA 21 (1981) 129-144). Theorem 4.1 of that paper says explicitly: Let P: A --> X be a semitopological functor over a compact category X. Then a monoidal structure on A is closed if and only if A \ten (-) preserves all colimits. The reason of course is that semitopological functors "lift" compactness (and so do monadic functors - Rattray 1975). The theorem below gives extra conditions to make sure that preservation of small colimits suffices. There is a lot more literature on that latter aspect, particularly by Kelly. Walter Tholen. Gaucher Philippe wrote:
Dear all,
I thank you very much for all the answers. I am going to read all these papers. But maybe it is simpler to explain my problem:
Consider Theorem 27.22 of the book "The joy of cats" <http://www.tac.mta.ca/tac/reprints/articles/17/tr17abs.html> : a well-fibered topological construct is cartesian closed iff for any object A, Ax- preserves colimits. Does this theorem have a generalization for other symmetric monoidal structures than the binary product ?
Thanks in advance. pg.