There are some 'standard' applications of the adjoint functor theorem (AFT) and the special adjoint functor theorem (SAFT), for example, the existence of a free $\tau$-algebra (where $\tau=$(operations,identities)) on a small set by the AFT,
Stone-Cech compactification by the SAFT, and, if I am not mistaken, the proof that the category of $\tau$-algebras is cocomplete (by using the AFT to establish a left adjoint to the appropriate diagonal functor).

However, I was not able to find any applications of the duals of the AFT and the SAFT, neither in MacLane, nor in the Joy of Cats.

The Joy of Cats contains the following intriguing remark on p. 311:

Since many familiar categories have separators but fail to have coseparators, the dual of the Special Adjoint Functor Theorem is applicable even more often than the theorem itself.

But what are the mentioned application of the dual of the SAFT?

So my questions is: What are the 'standard' applications of the duals of the AFT and the SAFT?

Googling for combinations of phrases like ''adjoint functor theorem'' and ''dual'' is not very useful, so I have tried ''dual of the adjoint functor theorem'' and ''dual of the special adjoint functor theorem.'' This resulted in a total of 7 papers/books, from which I was not able to get a clear answer to the current question. I have also tried in The Wikipedia article on adjoint functors, in nLab's article on the adjoint functor theorems, and in some MO questions, but without success.


One example is the construction of geometric morphisms. Any colimit-preserving functor between Grothendieck toposes has a right adjoint, so if it also preserves finite limits, then it is part of a geometric morphism. Of course, in many cases in practice, the right adjoint is also easy to write down explicitly.

  • $\begingroup$ Interesting! Thank you very much for this answer. $\endgroup$ – user2734 Mar 8 '10 at 5:50

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