## Background

In the course of reading Mac Lane linearly (currently in Chapter VI), I have seen again and again that duality can make life much easier. My problem is that I have almost no background in logic, and duality is a theorem in logic about category theory.

When I first read about duality in Chapter II of Mac Lane in the
context of the elementary theory of a single category, everything was
pretty clear even without knowing any logic. However, when I got to
the chapter on adjunctions, involving two categories and functors
between them, a bijection of hom-sets, and two natural
transformations, I got confused to the point that I wasn't even sure
how to *use* duality (let alone, why it is correct).

At this stage, I made a rather long pause and read the first three chapters of Ebbinghaus, Flum, and Thomas' ''Mathematical logic'' (so, I have read about the syntax and semantics of first-order logic). From this, I built my own (hopefully correct) ''poor man's proof of duality'' up to the situation of a single adjunction. This has both clarified the validity of duality for formulas involving adjunctions, and helped me understand how to use duality in such situations.

But a single adjunction is far from the most ''complicated'' situation one meets. There are composition of adjunctions, pointwise limits in functor categories, and many other situations in which I am still not totally convinced that I understand duality (both theoretically and practically).

For example, in one answer to a recent question on pointwise limits in functor categories, it was stated that the reference for limits is Mac Lane, while the reference for colimits is Mac Lane--Moerdijk. I really wanted to comment that the assertion on colimits is just the dual of the one on limits, but then I realized that I am not totally sure. I would be most grateful for some solid source that I can consult whenever I have doubts in what I get after doing the intuitive things (reverse arrows but not functors, etc.).

## Questions

- What is a good reference for an explicit, logic-based, statement of a duality theorem of category theory in ''complicated situations?''
- What are the prerequisites in logic? For example, up to which point of Ebbinghaus--Flum--Thomas should I read?

languagefor expressing certain formal arguments in a clean way which can be "plugged into" an interesting situation one cares about. So why bother with detours into logic, instead of applying it to situations with more substance (e.g., commutative algebra, not lattice theory)? I hope you have a situation in mind to which you plan to apply category theory. – BCnrd Apr 5 '10 at 0:07