6
$\begingroup$

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

  1. What is a good reference for an explicit, logic-based, statement of a duality theorem of category theory in ''complicated situations?''
  2. What are the prerequisites in logic? For example, up to which point of Ebbinghaus--Flum--Thomas should I read?
Improve this question
$\endgroup$
5
  • $\begingroup$ You might want to start with duality in lattice theory (e.g. George Grätzer's books). In that case, the idea is to replace ≤ with ≥, meets with joins, etc. (Note that a poset is a category where there is at most one arrow between any two objects.) $\endgroup$
    – François G. Dorais
    Commented Apr 4, 2010 at 22:45
  • $\begingroup$ Thank you for your comment. In the context of category theory, would you say that the equivalent of duality in lattice theory would be the duality of ETAC + duality for finite limits/colimits in a single category? $\endgroup$
    – user2734
    Commented Apr 4, 2010 at 23:04
  • 2
    $\begingroup$ All duality arguments work the same, it's a purely syntactic transformation! Setting it into a fixed global theory won't help you see this. I think that working through a few examples of moderate complexity (as found in lattice theory) should open the way for you... $\endgroup$
    – François G. Dorais
    Commented Apr 4, 2010 at 23:30
  • 9
    $\begingroup$ Is it really worth all of this effort? There is no serious content in these duality arguments, and at this level category theory itself hardly has any content beyond being a very efficient language for 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. $\endgroup$
    – BCnrd
    Commented Apr 5, 2010 at 0:07
  • $\begingroup$ +1 for well-written (etc.) post. $\endgroup$
    – Theo Johnson-Freyd
    Commented Apr 5, 2010 at 0:45

1 Answer 1

Reset to default
9
$\begingroup$

I would go even farther than the comments above, at least in the specific case you mention about computing (co)limits objectwise in a functor category. Once you know the statement for limits, deducing the statement for colimits is not even a syntactic transformation to the proof, and needs no arguments from formal logic at all. Instead, to prove that colimits are computed objectwise in the functor category [I, C], simply use the fact that limits are computed objectwise in the functor category [Iop, Cop] = [I, C]op, and the fact that colimits in a category are the same as limits in the opposite category.

I think this is the common case in this kind of argument, but perhaps someone can come up with an example where the argument really needs to be repeated in the dual situation.

Improve this answer
$\endgroup$
4
  • 2
    $\begingroup$ +1. It would perhaps be of interest to some people to have a precise general statement of the duality principle, but it sounds as if what the questioner really wants is a mechanical way of taking any given theorem and deducing its dual. And I think almost everyone does that in the way that Reid describes. $\endgroup$
    – Tom Leinster
    Commented Apr 5, 2010 at 1:08
  • 2
    $\begingroup$ If the argument needs to be repeated, then I don't think it deserves to be called a dual situation... maybe the only difficulty is figuring out which category to oppose. $\endgroup$
    – Qiaochu Yuan
    Commented Apr 5, 2010 at 2:10
  • $\begingroup$ +1. I think an additional difficulty is that (for good reason) nobody actually thinks about duality arguments, which might be disconcerting to a neophyte. Since it is never explicitly decomposed that way, it might be hard to encapsulate the pre-transform $[I^{op},C^{op}] = [I,C]^{op}$ and the post-transform "colimits are limits in the opposite category." $\endgroup$
    – François G. Dorais
    Commented Apr 5, 2010 at 3:47
  • 1
    $\begingroup$ @Reid Barton: Thank you very much for your answer. This does help. I suppose that my problem arises when it is not straightforward to find an opposite category for the dual statement. For example, it took quite a lot of effort to understand ''$\langle F,G,\varphi\rangle\ \colon X \rightharpoonup A$ is an adjunction'' is the dual of ''$\langle G,F,\varphi^{-1}\rangle\ \colon A \rightharpoonup X$ is an adjunction'' (an exercise on p. 92 of Mac Lane). $\endgroup$
    – user2734
    Commented Apr 5, 2010 at 7:44

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .