Sometime ago I had trouble remembering that right adjoints preserve limits (and so left adjoints preserve colimits). But ever since R. Vakil suggested that we use RAPL as a mnemonic to Right Adjoint Preserve Limits, it stuck on my mind and I never forgot. However I still have trouble remembering that

Inverse limits (= projective limit) are special cases of limits and direct limits (= inductive limit) are special cases of colimits. Also, the arrow (in $\varinjlim$) in direct limits go to the right and the arrow in inverse limits go to the left.

I would like to know you do to remember it.

  • 8
    $\begingroup$ Easy: always use limit and colimit, never anything else. A bit more seriously, limits come with canonical projections. As for the direction of the arrows, the easy solution is to always write lim or colim and simply ditch the arrow. $\endgroup$ – Ittay Weiss Jan 14 at 16:55
  • $\begingroup$ @IttayWeiss that's what I do for my own texts. But this doesn't work when reading a book or an article. Also I agree that I could just look for the direction of the morphisms, but this doesn't solve the direction of the arrow problem and makes the reading process a little longer. $\endgroup$ – Gabriel Jan 14 at 16:56
  • 2
    $\begingroup$ For the arrows, it seems easy to remember that the direct limits go left-to-right, which is the reading direction in English, so is the "direct" direction when you read and write. $\endgroup$ – Captain Lama Jan 14 at 18:35
  • $\begingroup$ And what about remembering projective and inductive limits? $\endgroup$ – Gabriel Jan 15 at 10:08

A key example of projective limits is given by diagrams $(\mathbb{N}, \leq)^{\mathrm{op}} \to \mathscr{C}$. Dually, key examples of inductive limits are given by $(\mathbb{N}, \leq) \to \mathscr{C}$. This gives an indication for all of those nasty conventions:

  • Inductive limits might remember us of induction which might remember us of $\mathbb{N}$. Arguably, the most natural ordering associated with $\mathbb{N}$ gives a natural diagram $(\mathbb{N}, \leq) \to \mathscr{C}$.
  • Projective limits are just the dual. In key examples, "projective" stands for "projection". One of the most important examples for $(\mathbb{N}, \leq)^{\mathrm{op}} \to \mathscr{C}$ is given by $\mathbb{Z}_p$. The diagram is given by $\mathbb{Z}/p^{i+1} \mathbb{Z} \to \mathbb{Z}/p^i \mathbb{Z}$ which are projections! More generally, you may refer to the $I$-adic completion of a ring.
  • The conventions $\varprojlim$ and $\varinjlim$ are also indicated by those diagrams. Of course, this depends on conventions again, but it at least seems vaguely natural to write $(\mathbb{N}, \leq)$ as $$1 \to 2 \to 3 \to 4 \to 5 \to \dots. $$ Dually, we get $\varprojlim$.

In general, however, I wholeheartedly agree with Ittay Weiss. I really wish that more people would also just write $\lim_I$ and $\mathrm{colim}_I$ and also just say (categorical) limit and colimit.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.