# Mnemonic to remembering that inverse limits are limits and direct limits are colimits

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.

• 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. – Ittay Weiss Jan 14 at 16:55
• @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. – Gabriel Jan 14 at 16:56
• 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. – Captain Lama Jan 14 at 18:35
• And what about remembering projective and inductive limits? – 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.