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

Question

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.

Answer 1

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.