My question is relatively simple: Why are there so many different notations for limits and colimits? Is there a reason people don't just use a common convention? Is there any benefit to using one notation over another? Here are some that I've seen.
- $\text{Lim }F$
- $\text{Colim }F$
- $\varprojlim F$ (The most common I've seen to denote a limit)
- $\varinjlim F$ (The most common I've seen to denote a colimit)
- $\text{colim}_{\rightarrow} F$ (I usually see the arrow under the "colim," but I don't know how to configure that on here, edits are welcome).
- $\text{colim}_{\leftarrow} F$
This isn't all, as I've also seen things like $\text{lim}_{x \in S} F$ and even $\varprojlim_{x \in S} F$ (and in these cases, I wonder why the arrow is/isn't included if it's already clear from context what is happening). In each case, I understand what is being written, but I personally find it puzzling and slightly annoying that such widely different notation is used for these concepts. Now, I am at a point where I need to decide on a convention for denoting limits and colimits in notes, but I think that I should first understand why people use such different limit and colimit notation before I decide, out of the above six, on what convention I will use (as I would like to not have to later go back and change all my notation).