Whilst studying some category theory, I was blindly using the notation $\lim_\leftarrow D$ for the limit of a diagram $D$ in some category $\mathcal{C}$ (as they are notated in MacLane, Awodey, my lectures etc.).

Why does the arrow go from right to left?

Is it because we think of the limit of a diagram as sitting on the left of the diagram (where the arrows in our diagram go from left to right)?

I sometimes think of a terminal object as being to the far right of a category, because everything can map into it. If limits had an arrow from left to right (like in analysis), I think I could justify this by saying "Well a limit is a terminal object in the category of cones, so it sits on the right". This makes me think that my 'reason' above is nonsense.

Any thoughts are appreciated. Thanks!


I don't know the reason for sure, but here is my guess.

The most basic (and historically important) examples of limits and colimits are those indexed by the natural numbers, ordered by $\geq$ (resp. $\leq$). In the case of limits, one takes diagrams of the form $$\dotsc \dotsc \to A_2 \to A_1 \to A_0$$ and the limit goes to the left direction and therefore(?) is denoted by $\varprojlim A_i$.

Dually, the colimit of $$B_0 \to B_1 \to B_2 \to \dotsc \dotsc$$ goes to the right direction and therefore(?) is denoted by $\varinjlim B_i$.

Notice that the limit of $A_0 \to A_1 \to \dotsc$ is $\varprojlim A_i = A_0$, which is "because" $A_0$ is already on the left.

  • $\begingroup$ also how do you Tex that $lim_\leftarrow$ so nicely? $\endgroup$ – Jack Davies Nov 1 '14 at 23:22
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    $\begingroup$ \varprojlim and \varinjlim $\endgroup$ – Martin Brandenburg Nov 1 '14 at 23:41

The universal property of the limit has certain arrows go into the limit.

Similarly, the universal property of colimits has certain arrows come out of the colimit.

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    $\begingroup$ I didn't know that this is the reason for the notation. Do you have a reference for this or is this just your mnemonic? Often one writes $\varprojlim$ (resp. $\varinjlim$) and here it is not clear if the arrow goes out or into the limit (resp. colimit). $\endgroup$ – Martin Brandenburg Nov 1 '14 at 8:50

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