# What's the point of “created limits”?

Seeing as there's an nLab page about creation of limits, I take it that at least some people think this is an important notion. There's also a discussion here about what the "correct" definition of this notion is. However, and maybe this is just my ignorance speaking, but I haven't actually found a reason to worry about this notion, so I basically just ignore it. I mean, okay, a diagram $D$ that lacks a limit in one category $\mathbf{C}$ might gain a limit in a category with more objects, or arrows, or both. More generally, given a functor $F : \mathbf{C} \rightarrow \mathbf{D}$, the composite $F \circ D$ might or might not have a limit, quite independently of whether or not $D$ does. But so what? I don't quite get what the notion "creation of limits" really teaches us.

Question. Can someone who finds this to be a helpful notion explain what they find helpful about it? Is there, for example, a theorem about creation of limits that's ubiquitous but impossible to see without this notion? Is there an important definition that can't be elegantly stated unless creation of limits is invoked? What's the point of this idea?

• The nLab entry mentions monadicity theorems. Does that not satisfy you? – Yemon Choi Sep 20 '18 at 7:06
• @Qfwfq What are you talking about? I really don't see the connection between "creating limits" in category theory and the completion of $\mathbb{Q}$ being $\mathbb{R}$... – Najib Idrissi Sep 20 '18 at 12:51
• @Qfwfq A categorical limit and a topological limit are distinct notions; 'creating limits' as the term is used in category theory doesn't correspond to taking equivalence classes of Cauchy sequences or Dedekind cuts etc. For a functor $F:\mathcal{C}\rightarrow\mathcal{C}'$ to preserve limits of type $\mathcal{I}$ means that for any diagram $D:\mathcal{I}\rightarrow\mathcal{C}$, if $F\circ D$ has a limit in $\mathcal{C}'$ then $D$ already had a limit in $\mathcal{C}$ and $F$ maps these limits to one another. – Alec Rhea Sep 20 '18 at 19:57
• @Qfwfq: I think Najib's objection was that creating limits has nothing to do with the fact that they didn't exist before and you somehow adjoin them. Limits in $\operatorname{\underline{Ab}}$ are created from the forgetful functor $\operatorname{\underline{Ab}} \to \operatorname{\underline{Set}}$ in the sense that properties of the functor prove that limits in $\operatorname{\underline{Ab}}$ exist, but they would still exist if we didn't use that functor. In no way is $\operatorname{\underline{Ab}}$ a completion of $\operatorname{\underline{Set}}$ (as I'm sure you're aware). – R. van Dobben de Bruyn Sep 20 '18 at 21:49
• @Qfwfq: of course you are right that the word "limit" in category theory was chosen in analogy with limits of functions or sequences, and that similarly there are various notions named "completion" in category theory following that same analogy. But first it's really just an analogy which motivates how those things are named, but taking it too seriously is often misleading. Also and more to the point, this is just NOT what the agreed meaning of "creating limits" is, even if this choice of terminology might perhaps make it sounds like it is, see Alec's comment and my answer below. – Adrien Sep 20 '18 at 22:11

The main point of “creating limits” is things like:

Theorem. If $C$ has all limits of shape $I$ and $F : D \to C$ creates such limits, then $D$ has all such limits and $F$ preserves and reflects them.

In other words, the definition is pretty much designed to be a tractable and concrete solution to the implication “(limits in $C$) + ???? $\Rightarrow$ (limits in $D$, and preservation/reflection)”

Pragmatically, it isolates the structure common to the construction/analysis of limits in groups, topological spaces, algebras for monads, and many other categories of “structured objects over some category where limits are already understood”. It does require constructing the limits in $D$ anyway, but it guides you to a particular tractable way of constructing them, and then shows that by constructing them this way, you get preservation + reflection for free.

Regarding the two different definitions of creation of limits, see Displayed Categories, Ahrens, Lumsdaine 2017, arXiv:1705.04296 for my personal (maybe slightly idiosyncratic) take on why the standard definition using equality of objects really is very reasonable and natural.

This is not what "creates limits" means, although I also find this terminology slightly confusing (in the sense that if I had to guess the definition just from the word "creates" my guess would probably have been the same as you, but it's just not that).

The definition (eg the one stated in the nLab page you mention) does require the limit of $D$ to exist in $\mathbf C$ in the first place, but what it says is basically that you can compute it in $\mathbf D$, ie roughly we have both:

• $F$ of a limit of $D$, is a limit of $F\circ D$
• somewhat conversely (and imprecisely), if some $F(x)$ happens to be a limit of $F\circ D$, then $x$ was already a limit of $D$.

Now as far as I know this definition is most often used for some shape of diagrams rather than for a single one, and as pointed out in the nlab is used in Beck's monadicity theorem. The main example of that phenomena (the colimit version) is that a direct sum of modules over some algebra is just the direct sum of the underlying vector spaces with the "obvious" module structure.