This question has its motivations in trying to understand more deeply the guiding intuitions for coining these terms. I am using the definitions of Joy of Cats.

A functor $F:\mathbf{A}\to\mathbf{B}$ is said to,

**lift limits**provided that for every diagram $D : \mathbf{I}\to \mathbf{A}$ and every limit $\mathcal{L}$ of $F \circ D$ there exists a limit $\mathcal{L}'$ of $D$ with $\mathscr{F}(\mathcal{L}') = \mathcal{L}$.**create limits**provided that for every diagram $D : \mathbf{I}\to \mathbf{A}$ and every limit $\mathcal{L}$ of $F \circ D$ there exists a unique source $\mathcal{S}$ in $\mathbf{A}$ with $F(\mathcal{S})=\mathcal{L}$ and that, moreover, $\mathcal{S}$ is a limit $D$.**detect limits**provided that a diagram $D : \mathbf{I}\to \mathbf{A}$ has a limit whenever $F\circ D$ has one.

I would like to know for example,

- In what sense does $F$ "lift" limits?
- In what sense does $F$ "create" limits?
- In what sense does $F$ "detect" limits?

It may look silly to try to understand terms in this way. An alternative is just to grasp the concept itself and forget about the terminology. However, many times I have seen that trying to understand the terminology often rewards back with beautiful and deep intuitions. I hope it turns out like that in this case as well. Apologies if that doesn't turn out to be so.

Hi, the wording of "in what sense" is a bit vague so I'm going to interpret your question as why have the English words "lift", "create" and "detect" been used. The terminology derives from the case where we have a category that we understand well, and a category which we don't. If we have a suitably well behaved functor we can hope to understand the behaviour of limits of in terms of the limits of . Firstly, clearly .

3) We say that detects limits because we can *detect* if a limit exists in by considering the image in .

1) We say that lifts limits because the limit structure in the image of *lifts *to structure on , so we can reason about in .

2) We say that creates limits because the data of uniquely determines limits in .

The classic example is given by forgetful functors into the category of sets. Consider the categories taken from [SGL, pg.62] (both topoi) and , that is the continuous and discrete representations of a topological group . Then the forgetful functors and both vacuously detect limits (as the categories , and have all limits). Also, creates limits but does so if and only if the intersection of any collection of open subgroups is again open.

I think this is exactly what I was looking for. However, although (3) and (1) makes sense to me, I am not sure (2) does justice to the term "create" because $F$ doesn't really create anything new in $\mathbf{A}$. But anyway, it's a very nice explanation. Many thanks.

I guess whether "creates" limits depends on whether those limits exist from before or after you observe them.

Probably, but I still can't shake the feeling of a lack of natural terminology to express this concept (if I am not missing anything). An idea (motivated by a private chat with Aleks Kissinger) is to consider "functorial equations". More specifically, if $F:\mathbf{A}\to\mathbf{B}$ is a functor then for $F$ all these concepts essentially reduces to finding certain "nice" solutions of the "functorial equation" of the form $F(\mathcal{X})=\mathcal{Y}$ (Aleks Kissinger told me to consider isomorphism instead of equality, but equality suffices for this case) for "nice" $\mathcal{Y}$. Here $\mathcal{X},\mathcal{Y}$ are respectively $\mathbf{A}$-source and $\mathbf{B}$-source variables (admittedly, all these are pretty vague right now, but I hope that I am able to convey the big picture).

I believe this goes well to what you have proposed in the beginning. Here $\mathcal{X}$ is the "unknown" and $\mathcal{Y}$ is the "known".