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# In what sense does a functor "lift/create/detect" limits?

(@sayantan-roy)
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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,

1. In what sense does $F$ "lift" limits?
2. In what sense does $F$ "create" limits?
3. 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.

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(@jwrigley)
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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 ${\bf B}$ that we understand well, and a category ${\bf A}$ which we don't.  If we have a suitably well behaved functor $F \colon {\bf A} \to {\bf B}$ we can hope to understand the behaviour of limits of ${\bf A}$ in terms of the limits of ${\bf B}$. Firstly, clearly $2) \implies 1) \implies 3)$.

3) We say that $F$ detects limits because we can detect if a limit exists in ${\bf A}$ by considering the image in ${\bf B}$.

1) We say that $F$ lifts limits because the limit structure in the image of $F$ lifts to structure on ${\bf A}$, so we can reason about ${\bf A}$ in ${\bf B}$.

2) We say that $F$ creates limits because the data of $F$ uniquely determines limits in ${\bf A}$.

The classic example is given by forgetful functors $U \colon {\bf A} \to {\bf Sets}$ into the category of sets.  Consider the categories taken from [SGL, pg.62] (both topoi) ${\bf B}G$ and ${\bf B}G^{\delta}$, that is the continuous and discrete representations of a topological group $G$.  Then the forgetful functors $U^1 \colon {\bf B}G \to {\bf Sets}$ and $U^2 \colon {\bf B}G^\delta \to {\bf Sets}$ both vacuously detect limits (as the categories ${\bf Sets}$, ${\bf B}G$ and ${\bf B}G^{\delta}$ have all limits).  Also, $U^2$ creates limits but $U^1$ does so if and only if the intersection of any collection of open subgroups is again open.

(@sayantan-roy)
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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.

(@jwrigley)
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I guess whether $F$ "creates" limits depends on whether those limits exist from before or after you observe them.

(@sayantan-roy)
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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".

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