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There are several definitions of what it means for a functor $F$ to create limits of a certain type.

There is the definition in MacLane's CWM:

Definition 1: A functor $F:\mathcal{C}\to \mathcal{D}$ creates limits of type $J$ if for every diagram $D\in \mathcal{C}^J$ such that $FD$ has a limit, there exists a unique cone $k$ for $D$ such that its image through $F$ is the limit of $FD$, and moreover $k$ is the limit of $D$.

There is the definition in the nLab:

Definition 2: A functor $F:\mathcal{C}\to \mathcal{D}$ creates limits of type $J$ if for every diagram $D\in \mathcal{C}^J$ such that $FD$ has a limit, we have that $D$ has a limit, and moreover $F$ preserves and reflects limits.

There is also the definition in the category theory notes for a course taught by Eugenia Cheng:

Definition 3: A functor $F:\mathcal{C}\to \mathcal{D}$ creates limits of type $J$ if for every diagram $D\in \mathcal{C}^J$ such that $FD$ has a limit, there exists a cone for $D$ such that its image through $F$ is the limit of $FD$, and moreover $F$ reflects limits.

What are the advantages or disadvantages of one over the other? Surely there was something that motivated the modern definitions (2 and 3) to change the older definition (1)

I'm aware of the remark on the nLab page on creation of limits: that probably answers why the need for a new definition. However, a functor that satisfies definition 2 also satisfies definition 3, but I don't see why a functor that satisfies definition 3 should preserve limits. So the question remains: why prefer definition 2 over definition 3 or viceversa?

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  • $\begingroup$ It's been a while since I posted in MO; I'm not sure this question really fits in here. I thought of asking it in math.SE but I also thought it might be more appropriate here even if it is a quite naïve question. Please indicate me if it is not appropriate and I will promptly move it over there. $\endgroup$ Jul 25, 2012 at 3:03
  • $\begingroup$ If $F$ satisfies definition 3, then any limit cone of $D$ is isomorphic to the one which is asserted to exist (which is a limit cone since it maps to a limit and $F$ reflects limits). Since that cone is mapped by $F$ to a limit of $F D$ by assumption, and $F$ preserves isomorphisms, any other limit cone of $D$ is also mapped to a limit of $F D$. Thus $F$ preserves limits. $\endgroup$ Jul 25, 2012 at 3:59
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    $\begingroup$ @Mike: Your argument seems to presuppose that $FD$ has a limit. As far as I can see, Definition 3 allows a situation where $D$ has a limit and $FD$ doesn't. Definition 2, on the other hand, doesn't allow that situation. $\endgroup$ Jul 25, 2012 at 4:49
  • $\begingroup$ Hmm. Okay, I suppose you're right. In practice, I've never heard the terminology "creates limits" used unless the codomain category has all limits of the type under consideration. So I think I would answer that the difference between 2 and 3 is just that the people writing them down forgot to consider that case carefully, or didn't care what their definition said in that case. Possibly the codomain category having limits ought even to be included in the definition. $\endgroup$ Jul 25, 2012 at 16:54
  • $\begingroup$ What is actually the definition which is "most accepted" in the ct community? $\endgroup$ Jun 4, 2021 at 17:21

2 Answers 2

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I just stumbled about the same problem and suggest the following solution:

All three definitions can be given in terms of the following terminology:

Let $F:A \to B$ be a functor, $D:J \to A$ a diagram in $A$ and $L$ a cone of $D$.

We say $F$

  1. preserves limits of $D$ iff: $\quad L$ is a limiting cone $\Rightarrow$ $FL$ is a limiting cone

  2. reflects limits of $D$ iff: $\quad FL$ is a limiting cone $\Rightarrow$ $L$ is a limiting cone

  3. reflects the existence of limits of $D$ iff: $\quad FD$ has a limiting cone $ \; \Rightarrow \;$ $D$ has a limiting cone.

  4. lifts limits of $D$ iff: $\quad$ Any limiting cone of $FD$ is the image of a limiting cone of $D$.

It should be clear what the respective terminology means, if we replace "limits of $D$" with "limits of shape $J$" or "all limits".

1.,2. and 4. are just the definitions of the nlab, and 3. seems to be fairly natural. Of course one might analog to 3. introduce "preservation of the existence of limits", but this is not needed for answering the question.

Neither $2 \Rightarrow 3 $ nor $3 \Rightarrow 2 $ is in general true.

It is true that $4 \Rightarrow 3$ but the opposite direction is wrong.

Now $F$ creates limits means according to

  • Definition 1: $F$ lifts limits, and the lift is unique even as a pure cone.

  • Definition 2: $F$ preserves and reflects limits and reflects the existence of limits.

  • Definition 3: $F$ reflects and lifts limits.

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  • $\begingroup$ @BrunoStonek: And in these terms we can more easily see the problem with definition 3: the property "$F$ lifts limits of $D$" isn't preserved by isomorphisms of $F$. (nor, I think, does it respect transferring the arrangement along equivalences of $A$ or $B$) $\endgroup$
    – user13113
    Feb 18, 2018 at 23:06
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    $\begingroup$ I am uncomfortable with "reflecting structure". A map $f:X\to Y$ can reflect a property: if $x$ defined over $X$ is such that $f x$ over $Y$ has the property then $x$ already had it. I don't see what it means if you don't have the $x$ before claiming it is the structure. So I think I prefer Saunders Mac Lane's definition, with all due respect to Eugenia Cheng and the author of the nLab page. $\endgroup$ Jun 4, 2021 at 10:22
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    $\begingroup$ @Paul Taylor, you could view "being a limiting cone" as a property of a cone, if this helps from your perspective. At the end of the day reflection of limits is just a well defined and pretty standard concept, no matter whether or not it fits nicely in some greater scheme of what "reflection of something" means in general. $\endgroup$ Jun 4, 2021 at 16:29
  • $\begingroup$ Yes, I accept that "being a limiting cone" is a property, but the cone has to be there first. $\endgroup$ Jun 4, 2021 at 17:29
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    $\begingroup$ Property (3) is called detection of limits (i.e. $F$ detects limits) by some authors (e.g. Adámek–Herrlich–Strecker's Abstract and concrete categories). $\endgroup$ May 31, 2022 at 23:24
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Definition. Let $F:\mathcal{C}\to\mathcal{D}$ be a functor, with $S:\mathcal{I}\to\mathcal{C}$ a diagram of shape $\mathcal{I}$ in $\mathcal{C}$. We say that $F$ creates the limit of $S$ iff $F\circ S$ having a limit $$\pi':\Delta L'\Rightarrow F\circ S$$ in $\mathcal{D}$ implies that there exists a unique (up to iso) source $$\hat L=\{\pi_I:L\to S(I)\}_{I\in{\bf Ob}_\mathcal{I}}$$ in $\mathcal{C}$ such that $$F(\hat L)\cong\pi'$$ as cones, and further that $\pi:\Delta L\Rightarrow S$ is a limit of $S$ in $\mathcal{C}$. We say that $F$ creates limits of shape $\mathcal{I}$ iff $F$ creates the limit of all functors $S:\mathcal{I}\to\mathcal{C}$, and that $F$ creates limits iff $F$ creates limits of shape $\mathcal{I}$ for all 'small' categories $\mathcal{I}$.

This definition is very similar to the one found in The Joy of Cats by Adamek, Herrlich, and Strecker (p. 227 definition 13.17), but they required uniqueness on the nose for $\hat L$ and that $F(\hat L)=\pi'$.

The discussion and lemmas in their book after their definition establish that it implies all other kinds of 'things you want a functor to do to limits' (c.f. the diagram in remark 13.38, p. 232, reproduced below), with the exception of preservation. The definition above also implies all the good behavior one could ask for (except preservation), and is 'respected by equivalence' better than the one in JoC since it makes no reference to uniqueness on the nose or equality -- equivalences create limits in both senses of the definition though.

EDIT: Unless I'm mistaken, their definition is satisfied by the forgetful functors they list but not equivalences, since $\hat L$ is only unique up to iso and we only have $F(\hat L)\cong\pi'$ for the obvious source $\hat L$ corresponding to $\pi'$ by essential surjectivity followed by fullness. Accordingly it seems this definition is better than the one in JoC, as it is satisfied by equivalences which should certainly 'create limits' however we define the term.

I would argue that the lattice of nice implications above (that includes properties explicitly listed in the definitions you reference) make both versions of this definition more satisfying than the other options available. This definition doesn't imply preservation of limits, though, so if preservation of limits matches our intuition for what 'creating a limit' should mean we ought add that $F$ preserves limits to the above definition.

As a note, the subtle difference between the Adamek, Herrlich, and Strecker definition and most other definitions encountered is that we don't a-priori impose the cone coherence conditions on the source in the domain when we assert that it is unique; it must be unique as a source, not as a cone. We then 'create' the cone coherence conditions using $F$; this matches nicely with what 'creating' a limit should mean, in my opinion.


What follows is a proof that equivalences satisfy the above definition of 'creating limits'.

Proof Let $F:\mathcal{C}\simeq\mathcal{D}$ be an equivalence with $S:\mathcal{I}\to\mathcal{C}$ be a diagram of shape $\mathcal{I}$ in $\mathcal{C}$, and suppose that $F\circ S$ has a limit $\pi':\Delta L'\Rightarrow F\circ S$. Since $F$ is essentially surjective there exists some object $L\in\mathcal{C}$ and an isomorphism $u:F(L)\cong L'$, so $\pi'\circ u:\Delta F(L)\Rightarrow F\circ S$ is also trivially a limit of $F\circ S$. Further, since $F$ is full we obtain a source $$\hat L=\{\pi_I:L\to S(I)\}_{I\in\mathcal{I}}$$ in $\mathcal{C}$ with $F(\pi_I)=\pi'_I\circ u$ for all objects $I\in\mathcal{I}$, thus $F(\hat L)\cong\pi'$ as cones since $u$ was an iso and a morphism of cones by the preceding equations for all $I$. If any other source $\hat L''=\{\pi'':L''\to S(I)\}_{I\in\mathcal{I}}$ satisfies $F(\hat L'')\cong\pi'\cong F(\hat L)$ then there exists a unique isomorphism of cones $$w':F(L)\to F(L''),$$ and since $F$ is full this arrow is the image of an arrow $w:L\to L''$ which is unique satisfying $F(w:L\to L'')=w':F(L)\to F(L'')$ by faithfulness of $F$. We then have that $$F(\pi''_I\circ w)=F(\pi''_I)\circ F(w)=F(\pi''_I)\circ w'=F(\pi_I)\implies\pi''_I\circ w=\pi_I,$$ since $F$ is faithful, and $w$ is an iso since $w'$ is with $w^{-1}:L''\to L$ the unique arrow such that $F(w^{-1}:L''\to L)=w'^{-1}:F(L'')\to F(L)$, so $\hat L''\cong\hat L$ and since $\hat L''$ was an arbitrary source $\hat L$ is the unique source up to isomorphism satisfying $F(\hat L)\cong\pi'$. We further have that $\pi:\Delta L\Rightarrow S$ is a limit of $S$; it is a cone since commutes for all arrows $f:I\to I'\in\mathcal{I}$ since $\pi'$ is a cone to $F\circ S$, thus commutes for all arrows $f:I\to I'\in\mathcal{I}$ since $F$ is faithful. This cone is further terminal, since any other cone $'\pi:\Delta'L\Rightarrow S$ gives rise to a cone $F('\pi):\Delta F('L)\Rightarrow F\circ S$ which induces a unique morphism of cones $'u:F('L)\to F(L)=L'$ with $F(\pi_I)\circ{'u}=\pi'_I\circ {'u}=F({'\pi_I})$, and since $F$ is full there exists an arrow $v:{'L}\to L\in\mathcal{C}$ which is unique satisfying $F(v:{'L}\to L)={'u}:F({'L})\to F(L)$ since $F$ is faithful, and $v$ is also a morphism of cones since $$F(\pi_I\circ v)=F(\pi_I)\circ F(v)=\pi'_I\circ{'u}=F({'\pi_I})\implies\pi_I\circ v={'\pi_I}$$ for all objects $I\in\mathcal{I}$, again by faithfulness of $F$.

It is not immediately apparent how to modify the above proof to get that equivalences satisfy the JoC definition of creating limits. We might be able to get around this (non-canonically in a universe with choice) by using the fact that two categories are equivalent iff they have isomorphic skeletons, then choosing a skeleton of $\mathcal{D}$ that already contains the limit $\pi':\Delta L'\Rightarrow F\circ S$ and a skeleton of $\mathcal{C}$ isomorphic to this skeleton -- we should then have that the JoC definition is satisfied as well, unless I'm mistaken.

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