What is the "correct" definition of creation of limits?

Question

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?

Answer 1

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.

Answer 2

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.

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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.