Creation of limits vs. reflection of limits

Question

Addition: by "creation of limits" I mean "strict creation of limits". Leinster just calls it "creation of limits".

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I'm not sure if I understand the difference between creation and reflection of limits of shape $I$. Suppose $D:I\to \mathscr A$ is a diagram and $F:\mathscr A\to\mathscr B$ is a functor. In my words, $F$ creates limits if any limit cone of the diagram $F\circ D$ "comes from" (under $F$) a limit cone of $D$ in a unique way. Reflection says that if we have a limit cone of $F\circ D$ of the form $(F(A)\to FD(i))_{i\in I}$ then "its preimage" (under $F$), which is $(A\to D(i))_{i\in I}$, is a limit cone on $D$.

Does reflection partially implies creation? If I have a limit cone $(F(A)\to FD(i))_{i\in I}$ from the definition of reflection, this meas that it also "comes from" (under $F$) a limit cone of $D$, namely from the limit cone $(A\to D(i))_{i\in I}$, but not necessarily in a unique way.

So is reflection the same as creation, except that uniqueness is not required?

Answer 1

The terminology is inconsistently used by different authors. Here is a discussion about the correct definition on math overflow: https://mathoverflow.net/questions/103065/what-is-the-correct-definition-of-creation-of-limits . Here is the definition which Emily Riehl uses in her book Categories in Context: A functor $F$ creates limits of a certain shape $\mathcal I$ if it satisfies two conditions:

• $F$ reflects limits of that shape. Whenever $F\lambda: Fl \Rightarrow FD$ is a limit cone in the codomain and $D$ is a diagram of shape $\mathcal I$, then $\lambda: l \Rightarrow D$ was a limit in the domain.
• $F$ lifts limits of shape $\mathcal I$. Whenever $FD$ has a limit $\lambda: l \Rightarrow FD$ in the codomain and $D$ is of shape $\mathcal I$, then there exists a cone that $\mu: x \Rightarrow D$ such that $F\mu$ is a limit cone of $FD$. Note that by the first point this automatically implies that $\mu$ is a limit cone. If you like to use strictly creates, then the condition is instead that there is a unique cone $\mu: x\Rightarrow D$ upstairs such that $F\mu = \lambda.$

It follows from your other question that if a functor $F$ creates limits of shape $\mathcal I$ and the codomain has those limits, then the domain has those limits and $F$ preserves and reflects them. This is the situation which is most relevant in practice. Note that If $F$ creates limits, then if the limit exists in the codomain then it does in the domain, hence the terminology creates. This is stronger than reflection.