# Creation of limits vs. reflection of limits

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

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?

• My memory is a bit fuzzy on this, so take this with a grain of salt - but I think the difference is reflecting limits requires a candidate cone in the domain category as a hypothesis where the image in the codomain is a limit cone, whereas creating limits only requires a limit cone to exist in the codomain category as a hypothesis. So creating limits has a weaker hypothesis, and thus is a stronger condition. Commented May 13, 2021 at 23:02

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.