Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: Creation of limits as lifting + reflection?

Ruby Khondaker (she/her) (Jan 18 2026 at 16:29):

I've been refactoring the nlab pages for the various ways in which functors can interact with limits - preservation, reflection, creation, lifting. In this post, I'll use the equivalence-invariant definition of lifting, which only requires lifts of limiting cones to exist up to isomorphism.

With regards to creation of limits - is it true that a functor creates limits if and only if it both reflects and lifts them? Spelling this out in detail:

• Lifting implies that, for a diagram $J : I \to C$ and $F : C \to D$, if there is a limiting cone for $F \circ J$, then there is a limiting cone for $J$ that is preserved by $F$.
• Reflection implies that any cone for $J$ that is sent to a limiting cone under $F$ must necessarily be limiting.

Thus, lifting and reflection together imply the following property:

• Suppose we have a limiting cone $(d, \eta)$ for $F \circ J$. Then there exists, up to isomorphism, a unique cone for $J$ whose image under $F$ is isomorphic to $(d, \eta)$. Moreover, this cone is a limit for $J$.

It's easy to see that this property implies lifting. Moreover, it also implies reflection - if we have a cone $(c, \alpha)$ for $J$ such that $(F c, F \cdot \alpha)$ is limiting for $F \circ J$, then we know there exists, up to iso, a unique $(c', \alpha')$ whose image under $F$ is isomorphic to $(F c, F \cdot \alpha)$ - in particular, $(c, \alpha)$ is necessarily isomorphic to $(c', \alpha')$. But then, since $(c', \alpha')$ is limiting, so is $(c, \alpha)$.

From what I can remember of my course, this property is equivalent to creation of limits. So I wanted to check - is it sensible to view "creation" as the conjunction of "lifting" and "reflection"? There's a separate issue as to whether you should require the limit to actually exist in the codomain.

Incidentally, MacLane's "strict" definition of creation of limits seems to be a strictified version of the property I laid out. It says:

• Suppose we have a limiting cone $(d, \eta)$ for $F \circ J$. Then there exists a unique cone $(c, \alpha)$ for $J$ such that $(F c, F \cdot \alpha) = (d, \eta)$. Moreover, this cone is limiting.

So this gives me some confidence that "reflection + lifting" is the morally correct definition for creation. I know that this terminology is used for other kinds of structure besides limits and colimits; in those cases, is it still sensible to think of creation as reflection + lifting? I'm curious whether it might be useful to have an nlab page dedicated to explaining "preservation, reflection, lifting" of structure in the general sense.