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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.

Morgan Rogers (he/him) (Jan 24 2026 at 17:42):

(looks right to me!)

Ruby Khondaker (she/her) (Jan 24 2026 at 18:49):

Thanks Morgan! I’ll update the nlab pages accordingly, and also make a dedicated page for “preservation, reflection, lifting of structure”

Aaron David Fairbanks (Feb 11 2026 at 01:51):

I noticed the nLab page defines what it means for a functor $F : C \to D$ to create limits of a specific diagram in the domain $J: I \to C$. My initial reaction was, it would be more intuitive to define what it means for $F : C \to D$ to create limits of a specific diagram in the codomain $J: I \to D$. There is no difference between these if we want to talk about functors creating a general class of limits. But in my head, we "lift" or "reflect" particular limit diagrams in the codomain to obtain limit diagrams in the domain. What do you think?

This is also how creation appears in practice. Monadicity theorems are about creation of specific colimits in the codomain, and (relatedly) monadic functors create colimits in the codomain preserved by the monad $M$ and its square $M \circ M$.

Actually, maybe it's best to have both options available. It could make sense to speak of a functor reflecting/lifting/creating limits of a diagram in the domain as well as to speak of a functor reflecting/lifting/creating limits of a diagram in the codomain.

Ruby Khondaker (she/her) (Feb 11 2026 at 07:41):

What definition do you suggest for a functor reflecting limits of a diagram in the codomain?

The definition of reflection in the domain is that you have $J \colon I \to C$, and that any cone for $J$ that is mapped to a limit cone for $F \circ J$ must already be a limit cone for $J$.

If we instead have $J \colon I \to D$, it’s not clear to me how we phrase reflection?

Aaron David Fairbanks (Feb 11 2026 at 16:52):

Ah, sorry for being unclear. I would just say limits of a diagram in the codomain are reflected/lifted/created if, in the current terminology, limits of all diagrams in the domain sent to it are reflected/lifted/created.

For example a diagram $J : I \to D$ is reflected if any cone to a diagram in the domain sent to a limit cone of $J$ is a limit cone.

The other definition is more general, but this definition matches my use cases more closely. I had defined creation this way in a paper I'm currently writing, but maybe that's not normal.

Ruby Khondaker (she/her) (Feb 11 2026 at 16:54):

Hm so unfolding this - you have a diagram in the domain $J' \colon I' \to C$ with a cone, and then applying $F$ gives a cone over $F \circ J' \colon I' \to D$. Presumably to say this is "sent to a limit cone of $J$" we need $I' = I$ and some specified natural transformation $F \circ J' \Rightarrow J$? But now it seems you have many ways to specify such a natural transformation.

Aaron David Fairbanks (Feb 11 2026 at 17:00):

I would put $F \circ J' = J$.

Aaron David Fairbanks (Feb 11 2026 at 17:08):

Maybe people don't do this because it's not respected by equivalence. You could have something respected by equivalence by only requiring $F \circ J' \cong J$, similarly to what you said. Do you see any problem with that?

Ruby Khondaker (she/her) (Feb 11 2026 at 17:37):

Hm, lemme think. So we have a diagram $J \colon I \to D$ and a functor $F \colon C \to D$. We say that $F$ reflects limits of $J$ if, whenever you have a diagram $J' \colon I \to C$ with $F \circ J' \cong J$ and a cone for $J'$ that maps under $F$ to a limiting cone, then it was already a limiting cone for $J'$.

In other words, $F$ reflects limits for $J$ if and only if any diagram $J'$ with $F \circ J' \cong J$ has its limits reflected by $F$. I guess that would be invariant under equivalence?

Maybe all you want is "F reflects limits of shape I"?

Aaron David Fairbanks (Feb 11 2026 at 17:57):

Yeah, that's what I'm thinking for the equivalence-invariant version. Though I had by default been using $F \circ J' = J$ on the nose.

It's very possible nobody else but me finds this way more intuitive!

Aaron David Fairbanks (Feb 11 2026 at 17:57):

My motivation is that when we use creation of colimits in situations related to monadicity, we're talking about creation of colimits of specific diagrams in the codomain. (In particular, not just all diagrams of a specific shape.)

Ruby Khondaker (she/her) (Feb 11 2026 at 20:42):

Hm, I think there’s a detail I’m missing - which diagrams for monadicity are we saying get created in the codomain?

Ruby Khondaker (she/her) (Feb 11 2026 at 20:50):

The main result I know is that monadic functors create all limits, and create all colimits that T and TT preserve

Aaron David Fairbanks (Feb 11 2026 at 21:29):

A right adjoint functor is monadic if and only if it creates split coequalizers in the codomain.

https://ncatlab.org/nlab/show/monadicity+theorem

Another formulation is that a right adjoint functor is monadic if and only if it creates absolute colimits in the codomain.

Ruby Khondaker (she/her) (Feb 11 2026 at 21:34):

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Right but the definition here is really still the standard one right? You have a parallel pair in $D$ which, if you set $I = \bullet \overset{\longrightarrow}{\longrightarrow} \bullet$, can be viewed as a diagram $J \colon I \to D$. Then you focus on a particular subset of these diagrams, those which are mapped to split coequalisers under $U \colon D \to C$.

Then the statement is that, if $J$ is $U$-split, then $U$ creates coequalisers of $J$, in the ordinary sense. Importantly I think "creates split coequalisers" could be misleading, since we're only guaranteed a coequaliser diagram in $D$, not a split one.

Ruby Khondaker (she/her) (Feb 11 2026 at 21:36):

The part that says "there exists a coequaliser $e$ of $f, g$ in $D$ which is preserved by $U$" is saying that $U$ lifts coequalisers (i.e. colimits) for $J$, and the part that says "any fork in $D$ whose image in $C$ is a split coequaliser must itself be a coequaliser" is saying that $U$ reflects coequalisers for $J$, if I understand correctly.

Aaron David Fairbanks (Feb 11 2026 at 21:55):

That page on the nLab is using the standard definition, right. In contrast, I stated these results using the non-standard definition of creation of colimits in the codomain from this thread.

Yeah, "creates split coequalizers in the codomain" could be misleading. Less concisely, "creates coequalizers that are split coequalizers in the codomain" might be better. To me it's more understandable than "creates $U$-split coequalizers", though others may disagree.

Kevin Carlson (Feb 11 2026 at 22:00):

I think it's definitely more understandable but people wanted some kind of plausible shorthand.