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Stream: learning: questions

Topic: Hom-functor preserves limits proof

Ruby Khondaker (she/her) (Feb 09 2026 at 09:40):

After understanding the definition of [[preserved limit]] better, I have a question about the current nlab article on [[hom-functor preserves limits]].

The proposition that the article proves is the following:

Let $X_\bullet \colon \mathcal{J} \to \mathcal{C}$ be a diagram. Then, if the limit $\lim_i X_i$ exists in $\mathcal{C}$ then for all $Y \in \mathcal{C}$ there is a natural isomorphism $\text{Hom}_\mathcal{C}(Y, \lim_i X_i) \cong \lim_i (\text{Hom}_\mathcal{C}(Y, X_i))$, where on the right we have the limit over the diagram of hom-sets given by $\text{Hom}_\mathcal{C}(Y, -) \circ X_\bullet \colon \mathcal{J} \overset{X}{\longrightarrow} \mathcal{C} \overset{\text{Hom}_\mathcal{C}(Y, -)}{\longrightarrow} \mathbf{Set}$.

However, I'm not entirely convinced this is the result we want to show. The page on preservation of limits emphasises that, for a functor $F$ to preserve a limit, it is not enough to show that $F(\lim_i X_i) \cong \lim_i F(X_i)$. Instead, the point is to preserve the entire universal cone. Another way to say this is that there is a canonical comparision map $F(\lim_i X_i) \to \lim_i F(X_i)$ obtained by applying $F$ to the legs of the limit cone; to preserve the limit, this comparison map must be an isomorphism.

Taking $F = \text{Hom}_\mathcal{C}(Y, -)$, shouldn't the proof involve explicitly applying $F$ to the legs of the limit cone, and showing that the resulting cone is a limit cone in $\mathbf{Set}$? Otherwise it seems like we're proving a weaker version of limit preservation than we actually want.

Here's an example of the phenomenon I mean. In $\mathbf{Set}$, we have that $\mathbb{N} \times \mathbb{N} \cong \mathbb{N}$ as sets, by a standard "snaking" argument. Consider the usual limit cone diagram for $\mathbb{N} \times \mathbb{N}$ over two copies of $\mathbb{N}$ with the usual cartesian product projections. Then the constant functor $\Delta_\mathbb{N} \colon \mathbf{Set} \to \mathbf{Set}$ sending every set to $\mathbb{N}$ (and every morphism to the identity) preserves this limit at the object level, since $\Delta_\mathbb{N}(\mathbb{N} \times \mathbb{N}) = \mathbb{N} \cong \mathbb{N} \times \mathbb{N} \cong \Delta_\mathbb{N}(\mathbb{N}) \times \Delta_\mathbb{N}(\mathbb{N})$. However, the limit cone with projections $\pi_1, \pi_2 \colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ gets mapped to an "identity" cone $\text{id}_\mathbb{N}, \text{id}_\mathbb{N} \colon \mathbb{N} \to \mathbb{N}$. And the latter is not a limit cone in $\mathbf{Set}$!

So in general, merely having an isomorphism $F(\lim_i X_i) \cong \lim_i F(X_i)$ is not enough to show $F$ preserves limits. Does the proof at the nlab page need to be updated, then?

Ruby Khondaker (she/her) (Feb 09 2026 at 09:50):

For reference, this would be my proposed proof:

Let $X_\bullet \colon \mathcal{J} \to \mathcal{C}$ be a diagram with a limit $(L, \lambda_i)$, with $\lambda_i \colon L \to X_i$ the legs of the limit cone. Then for all $Y \in \mathcal{C}$, the functor $\text{Hom}_\mathcal{C}(Y, -)$ preserves this limit.

Proof - we need to show that $(\text{Hom}_\mathcal{C}(Y, L), \text{Hom}_\mathcal{C}(Y, \lambda_i))$ is a limit cone in $\mathbf{Set}$. So, take a set $S$, and a family of maps $s_i \colon S \to \mathcal{C}(Y, X_i)$ that form a cone. This means that, for each $s \in S$, we have a family of morphisms $s_i(s) \colon Y \to X_i$ that form a cone in $\mathcal{C}$, by inspecting the definition of the diagram maps in $\mathbf{Set}$. By the universal property of the limit, there exists a unique $\alpha(s) \colon Y \to L$ such that $s_i(s) = \lambda_i \circ \alpha(s)$ for all $i \in \mathcal{J}$. But this is equivalent to saying that there is a unique $\alpha \colon S \to \text{Hom}_\mathcal{C}(Y, L)$ such that $s_i = \text{Hom}_\mathcal{C}(Y, \lambda_i) \circ \alpha$ for all $i \in \mathcal{J}$. Thus, $\text{Hom}_\mathcal{C}(Y, -)$ preserves this limit.

This is just my first attempt - if anyone has suggestions for how to make the proof more efficient, I'd be very open to hearing them!

David Wärn (Feb 09 2026 at 09:53):

What's missing from the claim on the nLab is simply the claim that the map $C(Y,\mathrm{lim}_i X_i) \to \mathrm{lim}_i C(Y,X_i)$ is the "expected one". This is a common "anti-pattern" when people state results like this: instead of saying that a certain easy-to-define map is invertible, they write "there exists a natural iso ...". You see this for example when people state the Yoneda lemma.

Ruby Khondaker (she/her) (Feb 09 2026 at 09:54):

Right, though I'm not actually sure the proof I provided explicitly shows that there is a natural iso, since I work with a fixed $Y$ throughout. I think the goal of showing that $\text{Hom}_\mathcal{C}(Y, -)$ preserves limits is separate to the goal of showing that there is a natural iso $\text{Hom}_\mathcal{C}(Y, lim_i X_i) \cong lim_i \text{Hom}_\mathcal{C}(Y, X_i)$. Neither immediately implies the other, I believe?

Josselin Poiret (Feb 09 2026 at 10:02):

@Ruby Khondaker (she/her) neither of your statements actually imply the other, but the “correct” general statement is indeed that the obvious map $\mathrm{Hom}(-, \mathrm{lim}_i F(i)) \rightarrow \mathrm{lim}_i \mathrm{Hom}(-, F(i))$ is an iso

Ruby Khondaker (she/her) (Feb 09 2026 at 10:04):

Hm, here's another idea I have for turning the nlab proof into a valid one:

We've shown there's a natural iso $\text{Hom}_\mathcal{C}(Y, \lim_i X_i) \cong \lim_i \text{Hom}_\mathcal{C}(Y, X_i)$. By yoneda, this is classified by an element of $\lim_i \text{Hom}_\mathcal{C}(\lim_j X_j, X_i)$, and I think by inspecting the iso used in the proof, this element is just the legs of the limit cone $\lambda_i \colon \lim_j X_j \to X_i$. Then, since natural isomorphisms are pointwise isomorphisms, we have that the map $\text{Hom}_\mathcal{C}(Y, \lim_i X_i) \to \lim_i \text{Hom}_\mathcal{C}(Y, X_i)$ given by postcomposition with the $\lambda_i$ is an iso. But I think that's precisely the "canonical map" we wish to show is an isomorphism?

Josselin Poiret (Feb 09 2026 at 10:05):

more generally, you can take this as the definition of a limit: a functor $F$ has a limit if there is an object $\mathrm{lim}_i F(i)$ equipped with an isomorphism $\mathrm{Hom}(-, \mathrm{lim}_i F(i)) \simeq \mathrm{lim}_i \mathrm{Hom}(-, F(i))$

Ruby Khondaker (she/her) (Feb 09 2026 at 10:05):

Right, that's defining limits representably by how they work in Set, if I understand correctly?

Josselin Poiret (Feb 09 2026 at 10:05):

yes

Josselin Poiret (Feb 09 2026 at 10:06):

don't think too hard about "fixing" existing proofs that actually do not achieve what they claim to do: it's often easier to just directly prove the correct statement

Ruby Khondaker (she/her) (Feb 09 2026 at 10:07):

In that case, which of the two proofs I offered do you think would suffice? The first one considering arbitrary sets S and showing we do indeed get a limit cone?

Josselin Poiret (Feb 09 2026 at 10:09):

Ruby Khondaker (she/her) said:

The first one considering arbitrary sets S and showing we do indeed get a limit cone?

yes, that's the most direct proof falling out of the definition of limit. You're missing the additional naturality proof, but it should go along the same line

Josselin Poiret (Feb 09 2026 at 10:10):

although i would probably factorize the proof by a general result that computes limits in $\mathbf{Set}$

Josselin Poiret (Feb 09 2026 at 10:13):

a limit in $\mathbf{Set}$ of a diagram $F : I → \mathbf{Set}$ is the set of "compatible families", ie. families $α : (i : I_0) → F(i)$ such that $F(f)(α(i)) = α(j)$ for any $f : i → j$

Ruby Khondaker (she/her) (Feb 09 2026 at 10:18):

Could you maybe spell out in more detail how your proposed proof would go?

David Wärn (Feb 09 2026 at 10:18):

I would also distinguish between two statements about the obvious map $\mathrm{Hom}(Y,\mathrm{lim}_i X_i) \to \mathrm{lim}_i \mathrm{Hom}(Y,X_i)$. One statement is that this is an isomorphism of sets. Another statement is that it is natural in $Y$, $X$, and anything else you can think of. These statements are independent of each other and the proof of one does not really involve the other. Together they give you the desired natural isomorphism

Ruby Khondaker (she/her) (Feb 09 2026 at 10:32):

Actually, how are we even defining the right-hand side? Limits are only defined up to isomorphism after all. Preservation of limits says that “(Hom(Y, lim_i X_i), Hom(Y, lambda_i)” is a limit cone in Set. Could we not just take the right-hand side to be the same as the left-hand side and have these maps all be identities?

Or is the RHS referring to some specific construction of the limit in Set…?

Ruby Khondaker (she/her) (Feb 09 2026 at 10:45):

image_28AB674D-6B2C-4B27-A265-3ED4B0200DFC_1770633862.jpeg

Ok, Riehl’s book resolved this confusion for me. There is an additional condition you need to check, that the maps we define commute with the projections to the product.

Ruby Khondaker (she/her) (Feb 09 2026 at 10:46):

Honestly, I think I would just prefer to show this result using the bare-bones “limit cone is mapped to limit cone” characterisation, rather than faffing around with natural isomorphisms…

David Wärn (Feb 09 2026 at 11:11):

Indeed, the data of an isomorphism $Z \cong \mathrm{lim}_i X_i$ (where the RHS denotes the apex of some limit cone, it doesn't matter which one) is precisely the data of a limit cone from $X$ to $Z$.

Ruby Khondaker (she/her) (Feb 09 2026 at 11:53):

Ok, I believe I've fixed the nlab proof of this now.

Josselin Poiret (Feb 09 2026 at 13:16):

Ruby Khondaker (she/her) said:

Limits are only defined up to isomorphism after all.

Maybe some will find me pedantic, but this is a non-question: limits are not "defined up to isomorphism", there just isn't a notion of "a" limit of a functor, just of what it means to be a limit cone

Ralph Sarkis (Feb 09 2026 at 15:07):

This is in contrast with [[subobjects]] which are defined up to isomorphism.

Ruby Khondaker (she/her) (Feb 10 2026 at 09:44):

Still, the yoneda-y proofs with manipulations of hom-functors feel cleaner than the manipulations “in components”. There should be a way to keep those kinds of proofs while also proving the true coherent statement of “preserves limits”, surely.

Ruby Khondaker (she/her) (Feb 10 2026 at 13:01):

Thinking about it a bit more, I believe you can keep the yoneda-y proof so long as you check that, upon following the identity, you obtain the desired canonical map.

Rémy Tuyéras (Feb 11 2026 at 00:44):

@Ruby Khondaker (she/her) The fact that hom-sets preserve limits does not follow from a proof, but rather directly from the definition of limits. See below.

Definition 1. Let $F:I \to \mathcal{C}$ be a functor and $\Delta_I$ be the obvious constant functor $\mathcal{C} \to \mathcal{C}^I$. A limit for a functor $F$ is a natural transformation $\eta_F: \Delta_I(L) \Rightarrow F$ such that every natural transformation $d:\Delta_I(X) \Rightarrow F$ in $\mathcal{C}$ factorizes uniquely as $d = \eta_F \circ \Delta_I(f)$ where $f: X \to L$ is an arrow in $\mathcal{C}$.

Definition 2. Given $\eta_F$, this is equivalent to requesting that the function $\mathcal{C}(X,L) \to \mathcal{C}^I(\Delta_I(X),F)$ is a bijection.

Remark 1. Now, it is quite straightforward to show that $\mathcal{C}^I(\Delta_I(X),F)$ is exactly the limit $\mathsf{lim}\,\mathcal{C}^I(X,F(-))$ in $\mathbf{Set}$ (hint: just describe the natural transformations as families of compatible arrows)