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Stream: learning: questions

Topic: Domains of naturality

Tobias Fritz (Jul 14 2025 at 23:56):

I'm interested in understanding in what ways a non-natural transformation between functors can fail to be natural. Or put differently, what are the possible domains on which naturality can hold?

Tobias Fritz (Jul 14 2025 at 23:56):

To formulate the question precisely, let $\mathsf{C}$ be any category. Then for any other category $\mathsf{D}$ and functors $F, G : \mathsf{C} \to \mathsf{D}$, let's say that the domain of naturality of a not necessarily natural transformation $t : F \Rightarrow G$ is the class of morphisms in $\mathsf{C}$ with respect to which $t$ is natural. Obviously this class is a subcategory, and it is wide (contains all identity morphisms). Now my question is, which wide subcategories can arise as domains of naturality?

Tobias Fritz (Jul 15 2025 at 00:00):

What I know so far is one additional necessary condition: if $\mathsf{N}$ is a domain of naturality and $m : X \to Y$ is any monomorphism in $\mathsf{N}$ that splits in $\mathsf{C}$, then we have

$$mf \in \mathsf{N} \quad \Longrightarrow \quad f \in \mathsf{N}$$

whenever the composite $mf$ exists. This is trivial if $m$ has a splitting in $\mathsf{N}$, but there are cases where it applies nontrivially.

fosco (Jul 15 2025 at 05:24):

I would expect one can take inspiration from the notion of an [[inserter]] and describe the domain of naturality of a family $t : \prod_X \hom(FX,GX)$ as a certain weighted limit

Amar Hadzihasanovic (Jul 15 2025 at 05:24):

Sorry if this is a too trivial observation, but domains of naturality in your sense should correspond exactly to functors that arise as pullbacks in $\mathrm{Cat}$ of the evident inclusions

$$\mathrm{Arr}(\mathsf{C}) \hookrightarrow \mathrm{Arr}_{nc}(\mathsf{C})$$

of the category of arrows and commutative squares into the category of arrows and not necessarily commutative squares in $\mathsf{C}$. So it seems to me like a good strategy would be to focus on these as “universal” domains of naturality, and then properties of arbitrary domains of naturality would be exactly those that are pullback-stable.

Amar Hadzihasanovic (Jul 15 2025 at 05:30):

(In particular this implies that the pasting law for pullbacks applies to domains of naturality, i.e. the pullback of a domain of naturality along an arbitrary functor is a domain of naturality, and if you have a domain of naturality such that the pullback square exhibiting it factors through another pullback square of categories, then the functor “in the middle” of the factorisation is also a domain of naturality.)

Tobias Fritz (Jul 15 2025 at 05:41):

Thanks both! Do you see any way to turn those observations into more concrete criteria that can be used to decide if a given subcategory $\mathsf{N} \subseteq \mathsf{C}$ is or is not a domain of naturality?

Tobias Fritz (Jul 15 2025 at 05:43):

I do have a situation with a transformation between two functors for which I can prove naturality on a subcategory but not yet on the whole category. So I wonder if there might be purely formal ways to do this by arguing that the smallest domain of naturality which contains my subcategory is the whole category; it does feel like a situation where this might be the case. Because of this I'm especially interested in necessary conditions for domains of naturality, like the one involving split monos I mentioned above.

fosco (Jul 15 2025 at 05:47):

What I have in mind is something like this:

Given $F,G : C\to D$ functors, and $t : \prod_X D(FX,GX)$, the "naturizer" $N_t(F,G)$ is the wide subcategory of C on all objects and only those arrows on which t is natural, i.e. a morphism in $N_t(F,G)$ is a pair $(\phi,p)$ where $\phi : X\to Y$ is an arrow of $C$ and $p$ a proof that $G\phi\circ t_X = t_Y\circ F\phi$.

fosco (Jul 15 2025 at 05:49):

The naturizer of F,G is a weighted limit; then, a domain of naturality is... a monomorphism into the naturizer?

Amar Hadzihasanovic (Jul 15 2025 at 07:05):

Another semi-trivial observation would be that, given a wide subcategory $\mathsf{W}$ of $\mathsf{C}$, one can concoct a "minimal" transformation whose domain of naturality contains it.

We let $I \square_\mathsf{W} \mathsf{C}$ be

• the "funny tensor product" of the arrow with $\mathsf{C}$, which is generated by two copies of $\mathsf{C}$ labelled 0 and 1, together with morphisms $t_a: (a, 0) \to (a, 1)$ for each object $a$ in $\mathsf{C}$,
• quotiented by the relation $t_b \circ (f, 0) = (f, 1) \circ t_a$ for all morphisms $f: a \to b$ in $\mathsf{W}$.

Then there is an evident transformation from the inclusion $(-, 0)$ to the inclusion $(-, 1)$, such that its domain of naturality contains $\mathsf{W}$, and this must be the smallest such domain of naturality.

Amar Hadzihasanovic (Jul 15 2025 at 07:09):

Then the following are equivalent:

• the smallest naturality domain containing $\mathsf{W}$ is the whole category,
• the canonical functor $I \square_\mathsf{W} \mathsf{C} \to I \times \mathsf{C}$ is an isomorphism

Amar Hadzihasanovic (Jul 15 2025 at 07:15):

I think this just makes precise the intuition that the domain of naturality "generated" by a wide subcategory $\mathsf{W}$ consists exactly of those morphisms for which $t \circ (f, 0) = (f, 1) \circ t$ is provable from the equations restricted to morphisms in $\mathsf{W}$ in the algebra of units and composition of morphisms

Amar Hadzihasanovic (Jul 15 2025 at 07:16):

So I doubt that there will be interesting properties beyond ones involving split monos and split epis like the one you already mentioned, and its dual?

Tobias Fritz (Jul 15 2025 at 10:13):

Amar Hadzihasanovic said:

Then the following are equivalent:

• the smallest naturality domain containing $\mathsf{W}$ is the whole category,
• the canonical functor $I \square_\mathsf{W} \mathsf{C} \to I \times \mathsf{C}$ is an isomorphism

That is very nice @Amar Hadzihasanovic!

Tobias Fritz (Jul 15 2025 at 10:14):

Amar Hadzihasanovic said:

So I doubt that there will be interesting properties beyond ones involving split monos and split epis like the one you already mentioned, and its dual?

That agrees perfectly with my own intuition about the problem.

Tobias Fritz (Jul 15 2025 at 10:23):

FWIW, here's the particular case that I'm dealing with: Let $\mathsf{C}$ be the category of countable compact Hausdorff spaces with all functions as morphisms and $\mathsf{N} \subseteq \mathsf{C}$ the wide subcategory consisting of the continuous funcctions. I have a transformation between two functors out of $\mathsf{C}$ which is natural on $\mathsf{N}$. I was wondering if the naturality on all of $\mathsf{C}$ might be automatically implied. At this point it seems rather unlikely that this would be the case. (Fortunately I have another approach to the problem that seems more promising, based on additional properties of the two functors involved.)

Morgan Rogers (he/him) (Jul 17 2025 at 06:30):

To expand on Amar's pullback law condition:
Suppose for $f\in C$ you have a jointly epic collection of morphisms in $N$ each of which stays in $N$ when postcomposed with $f$; then $f$ will be in $N$ too.
This does nothing for your example, though: if a map is discontinuous at some point then any jointly epimorphic family will have discontinuous preimages of the point.