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Stream: learning: questions

Topic: Parameter Theorems

Ruby Khondaker (she/her) (Feb 04 2026 at 21:46):

After finally reading "Categories for the Working Mathematician", I realised that the 'parameter theorems' mentioned in the text fall out as a general consequence of [[representability determines functoriality]], whose statement I will reproduce here:

Suppose $P : \mathcal{C}^\text{op} \times \mathcal{D} \to \mathbf{Set}$ is a profunctor that is representable in the $\mathcal{C}^\text{op}$ argument - meaning, for each $d \in \mathcal{D}$, the functor $P(-, d) \colon \mathcal{C}^\text{op} \to \mathbf{Set}$ is representable. Then, by choosing representations for each $P(-, d)$, we obtain a functor $F \colon \mathcal{D} \to \mathcal{C}$ with $P(c, d) \cong \mathcal{C}(c, Fd)$.

I.e. a bifunctor which is "pointwise representable" in a particular argument determines a functor. Here I am using "representable in an argument" in the same sense as "linear in an argument" for multilinear maps - if all other arguments are fixed, the resulting functor is representable.

Let me illustrate the ways in which various "parameter theorems" fall out as consequences of this general one:

• Adjunctions with a parameter. image.png. To prove this, we consider a trifunctor $H \colon X^\text{op} \times P^\text{op} \times A \to \mathbf{Set}$ defined on objects by $H(x, p, a) := \text{Hom}_A(F(x, p), a)$. The assumptions of the theorem imply that $H$ is representable in the $X^\text{op}$ argument - meaning, for each fixed $p \in P, a \in A$, the corresponding functor $X^\text{op} \to \mathbf{Set}$ is representable, by $G(p, a)$. Thus, by "representability determines functoriality", we may extend $G$ to a bifunctor $P^\text{op} \times A \to X$.
  
• Limits with a parameter. image.png. To prove this, we consider a bifunctor $H \colon X^\text{op} \times P \to \mathbf{Set}$ defined on objects by $H(x, p) := \text{Nat}(\Delta_x, T(-, p))$, where $\Delta_x \colon J \to X$ is the constant functor at $x \in X$. The hypotheses of the theorem imply that $H$ is representable in the $X^\text{op}$ argument, so we obtain a functor $P \to X$ mapping $p \mapsto \lim_j T(j, p)$.
  
• Ends with a parameter. image.png. To prove this, we consider a bifunctor $H \colon X^\text{op} \times P \to \mathbf{Set}$ defined on objects by $H(x, p) := \text{Wedge}(x, T(p, -, -))$. The hypotheses of the theorem imply that $H$ is representable in the $X^\text{op}$ argument, so we obtain a functor $P \to X$ mapping $p \mapsto \int_c T(p, c, c)$.
  

I've also found this notion of being "representable in an argument" helpful for understanding adjunctions and their variants:

• An adjunction between two categories $C$ and $D$ consists of a bifunctor $H \colon C^\text{op} \times D \to \mathbf{Set}$ that is representable in each argument, so that $\text{Hom}(Fc, d) \cong H(c, d) \cong \text{Hom}(c, Gd)$.
• A pair of "mutual right adjoints" is a bifunctor $H \colon C^\text{op} \times D^\text{op} \to \mathbf{Set}$ that is representable in each argument, so that $\text{Hom}(c, Fd) \cong H(c, d) \cong \text{Hom}(d, Gc)$. Similarly, a pair of "mutual left adjoints" is a bifunctor $H \colon C \times D \to \mathbf{Set}$ that is representable in each argument.
• A two-variable adjunction is a trifunctor $H \colon C^\text{op} \times D^\text{op} \times E \to \mathbf{Set}$ that is representable in each argument, providing isomorphisms $E(c \otimes d, e) \cong D(d, \text{hom}_l(c, e)) \cong C(c, \text{hom}_r(d, e)) \cong H(c, d, e)$.

In summary, I've found "multirepresentability" to be a helpful way to unify the various parameter theorems I've come across, as well as various notions of adjunctions. Is there a good reference that goes through this in more detail?

Nathanael Arkor (Feb 06 2026 at 11:01):

This is a good observation – unfortunately, I'm not aware of any expository reference for these ideas! There are several further related ideas that tie together very satisfactorily, but which I've never seen mentioned in the literature (in fact, I believe some of these connections have gone overlooked even by experts). The first observation is that the theorem you mention is a corollary of a more general result.

Theorem. Let $P : |\mathcal B| \times \mathcal A \to \text{Set}$ and $Q : \mathcal B^{\text{op}} \times |\mathcal A| \to \text{Set}$ be functors, equipped with a family of isomorphisms of sets $P(b, a) \cong Q(b, a)$. Then $P$ and $Q$ are uniquely equipped with isomorphic functor structures for which $P \cong Q$ is a natural isomorphism if and only if a single octagon condition holds.

I won't spell out the octagon condition for now, but I will explain how to derive it momentarily. This theorem says, in some sense, that "functoriality is for free". For instance, taking $L : |\mathcal A| \to |\mathcal B|$ and $R : |\mathcal B| \to |\mathcal A|$ to be functions and $P := \mathcal A(1, R)$ and $Q := \mathcal B(L, 1)$, this recovers Street's minimal presentation of an adjunction (as referenced on that nLab page). (In this case, the octagon condition reduces to a square.)

It may appear that the theorem above is very general, but it's a consequence of an elementary result: in fact, the very first proposition of Mac Lane's book!

Given functors $F : |\mathcal X| \times \mathcal Y \to \mathcal Z$ and $G : \mathcal X \times |\mathcal Y| \to \mathcal Z$ that agree on objects, they extend to a functor $\mathcal X \times \mathcal Y \to \mathcal Z$ if and only if a single square condition holds.

How can we prove the theorem from this proposition? Well, the isomorphism $P(b, a) \cong Q(b, a)$ equips $P$ with the structure of a functor $\mathcal B^{\text{op}} \times |\mathcal A| \to \text{Set}$, and in this case two sides of the square expand into a composable triple (apply the isomorphism, apply functoriality of $Q$, apply the isomorphism), recovering the octagon condition.

So these parameter theorems that Mac Lane proves (some of which take a fair amount of manipulation to establish) are actually all consequences of the very first result he proves, but this is never (to my knowledge) observed.

Nathanael Arkor (Feb 06 2026 at 11:04):

(There are analogues of everything I said in the context of enriched categories, but one must be a little more careful there, as it is not necessarily true that enriched distributors can be expressed as enriched functors, in which case the first theorem I mentioned does not reduce to Mac Lane's "bifunctor proposition", but must be established independently.)

Ruby Khondaker (she/her) (Feb 06 2026 at 12:02):

I think I recognise that result! It's essentially the bifunctor lemma covered in Awodey's book:

image.png

One thing I'm a little confused by is how this directly implies "representability determines functoriality", though. If I understand correctly, we start with $P : \mathcal{C}^\text{op} \times \mathcal{D} \to \mathbf{Set}$ and $Q : \mathcal{C}^\text{op} \times |\mathcal{D}| \to \mathbf{Set}$ given by $Q(c, d) := \mathcal{C}(c, Fd)$. We have this family of isomorphisms of sets $P(c, d) \cong Q(c, d)$, and using this we may extend $Q$ to a functor $\mathcal{C}^\text{op} \times \mathcal{D} \to \mathbf{Set}$. Presumably the naturality in $c$ is used to verify the octagon condition.

However, what I don't quite see is how we obtain the functoriality of $F$ from this? From what I remember, you need to use some amount of Yoneda to actually derive representability derives functoriality, whereas your lemma doesn't appear to use any of it.

Essentially, I can buy that $Q$ extends to a bifunctor, but what we want is the existence of a functor $F : \mathcal{D} \to \mathcal{C}$ for which $Q$ is the companion. That part should require using the Yoneda Lemma in the background, I believe.

Nathanael Arkor (Feb 06 2026 at 13:30):

Right, I was a little hasty when I wrote "the theorem you mention is a corollary of a more general result": the more general result is the one arising from combining "representability determines functoriality" with the theorem I mention, which allows you to eliminate the functoriality/naturality assumption entirely (replacing them both with the octagon condition).

Ruby Khondaker (she/her) (Feb 06 2026 at 13:34):

I see, that makes sense. I'd be interested to see applications of that more general result.

"Representability determines functoriality" is a bit of a mouthful, I'm realising. Do you think a suitable name for this might be the "Parameter Theorem"? Generalising the names that MacLane gave his results.

Ruby Khondaker (she/her) (Feb 06 2026 at 16:37):

Just to check I understand you correctly, the more general result you are referring to is the following:

Let $P : |\mathcal{C}^\text{op}| \times \mathcal{D} \to \mathbf{Set}$ and $F : |\mathcal{D}| \to |\mathcal{C}|$ be a functor. Suppose that we have a specified family of isomorphisms $P(c, d) \cong \mathcal{C}(c, Fd)$. Then so long as an "octagon" condition holds, we can extend $P$ to a bifunctor and $F$ to a functor such that these isomorphisms are natural.

Proof - Define $Q : \mathcal{C}^\text{op} \times |\mathcal{D}| \to \mathbf{Set}$ by $Q(c, d) = \mathcal{C}(c, Fd)$. The octagon condition is precisely what is needed to apply the bifunctor lemma to make $P$ and $Q$ naturally isomorphic bifunctors. Then, we can apply ordinary "representability determines functoriality" to deduce functoriality of $F$.

I suppose I would say that the bifunctor lemma and representability determines functoriality are two independent "pieces" for this more general result.

Nathanael Arkor (Feb 07 2026 at 07:59):

I see, that makes sense. I'd be interested to see applications of that more general result.

I have found the special case of Street's adjoint core result useful in establishing 2-adjointness, for instance, because it substantially reduces the number of conditions you have to check in that case, e.g. 2-functoriality of both adjoints.

Nathanael Arkor (Feb 07 2026 at 08:34):

I agree "representability determines functoriality" is too long a name. I'm not sure whether I feel "parameter theorem" is particularly evocative, because I don't get the impression there's a parameter in the statement, only in certain applications. (I also don't have a better suggestion, though.)

Nathanael Arkor (Feb 07 2026 at 08:35):

Just to check I understand you correctly, the more general result you are referring to is the following:

Yes, that's right.

Nathanael Arkor (Feb 07 2026 at 08:35):

I suppose I would say that the bifunctor lemma and representability determines functoriality are two independent "pieces" for this more general result.

Yes, I think that's accurate.

Ruby Khondaker (she/her) (Feb 07 2026 at 08:35):

In my original statement I guess “D” would be the parameter category, since you get a sort of “parametrised universal object” given by Fd.

Ruby Khondaker (she/her) (Feb 07 2026 at 08:39):

Here’s a rephrasing that makes that more apparent:

Let $H : P \times C^\text{op} \to \mathbf{Set}$ be a functor representable in the $C^\text{op}$ argument; meaning, for each fixed “parameter” $p \in |P|$, the functor $H(p, -) : C^\text{op} \to \mathbf{Set}$ is representable. Then, by choosing representations, we obtain a (essentially unique) functor $F : P \to C$ such that $H(p, c) \cong C(c, Fp)$, naturally in $c$ and $p$.

Nathanael Arkor (Feb 07 2026 at 08:40):

It does seem consistent to call this "parameterised representability", true.

Ruby Khondaker (she/her) (Feb 07 2026 at 08:42):

In that case, I may edit the nlab page for “representability determines functoriality” to “parameter theorem”. I’ll give the parametrised representability statement first, and then show how this implies various parameter theorems for universal objects like limits, ends, adjunctions.

Nathanael Arkor (Feb 07 2026 at 08:42):

Nathanael Arkor said:

Just to check I understand you correctly, the more general result you are referring to is the following:

Yes, that's right.

I have the feeling that in this case (i.e. where $Q$ is "representable"), it may be possible to reduce the coherence condition from an octagon to a square, but I would need to double check.

Nathanael Arkor (Feb 07 2026 at 08:45):

I think there's potentially room for an expository article on this topic, given that only fragments exist in the literature, and it's a bit awkward to cite an nLab article.

Ruby Khondaker (she/her) (Feb 07 2026 at 08:46):

I’ll add it to the list of my future articles! I did cover a version of the parameter theorem for preorders in my most recent article, too.

John Baez (Feb 07 2026 at 20:05):

I like "representability determines functoriality" as an nLab article title, because I instantly know what it means, while "parameter theorem" could be about differential equations, statistics... almost anything.

You could introduce the term "parameter theorem" in the article.

Ruby Khondaker (she/her) (Feb 09 2026 at 12:39):

Right, I'll keep that in mind.

Beyond "parametrised representability", this thread has gotten me thinking about other kinds of "parameter theorems" that may hold for bifunctors. For example, I did a quick calculation and believe the following result is true:

Let $J \colon I \to B$ be a diagram, and $F \colon A \times B \to C$ a bifunctor. Suppose that $F$ preserves limits of $J$ in the $B$ argument, meaning for each fixed $a \in A$, the functor $F(a, -) \colon B \to C$ preserves limits of $J$. Then the curried functor $F^T \colon B \to [A, C]$ preserves limits of $J$.

Does this result have an existing name? And what would be an efficient proof of this?

More generally, one could take existing properties of functors, like "representable, preserves limits, reflects limits, creates limits, conservative..." and consider multifunctors for which these properties hold "pointwise" in some argument - meaning, fixing all other arguments, the corresponding properties hold for the partially applied functor.

Representability determines functoriality is one example of a parameter theorem that holds when the property taken is "representable". My intuition is that there should be other kinds of parameter theorems for other properties of functors - the example I gave above would be for the property "preserves limits". Is there some kind of general theory of "parametrised functors with property P", and corresponding parameter theorems for these?

Ruby Khondaker (she/her) (Feb 09 2026 at 21:59):

Incidentally, I believe I managed to find a way to organise the proof of "parametrised limit preservation":

• The functors $F(a, -) \colon B \to C$ all preserving limits of $J$ is equivalent to the product functor $B \to C^{\text{ob}(A) }$ preserving limits of $J$, since limits in product categories are computed componentwise.
• Then, the forgetful functor $[A, C] \to C^{\text{ob}(A)}$ factorises this product functor, and reflects all limits.
• Hence, by prop 3.5 on [[preserved limit]], the curried functor $F^T \colon B \to [A, C]$ preserves limits of $J$.

Max New (Feb 11 2026 at 04:02):

I need to read this thread more carefully later, but I wrote the original [[representability determines functoriality]] article and I just made up the name because I didn't know of one at the time. More recently I had intended to merge it with [[functor comprehension principle]] (the second version on that page), and that's the name I use for this theorem lately.

Max New (Feb 11 2026 at 14:02):

Nathanael Arkor said:

I won't spell out the octagon condition for now, but I will explain how to derive it momentarily. This theorem says, in some sense, that "functoriality is for free". For instance, taking $L : |\mathcal A| \to |\mathcal B|$ and $R : |\mathcal B| \to |\mathcal A|$ to be functions and $P := \mathcal A(1, R)$ and $Q := \mathcal B(L, 1)$, this recovers Street's minimal presentation of an adjunction (as referenced on that nLab page). (In this case, the octagon condition reduces to a square.)

this is really nice I always wondered how to connect this result to Streets "core of adjoint functors".

Max New (Feb 11 2026 at 14:14):

Nathanael Arkor said:

Given functors $F : |\mathcal X| \times \mathcal Y \to \mathcal Z$ and $G : \mathcal X \times |\mathcal Y| \to \mathcal Z$ that agree on objects, they extend to a functor $\mathcal X \times \mathcal Y \to \mathcal Z$ if and only if a single square condition holds.

I'll point out that this is the essential lemma for proving that that currying of functors is an isomorphism. I.e., the data of these two functors $F : |\mathcal X| \times \mathcal Y \to \mathcal Z$ and $G : \mathcal X \times |\mathcal Y| \to \mathcal Z$ that agree on objects is exactly the same data as a functor $H : X \to \textrm{UnNat}(Y,Z)$ where $\textrm{UnNat}(Y,Z)$ is a category of functors and "unnatural" transformations, i.e., families of morphisms with no naturality square. Then the single square above is exactly saying that $H$ lands in the subcategory of natural transformations.

Ruby Khondaker (she/her) (Feb 11 2026 at 14:19):

I believe this is related to Awodey's "transcendental derivation of natural transformations":

image.png

The idea is that you can already determine the nerve of $[C, D]$ purely by exploiting the universal property, and this in particular gets you the definitions of what natural transformations and their (vertical) composites should be. Applying the bifunctor lemma then gets you the actual definition, from which you can then verify that the resulting $[C, D]$ indeed satisfies the universal property.