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Stream: theory: category theory

Topic: “Natural cotransformations”

Emily (May 02 2024 at 02:22):

The set of natural transformations between two functors $F,G\colon\mathcal{C}\to\mathcal{D}$ is given by

$$\mathrm{Nat}(F,G)\mathbin{\overset{\mathrm{def}}{=}}\int_{A\in\mathcal{C}}\mathrm{Hom}_{\mathcal{D}}(F(A),G(A)).$$

Define the set of “natural cotransformations” from $F$ to $G$ by

$$\mathrm{CoNat}(F,G)\mathbin{\overset{\mathrm{def}}{=}}\int^{A\in\mathcal{C}}\mathrm{Hom}_{\mathcal{D}}(F(A),G(A)).$$

Explicitly, $\mathrm{CoNat}(F,G)$ is the quotient of $\coprod_{A\in\mathcal{C}}\mathrm{Hom}_{\mathcal{D}}(F(A),G(A))$ by the equivalence relation $\mathord{\sim}$ generated by declaring two elements $(A,\alpha_{A}\colon F(A)\to G(A))$ and $(B,\beta_{B}\colon F(B)\to G(B))$ of this set equivalent if there exist:

1. A morphism $f\colon A\to B$ of $\mathcal{C}$;
2. A morphism $\phi\colon F(B)\to G(A)$ of $\mathcal{D}$;

making the diagrams

image.png

commute. A “natural cotransformation” from $F$ to $G$ is then an equivalence class $[(A,\alpha_{A})]$ under this relation.

Two important examples of this are the following:

1. When $F$ and $G$ are both given by the identity functor $\mathrm{id}_{\mathcal{C}}$ of a category $\mathcal{C}$, we have

$$\mathrm{CoNat}(\mathrm{id}_{\mathcal{C}},\mathrm{id}_{\mathcal{C}})\cong\mathrm{Tr}(\mathcal{C}),$$

where $\mathrm{Tr}(\mathcal{C})$ is the trace of $\mathcal{C}$. For instance, when $\mathcal{C}=\mathbf{B}G$ is the one-object category associated to a group $G$, the trace $\mathrm{Tr}(\mathbf{B}G)$ is precisely the set of conjugacy classes of $G$.

2. Given morphisms of monoids $f,g\colon\mathbf{B}A\to\mathbf{B}B$, the set of natural cotransformations from $\mathbf{B}f$ to $\mathbf{B}g$ is given by the quotient of $B$ by the equivalence relation generated by declaring $b\sim b'$ if there exist $a\in A$ and $k\in B$ such that we have

$$\begin{align*} b &= kf(a),\\ b' &= g(a)k. \end{align*}$$

From how natural the definition is and the importance of the example $\mathrm{CoNat}(\mathrm{id}_{\mathcal{C}},\mathrm{id}_{\mathcal{C}})\cong\mathrm{Tr}(\mathcal{C})$, I can't help but feel like this might be a kind of fundamental notion in category theory. However, I haven't really ever found anything written about them.

Has anyone here ever wondered about these "natural cotransformations" before, want to talk about them, or try to discuss/cook up some examples/results together in this chat?

Peva Blanchard (May 02 2024 at 04:00):

I find this very interesting. Here are some questions on the top of my head:

• We have a composition $Nat(F, G) \times Nat(G, H) \rightarrow Nat(F, H)$. Is there something similar for $CoNat$?
• What happens if $F$ or $G$ is representable? Is there an analog to the Yoneda lemma?
• What happens if, instead of taking $F = G = id$, we take $F = G$ to be naturally isomorphic to the identity?
• Is $CoNat$ a profunctor on the functor category $[\mathcal{C}, \mathcal{D}]$? If yes, how does it interact with $Nat$?

Emily (May 02 2024 at 17:51):

@Peva Blanchard I've thought about these questions, and it seems there are lots of interesting things going on. Here are some of the things I found:

On Composition I.
There doesn't seem to be a vertical composition map of the form

$$\mathrm{CoNat}(G,H)\times\mathrm{CoNat}(F,G)\to\mathrm{CoNat}(F,H)$$

since:

• An element of $\mathrm{CoNat}(F,G)$ will be an equivalence class $[(A,\alpha_{A})]$ with $\alpha_{A}\colon F(A)\to G(A)$;
• An element of $\mathrm{CoNat}(G,H)$ will be an equivalence class $[(B,\beta_{B})]$ with $\beta_{B}\colon G(B)\to H(B)$;

and in general there isn't a way to compose $\alpha_{A}\colon F(A)\to G(A)$ with $\beta_{B}\colon G(B)\to H(B)$.

That said, it seems that when both equivalence classes admit a different representative with the same object, e.g. we have $[(A,\alpha_{A})]=[(K,\alpha'_{K})]$ and $[(B,\beta_{B})]=[(K,\beta'_K)]$, we can define their composition as $[(K,\beta'_{K}\circ\alpha'_{K})]$, and this seems to be well-defined.

In particular, for one-object categories we do have such a composition.

On Composition II.
In a different direction, there seem to be composition maps of the form

$$\begin{align*} \mathrm{Nat}(G,H)\times\mathrm{CoNat}(F,G) \to \mathrm{CoNat}(F,H),\\ \mathrm{CoNat}(G,H)\times\mathrm{Nat}(F,G) \to \mathrm{CoNat}(F,H). \end{align*}$$

(There's a kind of analogy between co/ends and Hochschild co/homology, where the co/products and co/equalisers formula for computing co/ends look like a Hochschild co/simplicial set, and under this analogy these composition maps mixing natural transformations with "natural cotransformations" would correspond to a cap product on degree $(0,0)$.)

On Composition III.
When $\mathcal{C}$ is the one-object groupoid associated to a group $G$, these recover the action of the centre of $G$ on the set of conjugacy classes of $G$:

$$\begin{align*} \mathrm{Z}(G)\times\mathrm{Cl}(G) \to \mathrm{Cl}(G),\\ \mathrm{Cl}(G)\times\mathrm{Z}(G) \to \mathrm{Cl}(G). \end{align*}$$

On Composition IV: We Do Get a Profunctor :tada: .
In addition, a consequence of the existence of these maps (which I haven't checked yet, but plan to do in a bit) would be that the assignment $(F,G)\mapsto\mathrm{CoNat}(F,G)$ would define a functor

$$\mathrm{CoNat}(-_{1},-_{2})\colon\mathrm{Fun}(\mathcal{C},\mathcal{D})^{\mathsf{op}}\times\mathrm{Fun}(\mathcal{C},\mathcal{D})\to\mathsf{Sets},$$

i.e. it does define a profunctor on $[\mathcal{C},\mathcal{D}]$.

When $F$ or $G$ Are Representable.

I think there's a lot of things to explore here, but I've found a nice example that seems interesting:

• Take $\mathcal{C}=\mathbf{B}A^{\mathsf{op}}$ to be the opposite of the one-object category on a monoid $A$;
• Take $\mathcal{D}=\mathsf{Sets}$;
• A functor $F\colon\mathbf{B}A^{\mathsf{op}}$ then corresponds to a left $A$-set $X$, where we write $(a,x)\mapsto a\mathbin{\triangleleft_{X}}x$ for the left $A$-action $A\times X\to X$ of $A$ on $X$;
• The representable presheaf $h_{\bullet}$ on the unique object $\bullet$ of $\mathbf{B}A$ corresponds to the "left regular representation" of $A$, i.e. the left $A$-set consisting of $A$ with the left $A$-action $A\times A\to A$ of $A$ on itself given by multiplication $(a,b)\mapsto ab$.

Then:

1. The set $\mathrm{Nat}(h_{\bullet},X)$ is the set of morphisms of left $A$-sets from $A$ to $X$, corresponding to maps of sets $f\colon A\to X$ satisfying $f(ax)=a\mathbin{\triangleleft_{X}}f(x)$.
2. The set $\mathrm{CoNat}(h_{\bullet},X)$ is the quotient of $\mathsf{Sets}(A,X)$ by the equivalence relation generated by $f(ax)\sim a\mathbin{\triangleleft_{X}}f(x)$.

We know that the set $\mathrm{Nat}(h_{\bullet},X)$ is isomorphic as a left $A$-set to $X$; I wonder how $\mathrm{CoNat}(h_{\bullet},X)$ is related to $X$.

A Dual Notion of Morphisms of Left $A$-Sets.

More generally, taking $F$ and $G$ to be functors $\mathbf{B}A^\mathsf{op}\to\mathsf{Sets}$ corresponding to left $A$-sets $X$ and $Y$, we have another nice example illustrating natural transformations vs. natural "cotransformations":

• The set $\mathrm{Nat}(F,G)$ is the equaliser of $\mathsf{Sets}(X,Y)$ by the maps $f\mapsto[x\mapsto f(a\mathbin{\triangleleft_{X}}x)]$ and $f\mapsto[x\mapsto a\mathbin{\triangleleft_{Y}}f(x)]$, i.e., we have

$$\mathrm{Nat}(F,G)\cong\mathsf{Sets}^\mathrm{L}_{A}(X,Y),$$

the set of morphisms of left $A$-sets from $X$ to $Y$.

• The set $\mathrm{CoNat}(F,G)$ is the coequaliser of $\mathsf{Sets}(X,Y)$ by these maps, quotienting $\mathsf{Sets}(X,Y)$ by the relation generated by $f(a\mathbin{\triangleleft_{X}}x)\sim a\mathbin{\triangleleft_{Y}}f(x)$.

Whether the set

$$\mathsf{Sets}(X,Y)/(f(a\mathbin{\triangleleft_{X}}x)\sim a\mathbin{\triangleleft_{Y}}f(x))$$

is useful for working with left $A$-sets seems to be an interesting question by itself!

Emily (May 02 2024 at 20:16):

I've asked the last question above on MO here.

Tim Hosgood (May 03 2024 at 09:24):

(I don't have anything mathematical to add but I just wanted to say what a well-written post this is! thanks for taking the time to write everything out so cleanly)

John Baez (May 03 2024 at 10:19):

Emily said:

There doesn't seem to be a vertical composition map of the form

$$\mathrm{CoNat}(G,H)\times\mathrm{CoNat}(F,G)\to\mathrm{CoNat}(F,H)$$

Is there a "co-composition" map

$$\mathrm{CoNat}(F,H) \to \mathrm{CoNat}(G,H)\times\mathrm{CoNat}(F,G) \; ?$$

Morgan Rogers (he/him) (May 03 2024 at 11:19):

At first glance I don't think so. You can take $\mathcal{C}$ to be the one-object category and $\mathcal{D}$ to be a discrete two-object category. The two functors $0,1:\mathcal{C} \rightrightarrows \mathcal{D}$ have no conatural transformations between them, so we would have to have a map
$\{\mathrm{id}\} = \mathrm{CoNat}(0,0) \to \mathrm{CoNat}(1,0) \times \mathrm{CoNat}(0,1) = \emptyset$
These really don't behave as a dual of natural transformations in an intuitive way ;)

Tim Hosgood (May 03 2024 at 13:17):

Emily said:

I've asked the last question above on MO here.

looking at this question, it sounds a lot like (maybe actually equivalent to, in some sense) the following question: given a cosimplicial simplicial set $X_\bullet^\star$, we can construct its totalisation $\operatorname{Tot}X$ as the simplicial set given by some equaliser (which is essentially the same as the one in your MO question); what do you get if you take the coequaliser of this diagram instead?

Emily (May 03 2024 at 16:47):

Tim Hosgood said:

(I don't have anything mathematical to add but I just wanted to say what a well-written post this is! thanks for taking the time to write everything out so cleanly)

Hey Tim, thank you so much! :orange_heart:

Emily (May 03 2024 at 16:47):

Tim Hosgood said:

looking at this question, it sounds a lot like (maybe actually equivalent to, in some sense) the following question: given a cosimplicial simplicial set $X_\bullet^\star$, we can construct its totalisation $\operatorname{Tot}X$ as the simplicial set given by some equaliser (which is essentially the same as the one in your MO question); what do you get if you take the coequaliser of this diagram instead?

There is definitely co/simplicial stuff going on here: given a functor $D\colon\mathcal{C}^\mathsf{op}\times\mathcal{C}\to\mathcal{D}$, you can build a simplicial set $\mathrm{HH}^{\Delta}_{\bullet}(\mathcal{C};D)$ by precomposing $D$ with the opposite of the projection map $q\colon\int^{\Delta}\mathcal{C}\twoheadrightarrow\mathcal{C}^\mathsf{op}\times\mathcal{C}$ from the category of simplices of $\mathcal{C}$ to $\mathcal{C}^\mathsf{op}\times\mathcal{C}$, and then consider the left Kan extension of $D\circ q^\mathsf{op}$ along the projection $U^{\mathsf{op}}\colon(\int^{\Delta}\mathcal{C})^\mathsf{op}\to\Delta^{\mathsf{op}}$ from the opposite of the category of simplices of $\mathcal{C}$ to the opposite of the simplex category $\Delta$, giving a simplicial set $\mathrm{Lan}_{U^\mathsf{op}}(D\circ q^\mathsf{op})$.

Emily (May 03 2024 at 16:47):

Here's a picture from some old notes I wrote about this, with slightly different notation:

image.png

Emily (May 03 2024 at 16:47):

Then, as it turns out, the colimit of this simplicial set is the coend of $D$, since the inclusion of the category $[0]\mathbin{\overset{\leftarrow}{\scriptstyle\leftarrow}}[1]$ into $\Delta^\mathsf{op}$ is final, and the colimit of $\mathrm{Lan}_{U^{\mathsf{op}}}(D\circ q^{\mathsf{op}})$ precomposed with this inclusion gives the usual coproducts and coequalisers formula for coends.

Emily (May 03 2024 at 16:47):

This is mentioned in Item (B) of this paper from Rune Haugseng: PDF.

Emily (May 03 2024 at 16:47):

Taking $D=\mathrm{Hom}_{\mathcal{D}}(F,G)$ then gives a simplicial set $\mathrm{CoTrans}(F,G)$ whose colimit (i.e. its $\pi_0$) gives $\mathrm{CoNat}(F,G)$.

Emily (May 03 2024 at 16:48):

One of the things I've been wondering about for a while now is whether the $\pi_n$'s of this simplicial set are also interesting

Emily (May 03 2024 at 16:48):

The same goes for the $\pi_n$'s of the cosimplicial set $\mathrm{Trans}(F,G)$ having $\mathrm{Nat}(F,G)$ as its $\pi_0$.

Emily (May 03 2024 at 16:49):

Sadly, though, this would require a way to define $\pi_n$'s for arbitrary simplicial sets, which I'm not sure can be done (see here)

Emily (May 03 2024 at 17:00):

I wonder how the geometric realisation of $\mathrm{CoTrans}(F,G)$ relates (if at all) to $\mathrm{CoNat}(F,G)$, and similarly how the totalisation of $\mathrm{Trans}(F,G)$ relates (again, if at all) to $\mathrm{Nat}(F,G)$. These definitely sound like really interesting objects to study!

Tim Hosgood (May 03 2024 at 17:50):

Emily said:

The same goes for the $\pi_n$'s of the cosimplicial set $\mathrm{Trans}(F,G)$ having $\mathrm{Nat}(F,G)$ as its $\pi_0$.

could you just sketch out a bit how you define $\mathrm{Trans}(F,G)$ again? maybe it's obvious (you just pick $D$ to be something else?) but I'm very much not a fluent speaker of co-ends and Kans so I get a bit lost in the notation

Emily (May 03 2024 at 18:34):

Tim Hosgood said:

could you just sketch out a bit how you define $\mathrm{Trans}(F,G)$ again? maybe it's obvious (you just pick $D$ to be something else?) but I'm very much not a fluent speaker of co-ends and Kans so I get a bit lost in the notation

The quick definition is as $\mathrm{Ran}_{D\circ q}(U)$ for $D=\mathrm{Hom}_{\mathcal{D}}(F,G)$ again, but the description via $n$-cosimplices, cofaces, and codegeneracies is much more illuminating.

Also, spelling things out this way makes clear just how much $\mathrm{Trans}(F,G)$ looks like a “non-abelian Hochschild cosimplicial set of $\mathcal{C}$ with coefficients in the bimodule $\mathrm{Hom}_{\mathcal{D}}(F,G)$”.

For simplicity, I'll write $h^{A}_{B}$ for $\mathrm{Hom}_{C}(A,B)$ below, as well as $F_A$ and $G_A$ for $F(A)$ and $G(A)$ respectively.

(I'll also sketch things a bit instead of explicitly stating everything, but let me know if you'd like more detail!)

The Cosimplices in Low Degree.
We have

$$\begin{align*} \mathrm{Trans}(F,G)^0 &\cong \prod_{A\in\mathrm{Obj}(\mathcal{C})}\mathrm{Hom}_{\mathcal{D}}(F_A,G_A),\\ \mathrm{Trans}(F,G)^1 &\cong \prod_{A,B\in\mathrm{Obj}(\mathcal{C})}\mathrm{Sets}\left(h^{A}_{B},\mathrm{Hom}_{\mathcal{D}}(F_{A},G_{B})\right),\\ \mathrm{Trans}(F,G)^2 &\cong \prod_{A,B,C\in\mathrm{Obj}(\mathcal{C})}\mathrm{Sets}\left(h^{C}_{B}\times h^{B}_{A},\mathrm{Hom}_{\mathcal{D}}(F_{A},G_{C})\right),\\ \mathrm{Trans}(F,G)^3 &\cong \prod_{A,B,C,D\in\mathrm{Obj}(\mathcal{C})}\mathrm{Sets}\left(h^{D}_{C}\times h^{C}_{B}\times h^{B}_{A},\mathrm{Hom}_{\mathcal{D}}(F_{A},G_{D})\right). \end{align*}$$

The Codegeneracy Maps in Low Degree.
The explicit description is a little involved, but the idea is to project factors indexed by identities. For example, the $(0,0)$-th codegeneracy map

$$\sigma^{0}_{0}\colon\mathrm{Trans}(F,G)^{1}\to\mathrm{Trans}(F,G)^{0}$$

is the map

$$\prod_{A,B\in\mathrm{Obj}(\mathcal{C})}\mathrm{Sets}\left(h^{A}_{B},\mathrm{Hom}_{\mathcal{D}}(F_{A},G_{B})\right) \to \prod_{A\in\mathrm{Obj}(\mathcal{C})}\mathrm{Hom}_{\mathcal{D}}(F_{A},G_{A})$$

given by first writing the source set as

$$\prod_{\substack{A,B\in\mathrm{Obj}(\mathcal{C})\\f\colon A\to B}}\mathrm{Hom}_{\mathcal{D}}(F_{A},G_{B})$$

and then considering the collection of maps

$$\left\{ \mathrm{pr}_{A} \colon \prod_{\substack{A,B\in\mathrm{Obj}(\mathcal{C})\\f\colon A\to B}}\mathrm{Hom}_{\mathcal{D}}(F_{A},G_{B}) \to \mathrm{Hom}_{\mathcal{D}}(F_{A},G_{A}) \right\}_{A\in\mathrm{Obj}(\mathcal{C})}$$

given by projecting the factor of $\mathrm{Hom}_{\mathcal{D}}(F_{A},G_{B})$ in the product indexed by $(A,A,\mathrm{id}_{A})$; we then apply the universal property of the product to build $\sigma^{0}_{0}$ out of the $\mathrm{pr}_{A}$'s.

The $(1,0)$-th codegeneracy is built using projections indexed by $(A,B,B,f,\mathrm{id}_{B})$, while the $(0,1)$-th one has indices like $(A,A,B,\mathrm{id}_{A},g)$.

The Coface Maps in Low Degree.
The coface maps are again built using the universal property of the product, but this time we'll use either:

1. The "left/right actions of the bimodule $\mathrm{Hom}_{\mathcal{D}}(F,G)$", which correspond to precomposition by $F(f)$ or postcomposition with $G(g)$.
2. The composition of $\mathcal{C}$ to get maps like $h^{C}_{D}\times h^{B}_{C}\times h^{A}_{B}\to h^{B}_{D}\times h^{A}_{B}$.

For example, here's a picture of $\delta^{1}_{0}$ (again from the old notes I have on this), where $D^{A}_{B}=\mathrm{Hom}_{\mathcal{D}}(F,G)$:

image.png

Tim Hosgood (May 05 2024 at 14:02):

oh this is super interesting! is anything about this construction written down anywhere else?

Emily (she/her) (May 05 2024 at 19:11):

Tim Hosgood said:

oh this is super interesting! is anything about this construction written down anywhere else?

Not that I'm aware of =(

Emily (she/her) (May 05 2024 at 19:11):

I've tried asking about this on MO before (link), but my question kind of got dismissed as "just being subsumed by homotopy co/ends".

Emily (she/her) (May 05 2024 at 19:11):

(There is a precise connection between the two, but my question is broader in scope/includes other things in addition to those)

Emily (she/her) (May 05 2024 at 19:12):

I'm planning to try to eventually work out the theory surrounding this in detail, but I think that might take a little while

Emily (she/her) (May 05 2024 at 19:15):

By the way, I'll probably try working out a theory of such "natural cotransformations" in detail soon. Do you think it would make sense to update this thread once in a while if I end up finding anything interesting to discuss here?

Emily (she/her) (May 05 2024 at 19:16):

(In particular, would you be interested in this, Peva and Matteo? @Peva Blanchard @Matteo Capucci (he/him) )

Matteo Capucci (he/him) (May 05 2024 at 19:18):

I'm moderately interested because natural cotransformations pop out in ACT when one deals with feedback

Matteo Capucci (he/him) (May 05 2024 at 19:19):

Like the free feedback category construction on a monoidal category replaces each C(X,Y) with Conat(-X,-Y)

Matteo Capucci (he/him) (May 05 2024 at 19:21):

In general it seems this is related with Willerton's circular traces and more broadly with Barhite's coshadows

Emily (she/her) (May 05 2024 at 19:30):

Matteo Capucci (he/him) said:

Like the free feedback category construction on a monoidal category replaces each C(X,Y) with Conat(-X,-Y)

oh, that sounds super interesting! Is there somewhere I could read more about this?

Emily (she/her) (May 05 2024 at 19:32):

Matteo Capucci (he/him) said:

In general it seems this is related with Willerton's circular traces and more broadly with Barhite's coshadows

Yep! "Natural cotransformations" are definitely intimately connected to traces (e.g. $\mathrm{Tr}(\mathcal{C})\cong\mathrm{CoNat}(\mathrm{id}_{\mathcal{C}},\mathrm{id}_{\mathcal{C}}$))

Emily (she/her) (May 05 2024 at 19:32):

lately I've been trying to understand traces better; it seems there's a lot of stuff about them that hasn't been developed just yet

Emily (she/her) (May 05 2024 at 19:33):

e.g. these days I learned that for any presheaf $\mathcal{F}$, we have a map

$$\mathcal{F}\boxtimes\mathsf{O}(\mathcal{F})\to\mathrm{Tr}(\mathcal{C}),$$

where $\mathsf{O}$ is one of the functors appearing in Isbell duality and $\boxtimes$ is the functor tensor product.

Emily (she/her) (May 05 2024 at 19:34):

This map is analogous to the duality pairing $\mathrm{ev}\colon V\otimes_{k}V^*\to k$ for a vector space $V$ over $k$, but as far as I know it hasn't really been studied at all before

Emily (she/her) (May 05 2024 at 19:35):

all in all, I feel like there are lots and lots of interesting things to figure out around these notions :)

Peva Blanchard (May 05 2024 at 21:10):

Emily (she/her) said:

(In particular, would you be interested in this, Peva and Matteo? Peva Blanchard Matteo Capucci (he/him) )

Yes I would be interested :) But maybe I should warn you that I am on the beginner/learner's side on this forum, so I might not be able to follow the discussion in a manner that would be helpful for you.

Matteo Capucci (he/him) (May 06 2024 at 07:26):

You're at least an advanced beginner Peva :D

Matteo Capucci (he/him) (May 06 2024 at 07:27):

Emily (she/her) said:

Matteo Capucci (he/him) said:

Like the free feedback category construction on a monoidal category replaces each C(X,Y) with Conat(-X,-Y)

oh, that sounds super interesting! Is there somewhere I could read more about this?

p.41 here: https://arxiv.org/pdf/2010.10069

I recall @Mario Román showing a general definition with actegories instead but I can't remember where

Morgan Rogers (he/him) (May 06 2024 at 08:22):

With the context/related material that folks have mentioned in this thread, a study of these seems like it would make a good article, so do consider turning it into a formal write-up eventually if you weren't already planning to @Emily (she/her) !

David Corfield (May 06 2024 at 09:23):

Emily (she/her) said:

In a different direction, there seem to be composition maps of the form

$Nat(G,H) \times CoNat(F,G) \to CoNat(F,H)$, $CoNat(G,H) \times Nat(F,G) \to CoNat(F,H)$.​

Probably nothing, but I was reminded of

image.png

from Categorification of Negative Information using Enrichment.

David Corfield (May 06 2024 at 12:52):

But then the issue about $CoNat(0,0)$, $CoNat(1,0)$ and $CoNat(0,1)$ that Morgan raised above wouldn't match:
Screenshot-2024-05-06-13.49.21.png

Morgan Rogers (he/him) (May 06 2024 at 13:20):

I don't understand the notation well enough to be sure that that conflicts with what I wrote :thinking:

Emily (she/her) (May 13 2024 at 14:18):

Peva Blanchard said:

Yes I would be interested :) But maybe I should warn you that I am on the beginner/learner's side on this forum, so I might not be able to follow the discussion in a manner that would be helpful for you.

That's awesome to know! I'll be sure to post any interesting updates on these here then :)

Emily (she/her) (May 13 2024 at 14:18):

Matteo Capucci (he/him) said:

p.41 here: https://arxiv.org/pdf/2010.10069

I recall Mario Román showing a general definition with actegories instead but I can't remember where

Thanks, Matteo!

Emily (she/her) (May 13 2024 at 14:22):

Morgan Rogers (he/him) said:

With the context/related material that folks have mentioned in this thread, a study of these seems like it would make a good article, so do consider turning it into a formal write-up eventually if you weren't already planning to Emily (she/her) !

Thank you so much for the encouragement, Morgan! I think I'll definitely do so if these turn out to have an interesting theory (which certainly seems to be the case!)

Emily (she/her) (May 13 2024 at 14:27):

By the way, I've recently (kind of sneakily) asked about natural co/transformations between functors between one-object groupoids here: https://mathoverflow.net/questions/471017/

There $\mathrm{Z}_{H}(f,g)$ corresponds to $\mathrm{Nat}(\mathbf{B}f,\mathbf{B}g)$ and $\mathrm{Cl}_{H}(f,g)$ corresponds to $\mathrm{CoNat}(\mathbf{B}f,\mathbf{B}g)$.

It seems a careful development of these notions hasn't been carried just yet, but there are several special cases of these which have been studied and are pretty interesting!

Rémy Tuyéras (May 16 2024 at 11:41):

When I see the definition of natural cotransformations, I almost want to look at them as spaces that would be classified by natural transformations. More specifically, I feel like they could be studied with respect to a set of coefficients to bring them back to structures that we know better.

To expand on this, I think we would need to compare it to when the theory of categories is redefined relative to a given set, say $V$ (maybe this would make more sense in enriched categories, but I will stick to sets for simplicity).

Part 1

Specifically, for any graph $G$, we can define the graph $G_V$ as follows:

$Obj(G_V) = Obj(G)$

$G_V(a,b) = Hom(V,G(a,b))$

Now, let us call a category $D$ that is of the form $G_V$ a $V$-pointed category. If $G$ happens to be a category that induces the category structure of $G_V$, then we can say that $D$ is $V$-represented by $G$.

A functor $F:C \to G_V$ would then be said to be represented by a functor $F':C\to G$ if we have natural isomorphisms in $a$:

$G_V(F(a),-) \cong Hom(V,G(F'(a), -))$

and

$G_V(-,F(a)) \cong Hom(V,G(-,F'(a)))$

(hopefully I am not missing conditions here)

Now, for two functors $F,G:C \to D_V$ landing in a $V$-pointed category, we have

$Nat(F,G)=\int_a D_V(F(a), G(a))$

If $D$ is a category, and $F$ and $G$ are represented by functors $F',G':C\to D$, then we have:

$Nat(F,G) =\int_a Hom(V,D(F'(a),G'(a)))$

$\cong Hom(V,\int_a D(F'(a),G'(a)))$

$= Hom(V,Nat(F',G'))$

For example, we have the fact that $1$-pointed categories correspond to categories.

Part 2

Similarly, for any graph $G$, we can define the graph $G^V$ as follows:

$Obj(G^V) = Obj(G)$

$G^V(a,b) = Hom(G(a,b),V)$

Now, let us call a category $D$ that is of the form $G^V$ a $V$-valued category.

In contrast to part 1, it is not obvious how to formalize the relation being represented by. Still, it seems that this is where the concept of natural cotransformation belongs.

Specifically, for two functors $F,G:C \to D^V$ landing in a $V$-valued category, we have

$Nat(F,G)=\int_a D^V(F(a), G(a))$

Ideally, there would be a concept of representation that would allow us to say that:

$Nat(F,G) \cong Hom(CoNat(F',G'),V)$

Somehow, I feel like the concept of natural cotransformation belongs to a general framework copying Part 1. This framework would specifically be built on a definition that characterizes the structures $D$ to make $D^V$ a category

This structure should probably be categorized relative to the set $V$ (as with cohomologies and their coefficients) because it is intuitively easier to use functions landing in $V$ with a category structure on them. Studying colimit-based structures might be too "confusing" as these look less like operations-based structures.

Hopefully this can add another perspective to the discussion

Emily (she/her) (Nov 07 2024 at 16:22):

Hi Rémy!

Emily (she/her) (Nov 07 2024 at 16:22):

Sorry for taking so long to reply; shortly after my last reply I began facing a number of health issues which ended up with me unable to work for a few months.

Emily (she/her) (Nov 07 2024 at 16:22):

Seeking relations between natural transformations and cotransformations reminds me a bit of co/classifiers for dinatural transformations, which satisfy identities of the form

$$\mathrm{Nat}(\Gamma(F),G)\cong\mathrm{DiNat}(F,G)$$

I wonder if there is something similar for natural cotransformations...

Kevin Carlson (Nov 07 2024 at 18:23):

Sorry to hear about your health problems Emily, glad you're back!

Emily (she/her) (Nov 08 2024 at 17:47):

Kevin Carlson said:

Sorry to hear about your health problems Emily, glad you're back!

Thank you so much, Kevin :)

Rémy Tuyéras (Nov 08 2024 at 20:05):

Emily (she/her) said:

Seeking relations between natural transformations and cotransformations reminds me a bit of co/classifiers for dinatural transformations, which satisfy identities of the form

$$\mathrm{Nat}(\Gamma(F),G)\cong\mathrm{DiNat}(F,G)$$

I wonder if there is something similar for natural cotransformations...

Hi Emily!

(Sounds like you had quite a rough time, Emily -- it's good to hear from you)

I just wanted to clarify that, when I wrote my comment, my main thought was that it felt a bit strange to consider natural cotransformations with categories, so I was looking for a context in which developing natural cotransformations with cocategories would make more sense.

Since cocategories are not very "algebraic" (in the sense that they are not encoded by $n$-ary operations), I was considering a set of values (or "coefficients") that would allow me to handle cocategories in a way similar to categories, such that I would get some sort of duality between categories/transformations and cocategories/cotransformations.

Posina Venkata Rayudu (Nov 08 2024 at 21:59):

Upon reading "the set of natural transformations", one immediate question, which may be tangential to all that followed, is: with natural transformations as maps between functors, wouldn't the totality of maps between two functors be a map object of the functor category? The set of natural transformations between functors can be extracted as the set of points of the map object. Of course, if the category of functors is not cartesian closed, then one have to be content with a set of natural transformations (Lawvere & Schanuel, Conceptual Mathematics, p. 322).

Once I noticed that it is not about natural transformations but about natural cotransformations, I thought of moving on, but here I'm with my 5mBTC ;)

Natural cotransformation is to natural transformation what equivalence relation is to function (Arbib & Manes, Arrows, Structures, and Functors, pp. 16 - 23). A function and its corresponding [opposite] equivalence relation (speaking of which: is equivalence relation ever referred to as cofunction?, which in turn reminds me of co-element; Lawvere & Rosebrugh, Sets for Mathematics, pp. 127 - 128) structure both domain and codomain sets, and based on which corresponding categories can be constructed (Lawvere, Perugia Notes, pp. 9 - 11).

The above parallels raise a question: is there more, conceptually speaking, to:

natural transformation vs. natural cotransformation

than there is in the relatively simplistic dual:

function vs. equivalence relation.

In any case, if I may, I'd like to share a few things that I find fascinating about natural transformations.

(i) Maps of any category, beginning with functions between sets, can be represented as natural transformations (Lawvere & Schanuel, Conceptual Mathematics, p. 378).

(ii) "Naturality of the transformations between spaces and the transformations within thought" (Lawvere, Interview, p. 6), which can be approached via a relatively well-defined naturality relating geometry and algebra (Lawvere, Birkhoff's Theorem, p. 2).

(iii) Naturality figures prominently in the most precise scientific model of concept formation (Lawvere, Fifty Years of Functorial Semantics, p. 7).

(iv) The adequacy of one-dimensional figures needs to be explained because it occurs in many different examples.
Many kinds of cohesion (algebraic geometry, smooth geometry, continuous geometry) are well-expressed as a subtopos of the classifying topos of a finitary single-sorted algebraic theory. But often that algebraic theory is determined by its monoid M of unary operations via naturality only: for example, the binary operations, instead of being independently specified, are just the maps of right M-sets from M^2 to M. If a common explanation can be found for this adequacy of one-dimensional considerations in the determination of n-dimensional and infinite-dimensional functionals, in so many disparate cases, then it would further establish that the Eilenberg-Mac Lane notion of naturality is far more powerful than the mere tautology it is sometimes considered to be (Lawvere, Cartesian Closed Categories, p. 2).

In closing, given all that's going for natural transformations, how does its (dual?) natural cotransformation fare (in comparison).

Thanking you,
yours truly,
posina

P.S. I think this is the first time I came across natural cotransformation; I apologize in advance for anything that I shouldn't have written here and now.

Kevin Carlson (Nov 08 2024 at 22:19):

Please use paragraphs and organize your writing, Posina, it's very hard to work through a wall of technical text like that.

Emily (she/her) (Nov 08 2024 at 22:40):

Hi Rémy!

Rémy Tuyéras said:

(Sounds like you had quite a rough time, Emily -- it's good to hear from you)

Thank you so much! :orange_heart:

Rémy Tuyéras said:

I just wanted to clarify that, when I wrote my comment, my main thought was that it felt a bit strange to consider natural cotransformations with categories, so I was looking for a context in which developing natural cotransformations with cocategories would make more sense.

Since cocategories are not very "algebraic" (in the sense that they are not encoded by $n$-ary operations), I was considering a set of values (or "coefficients") that would allow me to handle cocategories in a way similar to categories, such that I would get some sort of duality between categories/transformations and cocategories/cotransformations.

I think there will likely be an analogue of both natural transformations and natural cotransformations for cocategories (though of course this doesn't preclude there being some kind of duality between both!)

The way I've been thinking about natural cotransformations lately is that ends may be viewed as a kind of zeroth non-Abelian Hochschild cohomology, and coends as [...] homology, and, under this analogy (which, incidentally, can be realised literally in the $\mathrm{Mod}_R$-enriched setting), natural co/transformations would correspond to zeroth non-Abelian Hochschild cohomology with coefficients in the profunctor (i.e. bimodule!) $\mathrm{Hom}(F(-_1),G(-_2))$

I don't have yet anything too neat written about this, but it's something I'm hoping to explore soon(-ish)

Emily (she/her) (Nov 08 2024 at 22:45):

Posina Venkata Rayudu said:

Upon reading "the set of natural transformations" [...]

I don't have much to add right now, just that under this Hochschild co/homology analogy, there's actually a cosimplicial set $\mathrm{Trans}^\bullet(F,G)$ with $\pi_0(\mathrm{Trans}^\bullet(F,G))\cong\mathrm{Nat}(F,G)$ (and dually, a simplicial set for cotransformations recovering natural cotransformations as its $\pi_0$), so there does seem to be something deeper going on.

Posina Venkata Rayudu (Nov 09 2024 at 00:16):

Kevin Carlson said:

Please use paragraphs and organize your writing, Posina, it's very hard to work through a wall of technical text like that.

I am sorry about that! I'll revise it right now.