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Stream: learning: questions

Topic: preliminaries to Yoneda Lemma

Eduard BABKIN (Sep 29 2020 at 05:48):

Consequent reflection upon the Yoneda lemma moved me to the following case and a basic question:

Given a discrete category $C$ with a single object $a$ and a single morphism $id_a$, we can define two functors F, G, both $C \rightarrow Set$.
Now imagine, that F maps the object $a$ to a set with $m$ elements, while G maps $a$ to the set with $n$ elements.
Let's consider natural transformation between F and G. its single component $\alpha_a$ is a morphism in Set between the images of F and G correspondingly (Fa ->Ga ). Any morphism in Set category is a function between sets. So, in principle we are able to define $n^m$ different mappings between the image of F and the image of G.

And now the question: what is the correct choice: (1) the single component $\alpha_a$ denotes a whole set of defined mappings Fa->Ga or (2) actually
there is a plenty of components of natural transformation, and each of them corresponds to one particular mapping ?

Morgan Rogers (he/him) (Sep 29 2020 at 08:49):

There are actually $n^m$ natural transformations between the functors you describe. Each of the possible functions describes a valid natural transformation, because the naturality condition is trivial.

Morgan Rogers (he/him) (Sep 29 2020 at 08:51):

However, the Yoneda lemma refers to representable functors. In this case, the representable functor $y(a):C=C^{\mathrm{op}} \to \mathbf{Set}$ sends the unique object of $C$ to the $1$-element set. And indeed, you'll notice that if $G$ is as you describe, there are exactly $n = |G(a)|$ natural transformations from $y(a)$ to $G$.

Eduard BABKIN (Sep 30 2020 at 07:07):

[Mod] Morgan Rogers said:

There are actually $n^m$ natural transformations between the functors you describe. Each of the possible functions describes a valid natural transformation, because the naturality condition is trivial.

Thank you for clarification, Morgan

In that case I have a technical question: how we can denote each component of natural transformation ?
Because we have a discrete category with a single element $a$, could each component be named as $\alpha_{a_i}$, where $i$ runs from 1 to $n^m$ ?

Or for that case we can say, that the set of components contains sets as elements, so, we have a singleton set of the components of natural transformation ($\alpha_a$), but the element of that set is the set itself (with $n^m$ elements -- all possible functions) ?

Eduard BABKIN (Sep 30 2020 at 07:47):

[Mod] Morgan Rogers said:

However, the Yoneda lemma refers to representable functors. In this case, the representable functor $y(a):C=C^{\mathrm{op}} \to \mathbf{Set}$ sends the unique object of $C$ to the $1$-element set. And indeed, you'll notice that if $G$ is as you describe, there are exactly $n = |G(a)|$ natural transformations from $y(a)$ to $G$.

Am I right that when you speak about the representable functor in the context of Yoneda Lemma, You actually mean only the functor $y$, while the functor $G$ can be not representable ?

Now after preliminaries I can come closer the main subject -- the confusion about Yoneda lemma:
For the main case I wish to use the category 2 ( objects: $a$ and $b$, morphism $f : a \rightarrow b$) and a covariant type of Yoneda lemma $Nat(Hom(a,-), G) \sim Ga$ , $G : 2 \rightarrow Set$ .
Let define a simple case of functor $G$ when it sends $a$ to one singleton set in $Set$, and $b$ to another singleton set.

if I'm not mistaken, the Hom-functor $Hom(a,-)$ sends $a$ from category 2 to a singleton set ($Hom(a,a)$), while $b$ is also sent to another singleton set ($Hom(a,b)$). For such a trivial case we can determine two components of natural transformation between $Hom(a,-)$ and $G$:
$\alpha_a$ and $\alpha_b$ correspondingly. These components are simple functions on sets.

Now I recall that Yoneda lemma says about bijection between the set of components of natural transformation ($Nat(..)$) and the $Ga$ (also the set).
However the set of the components contains 2 elements ($\alpha_a$ and $\alpha_b$ ), while $Ga$ has only one element in the set.
That is a point of my confusion.

Where is a flaw in the speculations above ?

Morgan Rogers (he/him) (Sep 30 2020 at 08:50):

Eduard BABKIN said:

I have a technical question: how we can denote each component of natural transformation ?
Because we have a discrete category with a single element $a$, could each component be named as $\alpha_{a_i}$, where $i$ runs from 1 to $n^m$ ?

Or for that case we can say, that the set of components contains sets as elements, so, we have a singleton set of the components of natural transformation ($\alpha_a$), but the element of that set is the set itself (with $n^m$ elements -- all possible functions) ?

Hmm it seems there is some confusion about the data of natural transformations here.
Given functors $F,G: \mathcal{C} \to \mathcal{D}$, a natural transformation $\alpha$ from $F$ to $G$ consists of a morphism $\alpha_A: F(A) \to G(A)$ for each object of $\mathcal{C}$, subject to a naturality condition for each morphism of $\mathcal{C}$. In your example, $\mathcal{C}$ has just one object, so a natural transformation consists of a single morphism.

As we have seen, there may be many (distinct!) natural transformations between a pair of functors, but they are separate, not all part of the same transformation.

Morgan Rogers (he/him) (Sep 30 2020 at 08:58):

Eduard BABKIN said:

Now I recall that Yoneda lemma says about bijection between the set of components of natural transformation ($Nat(..)$) and the $Ga$ (also the set).
However the set of the components contains 2 elements ($\alpha_a$ and $\alpha_b$ ), while $Ga$ has only one element in the set.
That is a point of my confusion.

Where is a flaw in the speculations above ?

The same confusion is happening here. Yoneda talks about natural transformations, not their components.
Here there is exactly one natural transformation $\mathrm{Hom}(a,-) \to G$, and its components are the ones you identify, the unique functions between these one-element sets. And indeed, we see that the Yoneda lemma is verified: $G(a)$ has exactly one element.

Eduard BABKIN (Sep 30 2020 at 11:02):

[Mod] Morgan Rogers said:

Eduard BABKIN said:

I have a technical question: how we can denote each component of natural transformation ?
Because we have a discrete category with a single element $a$, could each component be named as $\alpha_{a_i}$, where $i$ runs from 1 to $n^m$ ?

Or for that case we can say, that the set of components contains sets as elements, so, we have a singleton set of the components of natural transformation ($\alpha_a$), but the element of that set is the set itself (with $n^m$ elements -- all possible functions) ?

Hmm it seems there is some confusion about the data of natural transformations here.
Given functors $F,G: \mathcal{C} \to \mathcal{D}$, a natural transformation $\alpha$ from $F$ to $G$ consists of a morphism $\alpha_A: F(A) \to G(A)$ for each object of $\mathcal{C}$, subject to a naturality condition for each morphism of $\mathcal{C}$. In your example, $\mathcal{C}$ has just one object, so a natural transformation consists of a single morphism.

As we have seen, there may be many (distinct!) natural transformations between a pair of functors, but they are separate, not all part of the same transformation.

It seems, I catch the idea. So, am I right, that in the case of the simple discrete category and $n^m$ functions between the image-sets of $F$ and $G$ there are actually $n^m$ distinct natural transformations, and their components may be named as $\alpha_a , \beta_a, ...$ (up to $n^m$-th symbol in the component's name ) ?

Eduard BABKIN (Sep 30 2020 at 11:14):

[Mod] Morgan Rogers said:

Eduard BABKIN said:

Now I recall that Yoneda lemma says about bijection between the set of components of natural transformation ($Nat(..)$) and the $Ga$ (also the set).
However the set of the components contains 2 elements ($\alpha_a$ and $\alpha_b$ ), while $Ga$ has only one element in the set.
That is a point of my confusion.

Where is a flaw in the speculations above ?

The same confusion is happening here. Yoneda talks about natural transformations, not their components.
Here there is exactly one natural transformation $\mathrm{Hom}(a,-) \to G$, and its components are the ones you identify, the unique functions between these one-element sets. And indeed, we see that the Yoneda lemma is verified: $G(a)$ has exactly one element.

Thank you for pointing a correct view point. Now I see, that Yoneda lemma in our second case tells about bijection between the set $Ga$ and a rather "special" set: the set of morphisms between the object $Hom(a,-)$ and the object $G$ in the category $[C,Set]$. Indeed these sets are equivalent and have one element. The single morphism $Hom(a,-) \rightarrow Ga$ can "cover under the hood" multiple components of natural transformation, but Yoneda lemma keeps silence about that subject.

Morgan Rogers (he/him) (Sep 30 2020 at 11:26):

Indeed, natural transformations are actually a neat package of information, and when the categories get more complicated it's a lot more convenient to think of them as "just morphisms in a functor category" without worrying about the information they carry. This is the same kind of information hiding for clarity that pops up all over category theory, like thinking about sets or vector spaces or groups as "just objects" in their respective categories.