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Stream: learning: questions

Topic: viewing objects as categories

David Egolf (Sep 19 2022 at 15:53):

Some objects can be viewed as categories themselves, for example:

• Each set can be viewed as a discrete category
• Each monoid can be viewed as a category with one object
• Each group can be viewed as a category with one object, where all morphisms are isomorphisms
• Each groupoid can be viewed as a category where each morphism is an isomorphism
• Each preorder can be viewed as a category, with up to one morphism between each pair of objects

However, is there a way to view some functors as categories? Functors can be viewed as objects, in categories where functors are the objects and the morphisms are natural transformations. However, I don't know how to "zoom in" on these objects and view them as categories in their own right.

If we could view functors as categories, then we could view a lot of things as categories, including functions and group representations. If this isn't possible to do this with all functors, is it maybe possible to view some special functors as categories?

Mike Shulman (Sep 19 2022 at 15:55):

Well, there is the [[graph of a functor]] and the [[cograph of a functor]]...

David Egolf (Sep 19 2022 at 16:04):

I see, interesting! Thanks for sharing those links.

Looking at the link on the graph of a functor:
I had read about the category of elements of a functor before, but hadn't connected it to this kind of idea.
So, we can associate a functor with a category, namely its category of elements(?)... Actually, I guess it's a bit more complicated than this, for functors to categories other than $\mathsf{Set}$. That link seems like a good reference for understanding this better, though.

John Baez (Sep 19 2022 at 17:44):

To unpack Mike's answer a little bit: if you're trying to view functors as categories, David, you should start by "decategorifying" and try to view functions as sets. Any function $f: X \to Y$ gives a set called its graph:

$\{(x,y) \in X \times Y \; | \; y = f(x) \}$

and in fact in some axiomatic approaches to math we say a function from $X$ to $Y$ is its graph: any subset of $X \times Y$ with certain well-known properties counts as a function from $X$ to $Y$.

John Baez (Sep 19 2022 at 17:47):

You might try to copy this idea and define the graph of a functor $F: C \to D$ to be some subcategory of $C \times D$.

John Baez (Sep 19 2022 at 17:49):

However, that's not quite right.

It turns out that what we get is not a subcategory of $C \times D$, but a category equipped with a functor to $C^{\text{op}} \times D$. So there are two subtleties: replacing $C$ with its opposite, and replacing "subcategory of...." with "category with a functor to...".

Mike Shulman (Sep 19 2022 at 17:54):

It is important, though, that a function (resp. functor) is not determined by its (co)graph as a standalone set (resp. category), but only together with the (co)projections relating it to its domain and codomain.

John Baez (Sep 19 2022 at 17:59):

Yeah - so for example I didn't wind up seeing a function from $X$ to $Y$ as a mere set, but as a subset of $X \times Y$. (I'm trying to say what Mike said in a less intimidating way.)

Mike Shulman (Sep 19 2022 at 18:02):

I'm trying to say what Mike said in a less intimidating way.

We make a good team that way...

Mike Shulman (Sep 19 2022 at 18:03):

John Baez said:

It turns out that what we get is not a subcategory of $C \times D$, but a category equipped with a functor to $C^{\text{op}} \times D$. So there are two subtleties: replacing $C$ with its opposite, and replacing "subcategory of...." with "category with a functor to...".

Well, actually there are a bunch of different "graphs" of a functor. The nLab page emphasizes the one that comes with a functor to $C^{\text{op}} \times D$, but (as noted in the old query box on that page) you can get other graphs that have functors to $C \times D^{\text{op}}$, or $C\times D$, or even $C^{\text{op}}\times D^{\text{op}}$. I happen to prefer the one that lives over $C\times D$.

Mike Shulman (Sep 19 2022 at 18:57):

I've edited [[graph of a functor]] to discuss all eight (!) graphs.

David Egolf (Sep 19 2022 at 22:32):

Mike Shulman said:

I've edited [[graph of a functor]] to discuss all eight (!) graphs.

Wow!

And thanks to both of you for your explanations.

David Egolf (Sep 19 2022 at 22:43):

By the way, for context, I was reading about group representations. In particular, viewing a group representation as a functor, I was reading about natural transformations from a group representation to itself. I have the intuition that in some settings endofunctors correspond to a certain class of "valid reasoning strategies", as they turn true equations into true equations. I was wondering if somehow a group representation could be viewed as a category, so that an endomorphism of a group representation could be seen as an endofunctor of the corresponding category, thereby corresponding to some kind of reasoning strategy (with regards to equations) in that category.

David Egolf (Sep 20 2022 at 01:51):

A possibly related thought - we can sometimes think of maps into an object as generalized elements of that object.
So, maybe one could try and view natural transformations to a functor as "elements" of that functor? One could then hope that these elements could relate to objects in a category corresponding to the functor.
(This might be related to the "graph of the functor" concept above too(?), but I don't understand it well enough yet).

Mike Shulman (Sep 20 2022 at 02:12):

David Egolf said:

A possibly related thought - we can sometimes think of maps into an object as generalized elements of that object.
So, maybe one could try and view natural transformations to a functor as "elements" of that functor? One could then hope that these elements could relate to objects in a category corresponding to the functor.

Well, there's this nifty thing called the Yoneda Lemma...

David Egolf (Sep 20 2022 at 02:41):

Mike Shulman said:

David Egolf said:
Well, there's this nifty thing called the Yoneda Lemma...

I'm not sure exactly what you were hinting at by this, but it does seem interesting to consider in this context. Maybe something along the lines of associating a functor with its image under the Yoneda embedding, which holds information about how other functors relate to it? Let me see what happens if I try this.

Assume we have a functor $F$ in a category $\mathsf{C}$ where the objects are functors and the morphisms are natural transformations. Then we can consider the functor $\mathsf{C}(-,F): \mathsf{C}^{op} \to \mathsf{Set}$, which sends a functor $G$ to the set of morphisms $\mathsf{C}(G,F)$, which is the set of natural transformations from $G$ to $F$. Morphisms in $\mathsf{C}$ - natural transformations - get mapped to appropriate functions between such sets. Following the intuition from above, this functor would seem to contain information about all the generalized elements of $F$ and how they relate to one another. So, we would hope we could somehow make a category from this information.

I notice that we now we have a functor to $\mathsf{Set}$ associated with $F$, so we can now make a category from $F$ which I think roughly corresponds to the intuition above. Namely, associate a functor $F$ with the category of elements of the functor $\mathsf{C}(-,F): \mathsf{C}^{op} \to \mathsf{Set}$! I haven't checked properly, but I am guessing that objects in this category are natural transformations to $F$, and morphisms are commutative triangles of natural transformations to $F$ - which I am thinking of as relating different generalized elements.

Mike Shulman (Sep 20 2022 at 03:25):

Sorry for being mysterious, I was in a hurry. The Yoneda lemma says that for a functor $F:C\to \rm Set$, maps into $F$ out of a representable functor $C(c,-)$ are uniquely determined by elements of the set $F(c)$. That is, the generalized elements of $F$ at "stages" that are representable are the same as the elements of the sets $F(c)$ as $c$ varies, and these are indeed the same as the objects of the category of elements of $F$.

Mike Shulman (Sep 20 2022 at 03:27):

For a more general functor $F:C\to D$, you can fix any object $d\in D$ and consider the composite $C \xrightarrow{F} D \xrightarrow{D(d,-)} \rm Set$. If this functor is called $G_d$, then an element of $G_d$ in the previous sense — meaning either a natural transformation $C(c,-) \to G_d$ or an object of the category of elements of $G_d$ — is an object $c\in C$ and a morphism $d\to F c$. These are the objects of some of the graphs of $F$.

David Egolf (Sep 20 2022 at 03:33):

I'll have to let things percolate a bit. Thanks for elaborating! It's all very interesting.

John Baez (Sep 20 2022 at 13:34):

Playing my now well-established role as "simplifier of Mike", one thing he's doing here is building up some ideas starting from these two facts:

1) any functor $F: C \to \mathsf{Set}$ can be understood as a collection of sets $F(c)$ where $c$ ranges over all objects of $C$, together with maps between these coming from morphisms in $C$,

and

2) any functor $G : C \to D$ can be understood as a collection of sets $\mathrm{hom}(d,Gc)$ where $c$ ranges over all objects of $C$ and $d$ ranges over all objects of $D$, together with maps between these coming from morphisms in $C$ and $D$.

John Baez (Sep 20 2022 at 13:38):

The first is utterly obvious and the second reduces to the first if we consider all the functors $G_d : C \to \mathsf{Set}$ given by $G_d = \mathrm{hom}(d, G-)$.

John Baez (Sep 20 2022 at 13:41):

But it's very good to build on these ideas using the Yoneda lemma and the concepts of "representable functor", "generalized elements" and "graphs" as Mike has outlined.

David Egolf (Sep 20 2022 at 15:24):

John Baez said:

2) any functor $G : C \to D$ can be understood as a collection of sets $\mathrm{hom}(d,Fc)$ where $c$ ranges over all objects of $C$ and $d$ ranges over all objects of $D$, together with maps between these coming from morphisms in $C$ and $D$.

I think here the functor we are talking about is $F: C \to D$, right?

Mike Shulman said:

Sorry for being mysterious, I was in a hurry. The Yoneda lemma says that for a functor $F:C\to \rm Set$, maps into $F$ out of a representable functor $C(c,-)$ are uniquely determined by elements of the set $F(c)$. That is, the generalized elements of $F$ at "stages" that are representable are the same as the elements of the sets $F(c)$ as $c$ varies, and these are indeed the same as the objects of the category of elements of $F$.

It is very cool to understand the Yoneda lemma as a determining relationship between elements of $F(c)$ and the generalized elements of the functor $F$ of "kind" corresponding to natural transformations from $C(c,-)$!

Let me see if I can understand this in terms of a "graph" of $F$. If $F$ was a function, we would want its graph to correspond to a collection of points $(c, F(c))$. However, $F(c)$ is often a set with more than one element, and we can break it up into smaller pieces by considering its elements. Then can imagine adding the points $(c, x)$ for $x \in F(c)$ to our graph for $F$. Each "point" $(c,x)$ corresponds to a specific element of $F(c)$, and therefore to a specific natural transformation from $C(c,-)$ to $F$, which corresponds to a specific generalized element of $F$.

David Egolf (Sep 20 2022 at 15:43):

We can also understand $F$'s action on morphisms in terms of generalized elements, I think. Let $u: c \to d$ be a morphism in $C$. Then $F$ sends this to a function $F(u): F(c) \to F(d)$. For each $x \in F(c)$, it specifies a $F(u)(x) \in F(d)$. We've seen that $x \in F(c)$ can be associated uniquely with a natural transformation from $C(c,-)$ to $F$, and we can do a similar thing with $F(u)(x) \in F(d)$ to get a natural transformation from $C(d,-)$ to $F$. So, $F(u)$ determines a function from $Hom(C(c,-),F)$ to $Hom(C(d,-),F)$. So the output of $F$ on a morphism $u$ can be thought of as specifying a function relating generalized elements of $F$.

Mike Shulman (Sep 20 2022 at 15:46):

Right -- this is one of the naturality properties of the Yoneda lemma.

David Egolf (Sep 20 2022 at 15:54):

Now, let's see if we can get a category relating to $F$ from this in terms of generalized elements and the relationships between them discussed above.
Choose objects to be all individual natural transformations from representable functors $C(c,-)$ to $F$, with $c$ allowed to vary over the objects of $C$. These are specific generalized elements of $F$.
I'm getting a bit confused when thinking about morphisms. I think it relates closely to what I was saying above. Time to take a break though and let things settle a bit.

Thanks again to both of you for your thoughts!

David Egolf (Sep 20 2022 at 16:54):

I was thinking a bit about generalized elements of $F$, and I think there are too many of them - I shouldn't be trying to make each generalized element an object in a category associated with $F$. For example, if we consider a set $X$, then the objects of $X$ when we view it as a category are its elements, corresponding to functions specifically from $1$, a set with one object. So, it's a specific kind of generalized element of $X$ that ends up getting viewed as an object of $X$ when we think of $X$ as a category.

David Egolf (Sep 20 2022 at 16:57):

I know that $1$ is a terminal object in the category of sets - there is a single function to it from every other set. Is there a terminal object in the category $[C, Set]$ of functors from $C$ to $Set$? We could try $T: C \to Set$ defined by $T(c)= 1$ for each object $c \in C$ and $T(u)=id_1$ for each morphism $u \in C$. Then I think there is one natural transformation to $T$ from any functor in $[C, Set]$.

David Egolf (Sep 20 2022 at 17:02):

We could then try to view an "object" of a functor $F: C \to Set$ as a natural transformation $\eta: T \to F$.
natural transformation to T to F

David Egolf (Sep 20 2022 at 17:09):

We need $F(u)(\alpha_c) = \alpha_d$, where we view $\alpha_c$ as an element of $F(c)$ and $\alpha_d$ as an element of $F(d)$.
From this perspective, an "object of $F$" is one element of $F(c)$ for each $c$, so that the selected elements map onto one another through any morphism relating them in the image of $F$.

David Egolf (Sep 20 2022 at 17:26):

As an example, say $\pi_0: Top \to Set$. Let $\pi_0(X)$ be the set of path-connected components of a topological space $X$. Then $\pi_0(f)$ for $f$ a continuous function $f: X \to Y$ is an induced map $\pi_0(f):\pi_0(X) \to \pi_0(Y)$ on path-connected components, and it makes sense because all the points that start in the same path connected component must end up in the same path connected component.
Then an "object" of $\pi_0$ is a natural transformation from $T: Top \to Set$ to $\pi_0$. This corresponds to an element $\alpha_X$ of $\pi_0(X)$ for each $X$, which is a specific path-connected component of $X$. And we require that $F(f)\alpha_X = \alpha_Y$ for each $f: X \to Y$. But this seems very restrictive, as different continuous functions could send different connected components to different connected components. It seems like these "objects" might not exist as often as I would like...

David Egolf (Sep 20 2022 at 17:33):

I started out by complaining that there are too many generalized elements of objects in general, for purposes of viewing them as categories. But it seems like there are too few of this specific form that I was intuitively viewing as "objects".

David Egolf (Sep 20 2022 at 17:40):

...One last thought, the over category associated with an object $F$ I think is related to what we were talking about above, with generalized elements and relationships between them. The objects of the over category of a functor $F$ are natural transformations to $F$ - generalized elements of $F$ - and the morphisms of commutative triangles relate these generalized elements to one another. Maybe this is a simple enough answer to "How can we make a category from a functor?" to make me happy for now.

Mike Shulman (Sep 20 2022 at 17:42):

Right! For a general category, you often want to consider generalized elements whose domains (sometimes called "stages") are not completely arbitrary, but more general than the terminal object. Specifically, you want to consider a set of domain objects that "generate" the category in some sense. There are a bunch of such senses; the weakest is a [[separator]] and the strongest is a [[dense subcategory]]. The singleton $\{1\}$ in $\rm Set$, and the collection of representables in a presheaf category, are both dense; but the singleton $\{T\}$ in a functor category is not in general even a separator.

David Egolf (Sep 20 2022 at 17:43):

Oh, wow, very interesting! I will enjoy looking at those links!

David Egolf (Sep 24 2022 at 18:35):

I recently ran across another way of getting a category from a functor (from the paper "So, what is a derived functor?"):
another category from a functor

This category appears to be like two categories $C$ and $D$ sitting side by side, except with some "cross-over" morphisms provided by a functor $F: C \to D$.

As an example, for a function between two finite sets, I think the resulting category looks like two collections of dots, with some arrows from the first collection to the second, corresponding to the mapping of the function. This seems cool - maybe it could be used to define some kind of equivalence relation on functions, corresponding to them having isomorphic (or equivalent) categories under this construction.

Mike Shulman (Sep 25 2022 at 00:15):

This is also called the [[cograph of a functor]].

John Baez (Sep 28 2022 at 10:33):

David Egolf said:

John Baez said:

2) any functor $G : C \to D$ can be understood as a collection of sets $\mathrm{hom}(d,Fc)$ where $c$ ranges over all objects of $C$ and $d$ ranges over all objects of $D$, together with maps between these coming from morphisms in $C$ and $D$.

I think here the functor we are talking about is $F: C \to D$, right?

I was talking about a functor $G : C \to D$ and reducing it to our previous study of functors $F: C \to \mathsf{Set}$. But I made a bunch of typos because of too many edits.

John Baez (Sep 28 2022 at 10:33):

This is what I was trying to say:

1) any functor $F: C \to \mathsf{Set}$ can be understood as a collection of sets $F(c)$ where $c$ ranges over all objects of $C$, together with maps between these coming from morphisms in $C$,

and

2) any functor $G : C \to D$ can be understood as a collection of sets $\mathrm{hom}(d,Gc)$ where $c$ ranges over all objects of $C$ and $d$ ranges over all objects of $D$, together with maps between these coming from morphisms in $C$ and $D$.

John Baez (Sep 28 2022 at 10:34):

The first is utterly obvious and the second reduces to the first if we consider all the functors $G_d : C \to \mathsf{Set}$ given by $G_d = \mathrm{hom}(d, G-)$.