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Stream: learning: questions

Topic: defining a "diagram limit" for a diagram

David Egolf (Nov 04 2023 at 18:13):

I've been working on better understanding limits (in my ongoing quest to understand how limits are preserved by certain representable functors!). My current understanding is that a limit of a diagram $D: J \to C$ consists of two pieces of data $(x,f)$, where:

• $x$ is an object of $C$
• and $f: C(-,x) \cong Cone(-,D)$ is a natural isomorphism

David Egolf (Nov 04 2023 at 18:13):

Here $Cone(-,D): C^{\textrm{op}} \to \mathsf{Set}$ sends an object $c \in C$ to the set of cones over $D$ with apex of $c$. And $Cone(-,D)(f: c \to d)$ is a function $f^*:Cone(d,D) \to Cone(c,D)$ which acts by precomposition "at the apex", so $g \mapsto g \circ f$ for each "leg" $g$ of a cone with apex $d$.

If we view $x$ as a diagram with one object, I think that $Cone(-,x)$ is very similar to the functor $C(-,x)$. That is because a cone over a diagram with a single object $x$ is just a morphism to $x$.

David Egolf (Nov 04 2023 at 18:13):

Inspired by this, I was wondering if it could be interesting to consider the concept of a "diagram limit". A diagram limit for a diagram $D: J \to C$ of "shape" $F: I \to C$ would consist of two pieces of data $(F,f)$, where:

• $F: I \to C$ is a diagram in $C$
• and $f: Cone(-,F) \cong Cone(-,D)$ is a natural isomorphism

David Egolf (Nov 04 2023 at 18:13):

My hope is that even if a diagram $D: J \to C$ doesn't have a limit (in the standard sense), it might have a diagram limit with respect to some $F: I \to C$ where $I$ is much "smaller" then $J$. Intuitively, this might allow one to "compress" one's understanding of cones over $D: J \to C$ to be in terms of cones over a smaller diagram $F: I \to C$.

Is this something that people do sometimes?

I'd also be interested in finding examples of diagrams $D: J \to C$ that don't have a limit, but have a "diagram limit" ($F: I \to C,f)$ where $I$ is "smaller" than $J$. (To determine if $I$ is "smaller" than $J$, we might ask if $I$ has fewer morphisms than $J$, for example).

Todd Trimble (Nov 04 2023 at 18:47):

You and I had begun discussing the preservation of limits by representable functors, and also the connection between this and the Yoneda lemma, but the discussion was dropped at some point.

Todd Trimble (Nov 04 2023 at 18:47):

In any case, let me encourage you to look more closely into how "$Cone(-, x)$ is very similar to the functor $C(-, x)$". (Did you mean to write $Cone(-, D)$?) More precisely, let me encourage you to understand exactly how $Cone(-, D)$ is defined in terms of categories, functors, and natural transformations. A cone is a certain natural transformation, but between which functors?

Todd Trimble (Nov 04 2023 at 18:49):

With regard to your idea of diagram limits, you might also be interested in the notion of "final functor", if you haven't bumped into this already.

David Egolf (Nov 04 2023 at 18:57):

Todd Trimble said:

You and I had begun discussing the preservation of limits by representable functors, and also the connection between this and the Yoneda lemma, but the discussion was dropped at some point.

Yes, and I am still interested in talking about this! I've been slowly working on better understanding some of these things before posting again in that thread. (And I've also been distracted by other things!) But I'm happy to connect this conversation to that one.

David Egolf (Nov 04 2023 at 19:03):

Todd Trimble said:

In any case, let me encourage you to look more closely into how "$Cone(-, x)$ is very similar to the functor $C(-, x)$". (Did you mean to write $Cone(-, D)$?) More precisely, let me encourage you to understand exactly how $Cone(-, D)$ is defined in terms of categories, functors, and natural transformations. A cone is a certain natural transformation, but between which functors?

Thanks for the suggestion! It will take me some time to think about this.

Although I can immediately clear this up: I did mean to write $Cone(-,x)$, and not $Cone(-,D)$. The idea is that $Cone(-,x)$ is shorthand for $Cone(-,F)$ where $F: 1 \to C$ is the diagram that picks out the element $x \in C$. (Here $1$ is a category with a single object and just the identity morphism).

David Egolf (Nov 04 2023 at 19:08):

I think I can start to clarify my understanding of how $Cone(-,D)$ is defined in terms of categories, functors, and natural transformations. A cone over $D$ with apex $c$ is a natural transformation from a functor that is constant at $c$ to the functor $D: J \to C$. We might call the constant functor $\Delta_c: J \to C$.

So, we can consider the category of functors from $J$ to $C$ and the natural transformations between them, $[J,C]$. The set of cones over $D$ is I think then the set of morphisms from $\Delta_c: J \to C$ to $D: J \to C$. So, $Cone(c,D) \cong [J,C](\Delta_c, D)$.

David Egolf (Nov 04 2023 at 19:12):

We still have to consider how $Cone(-,D)$ acts on morphisms. It is tempting at this point to guess that $Cone(-,D)$ is $[J,C](\Delta-, D)$, where $\Delta:C \to [J,C]$ is some functor that sends an object $c$ to the functor $\Delta_c: J \to C$ that is constant at $c$.

However, it will take me some work to properly figure this out!

Todd Trimble (Nov 04 2023 at 19:15):

In that case, $Cone(-, x) \cong C(-, x)$ looks pretty tautological, if your shape category is the category $1$ with exactly one object and morphism.

Todd Trimble (Nov 04 2023 at 19:17):

Your understanding of cones as natural transformations is correct!

David Egolf (Nov 04 2023 at 19:19):

Todd Trimble said:

In that case, $Cone(-, x) \cong C(-, x)$ looks pretty tautological, if your shape category is the category $1$ with exactly one object and morphism.

To be honest, I hadn't thought about the role that the shape category plays here. I mostly just wanted the diagram corresponding to $x$ to only have the object $x$ and its identity morphism.

Todd Trimble (Nov 04 2023 at 19:20):

So if $D$ has a limit, then $[J, C](\Delta-, D) \cong C(-, \lim D)$.

Todd Trimble (Nov 04 2023 at 19:22):

We would say in that case that $\lim D$ "represents" the functor $[J, C](\Delta -, D)$.

David Egolf (Nov 04 2023 at 19:25):

Yes, that all makes sense!

Todd Trimble (Nov 04 2023 at 19:29):

There's a good chance you know this already, but if every diagram $D$ of shape $J$ in $C$ has a limit, then the natural isomorphism

$[J, C](\Delta x, D) \cong C(x, \lim D)$

says that $\Delta: C \to [J, C]$ has $\lim: [J, C] \to C$ for its right adjoint. But there's a nice little bit of categorical wisdom lurking there:

Todd Trimble (Nov 04 2023 at 19:37):

Namely, that even though we haven't quite defined how $\lim$ is defined as a functor, the effect of $\lim$ on morphisms is automatically determined by exploiting universal properties. More generally, the wisdom is that in order for a functor $F: C \to D$ to have a right adjoint $G: D \to C$, it is sufficient that every functor $D(F-, d)$ be representable (by an object of $C$). It's highly recommended that every student of category theory goes through this at one time in their life!

David Egolf (Nov 04 2023 at 19:42):

I think I have seen the natural isomorphism you describe above (in Leinster's book "Basic Category Theory"), although I did not understand it. I still don't understand why the isomorphism is natural in $D$, or exactly how $\lim$ is a functor.

But I see that you indicate below that there is a way to figure out how $\lim$ needs to act on morphisms using universal properties! That sounds pretty cool! The more general insight you share also sounds interesting. (Although I don't quite see how the pieces fit together though, just yet.)

Todd Trimble (Nov 04 2023 at 19:49):

Take your time with this! It's really worthwhile fitting all these pieces together. I don't remember how long it took me; I only remember that it was well worth the effort.

David Egolf (Nov 04 2023 at 20:00):

I will plan to chip away at this then! If I figure things out or get too stuck I'll hopefully post here about that.

(And thanks again for your comments in both this thread and the previous one on representable functors. Once I get a bit further with understanding preservation of limits, I am hopeful that I'll be able to post a bit again in that other thread too!)

Ralph Sarkis (Nov 04 2023 at 20:50):

Not sure if I should let you make that step on your own, but your question at the start (which I find quite interesting) essentially amounts to finding another diagram $F:I \to C$ such that

$$[J, C](\Delta^J-, D) \cong [I, C](\Delta^I-, F)$$

This makes sense when $J$ and $I$ are small and $C$ is locally small (otherwise the functor categories involved might not be locally small). When $D$ has a limit, you can pick $F: \mathbf{1} \to C$ to be the constant functor at $\lim D$. Otherwise, what does it mean for $F$ and $D$ to have the same cones?

There are some trivial cases. For instance when $D$ is a diagram with one or more instances of a terminal object from $C$, you can remove them to get a diagram $F$, and cones over both diagrams will be the same thing (because a morphism from the tip of the cone to a terminal object adds no information EDIT: I was too quick en besogne, see James' answer below).

I am sure there are interesting cases, but I don't see any at the moment.

James Deikun (Nov 05 2023 at 03:49):

When D is itself a cone, it has the apex as a limit, but if it's a limiting cone you can also remove the apex and get a "diagram limit".

James Deikun (Nov 05 2023 at 03:54):

In fact whenever there's a limiting cone contained in D you can "safely" remove its apex; the above fact about terminal objects is a special case as $\bold{1}$ by itself is a limiting cone.

James Deikun (Nov 05 2023 at 04:10):

There's kind of a trick to the sense of "contains" though. You can remove arrows into the apex, and you can remove arrows out of the apex that are part of the limiting cone, but arrows out of the apex that are not part of the limiting cone can't always be removed; a simple example is the diagram $\bold{1} \to \bold{1} + \bold{1}$ (either one) where the $\bold{1}$ can't be removed; with it the limit is $\bold{1}$ and without it is $\bold{1} + \bold{1}$.

David Egolf (Nov 06 2023 at 20:02):

Thanks @Ralph Sarkis and @James Deikun for your comments! I don't yet understand the point that James Deikun is making though.

The situation I'm imagining at the moment, while trying to understand your comments, is the following:
adding or removing a terminal object

On the right, we have some cone with apex $x$ over a diagram $D'$. I have denoted this cone using $a: x \to D'$. On the left, we have augmented the diagram $D'$ with a terminal object $1$. This requires us to augment the cone by adding in a (uniquely specified) morphism to the terminal object.

Above, I have drawn $f:y \to x$ to facilitate thinking about the naturality square below. We get this square by evaluating each cone functor (which goes from $C^{\mathrm{op}}$ to $\mathsf{Set}$) on a morphism $f^{op}: x \to y$, which has a corresponding morphism $f: y\to x$ in $C$.

naturality square

David Egolf (Nov 06 2023 at 20:05):

Here $D_L$ means "the diagram on the left" (in the picture above, it is the diagram $D'$ augmented with the terminal object 1) and $D_R$ means "the diagram on the right", which is $D'$ in this case. By $f^*$ I mean the function that produces a new cone by precomposing each leg of cone by $f$. And by $(!,a) \mapsto a$ I mean the function that takes a cone over the augmented version of $D'$ and deletes the leg going to the terminal object, so that we get a cone over the un-augmented version of $D'$.

It seems like the naturality square should commute, and that each of the vertical morphisms are isomorphisms. So I am thinking that $Cone(-,D_R) \cong Cone(-,D_L)$.

However, you both seem to be indicating there is some additional subtlety here, so I suspect I am missing something!

David Egolf (Nov 06 2023 at 20:17):

Setting aside my confusion on this point for the moment, I am wondering if we can get interesting examples of "diagram limits" in a certain way. (The way I describe below might be what James Deikun mentioned above, but I am not sure).

The idea is that if you have a subdiagram in $D$ that has a limit, then you can replace that subdiagram by (the object part of) its limit. Then it intuitively seems like the cones over the original diagram $D$ and over the smaller resulting diagram $F$ should be in bijection with one another.

Here is an illustration of the idea:
two diagrams with "equivalent" cones

The picture above shows two cones, each with apex $x$. I think there should be a bijection between the cones over the diagram with $a,b,c$ on the left and the cones over the diagram with $a \times b,c$ on the right. I haven't checked, but I am guessing this is a natural bijection so that the cone functors over these two diagrams are naturally isomorphic. That is, $Cone(-,\{a,b,c\}) \cong Cone(-\{a \times b, c\})$ where $\{a,b,c\}$ means the discrete diagram with objects $a$ ,$b$, and $c$; and $\{a \times b, c\}$ means the discrete diagram with objects $a \times b$ and $c$.

Ralph Sarkis (Nov 06 2023 at 20:21):

The subtlety can arise only when the terminal object you remove has some morphisms going out of it in the diagram (which is not the case in your $D_L$). The example James gave is below $D_L$ is in green and $D_R$ is in purple. A cone over $D_L$ is not the same thing as a cone over $D_R$. It is true that the morphism ${!}: X \to \mathbf{1}$ is uniquely determined (by terminality), but the morphism $\mathsf{inl}:\mathbf{1} \to \mathbf{1}+\mathbf{1}$ (which is the left coprejction) enforces that the morphism $X \to \mathbf{1}+\mathbf{1}$ must be $\mathsf{inl} \circ {!}$ whereas it could be anything for the cone over $D_R$.

image.png

Ralph Sarkis (Nov 06 2023 at 20:23):

Your example with $a \times b$ works again because there are no morphisms going out of $a$ or $b$.

David Egolf (Nov 06 2023 at 20:25):

Ah, thanks for explaining @Ralph Sarkis ! That makes a lot of sense. So things become more tricky when our subdiagram of interest has morphisms coming out of it that land in other parts of the diagram!