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Stream: learning: questions

Topic: What is a D-filtered category?

Julius Hamilton (May 19 2024 at 15:10):

$(*)$ Let $\mathbb D$ be a collection of (small) categories. A (small) category $\mathcal C$ is called $\mathbb D$-filtered when $\mathcal C$-colimits commute with $\mathbb D$-limits in the category of sets.

A limit/colomit is a universal cone/cocone. A cone/cocone is a family of morphisms (with some more details).

How can a colimit commute with a limit? Since the limit of a diagram can be thought of as an object in a category with morphisms to all the objects in the diagram, it could make sense to say those morphisms commute with certain other morphisms. However, $\mathcal C$ is a separate category from any category $D$ in $\mathbb D$. In order to commute, there has to be a way to compose morphisms from these two categories. Since they say “in the category of sets”, it sounds like there is an assumed functor into $\mathbf Set$. Does this mean any functors?

David Egolf (May 19 2024 at 19:28):

I don't understand this stuff, myself. I am aware that one can talk about a functor "preserving" or "commuting with" a limit of a diagram $D$. In that case, we have that $F( \lim D) \cong \lim F \circ D$ for a functor $F$ and a diagram $D$. Maybe there is some way to view "taking colimits" as a functor, maybe from some category of diagrams? Then I'm hoping that the notion of a functor commuting with a limit can be applied to this case, so we can talk about a colimit-taking functor that commutes with a limit.

(Riehl's "Category Theory In Context" around page 90 might be a helpful for some related concepts.)

Hopefully someone else will chime in, too!

Tim Hosgood (May 19 2024 at 19:35):

just to answer the "how can a colimit commute with a limit" part, imagine you have a diagram of diagrams: let's say you had $D\colon I\times J\to\mathcal{C}$ for some small categories $I$ and $J$; you can equivalently (by "currying") describe this a either a functor $D_{IJ}\colon I\to[J,\mathcal{C}]$ or $D_{JI}\colon J\to[I,\mathcal{C}]$ (where I'm using square brackets to denote the category of functors and natural transformations). That is, I can think of it as an $I$-shaped diagram with values in $J$-shaped diagrams (or vice versa).

Tim Hosgood (May 19 2024 at 19:36):

now I could take e.g. the limit over the $I$-shaped part and the colimit over the $J$-shaped part ($\lim_I\operatorname{colim}_J D_{IJ}$) or I could do the same but in the opposite order ($\operatorname{colim}_J\lim_I D_{JI}$). by looking at the universal properties, I can show that there is a canonical morphism from the latter to the former, but nothing tells me that it has to be an isomorphism!

Tim Hosgood (May 19 2024 at 19:40):

you can consider a more explicit example if you pick your favourite limits and colimits! e.g. is it true that a products and disjoint unions commute in the category of sets? i.e. does $\coprod_i \prod_j X_{ij}\cong\prod_j\coprod_i X_{ij}$ hold?

John Baez (May 20 2024 at 11:09):

@Tim Hosgood's explanation is great, and I find his last example a great way to bring this subject down to earth. I would only add a similar example: is it true that products and coproducts commute in the category of vector spaces?

Julius Hamilton (Jun 08 2024 at 16:42):

I was busy last few weeks so can finally come back to this now.

Julius Hamilton (Jun 08 2024 at 16:49):

@David Egolf

I am aware that one can talk about a functor "preserving" or "commuting with" a limit of a diagram $D$.

Let me think aloud about that. I want to try to remind myself of the definition of a limit by figuring it out instead of looking it up.

Julius Hamilton (Jun 08 2024 at 16:50):

Exhausted right now, excuse any seeming laziness. I like to be able to continue learning even when my presentation isn’t at its sharpest.

Julius Hamilton (Jun 08 2024 at 16:51):

Maybe we can say a limit is an object which “subsumes” a lot of other objects. Since I have a literary background, sometimes I like a good conceptual metaphor, even if it’s not mathematically precise.

Julius Hamilton (Jun 08 2024 at 16:52):

But, I don’t think we can take any random collection of objects in a category and ask, “what is their limit”?

Julius Hamilton (Jun 08 2024 at 16:53):

Or maybe we can. I believe the full phrasing of a “limit” is “a limit of a diagram”.

Julius Hamilton (Jun 08 2024 at 16:54):

It’s said that products, pullbacks, and pushouts are all special cases of limits.

Julius Hamilton (Jun 08 2024 at 16:57):

I’ve been thinking about how a lot of constructions are called “universal” because you have some pattern or structure, and then you can say how any other such structure factors through the first one. For example:

Julius Hamilton (Jun 08 2024 at 16:58):

A “product” can be thought of very simply - it’s just an object with an arrow to two other objects:

Julius Hamilton (Jun 08 2024 at 17:01):

$$\begin{CD} A @>>> B \\@VVV \\C \end{CD}$$

Julius Hamilton (Jun 08 2024 at 17:03):

Except we add on that for any other such structure, in a way, $A$ is more “maximal”:

Julius Hamilton (Jun 08 2024 at 17:04):

3F20EC13-4083-4F40-B96E-AB86FD15508E.jpg

Julius Hamilton (Jun 08 2024 at 17:04):

And I feel like this is the same idea behind a natural numbers object:

Julius Hamilton (Jun 08 2024 at 17:06):

We can express the idea of a successor function like this:

$$\begin{CD} 1 @>>> N @>>> N \end{CD}$$

Julius Hamilton (Jun 08 2024 at 17:09):

But in the normal definition, we add in that this structure is “unique” or “universal” by showing how any other such structure factors through that one:

Julius Hamilton (Jun 08 2024 at 17:10):

5B844F66-86F1-4590-B072-33F89DAEBE73.jpg

Julius Hamilton (Jun 08 2024 at 17:14):

Diagrams are defined as functors. We choose objects and arrows from a category, by indexing them with some other category.

Julius Hamilton (Jun 08 2024 at 17:15):

I don’t see it clearly now, but I think the limit of a diagram can be expressed as a property of a functor.

Julius Hamilton (Jun 08 2024 at 17:20):

I’ll draw a picture now, but I think the idea is, for any diagram we can select, from a category, we can find ways of varying the objects in that diagram to still retain the same shape. The limit is the choice of objects where all the other choices can factor through it, but it has nothing else to factor through.

Julius Hamilton (Jun 08 2024 at 17:22):

I think it’s called localization when you find ways to map diagrams of a category to a specific object in the category, with an endofunctor. Maybe by simplifying a category to its limit objects, we can see the structure more clearly.

Julius Hamilton (Jun 08 2024 at 17:24):

So when someone says a functor commutes with the limit of a diagram, if we think of taking the limit as an endofunctor mapping a diagram to its limit, then we’re just saying two functors commute with each other.

Julius Hamilton (Jun 08 2024 at 17:25):

That makes me think about how since each category has a set of endofunctors, each functor should implicitly map the endofunctors of the source category to a set of endofunctors for the target category.

Kevin Carlson (Jun 08 2024 at 17:29):

No, functors don’t actually generally induce a mapping on endofunctors! If $f:x\to y$ is a morphism in an arbitrary category, there’s generally no way to use it to turn endomorphisms of $x$ into endomorphisms of $y.$ In other words, mapping an object to its endomorphism monoid is not a functor. This is for whatever reason a very tempting misconception, so you’re in good company!

Kevin Carlson (Jun 08 2024 at 17:32):

Taking the limit also isn’t really an endofunctor. A $J$-shaped diagram in $C$ is not an object of $C,$ after all; it’s an object of the functor category $C^J.$ Thus limit is a functor $\mathrm{lim}:C^J\to C,$ and a functor $F$ commuting with such limits means $F\circ \mathrm{lim}=\mathrm{lim}\circ F^J,$ where perhaps you can see what $F^J$ has to mean.

Julius Hamilton (Jun 08 2024 at 17:36):

Ok. I’m going to try to draw some pictures, formulate some small, precise conjectures, and prove or disprove them.

David Egolf (Jun 08 2024 at 17:47):

Julius Hamilton said:

But, I don’t think we can take any random collection of objects in a category and ask, “what is their limit”?

Given any diagram, you can ask if it has a limit! But note that there's no guarantee in general that a given diagram actually has a limit.

David Egolf (Jun 08 2024 at 17:53):

Julius Hamilton said:

Maybe we can say a limit is an object which “subsumes” a lot of other objects. Since I have a literary background, sometimes I like a good conceptual metaphor, even if it’s not mathematically precise.

For myself, I like metaphors that relate to imaging or observation. I sometimes think of a cone over a diagram with apex $A$ as a collection of related observations of $A$. Then, if that diagram has a limit, each morphism from $A$ to the corresponding limit object induces (and is induced by) one such collection of related observations of $A$. So, in a way, the limit "condenses" the multiple related "perspectives" provided by the diagram: roughly speaking, the limiting object "sees the world" in a way equivalent to how our original collection of related objects (our diagram) "sees the world".

This intuition corresponds to the fact that we have a bijection: $C(A, \mathrm{lim} F) \cong \mathrm{Cone}(A, F)$. Here $C$ is a category, $F$ is a diagram in $C$, by $C(A, \mathrm{lim} F)$ we mean the set of morphisms in $C$ from $A$ to the object corresponding to a limit of $F$, and $\mathrm{Cone}(A,F)$ is the set of cones over the diagram $F$ with apex $A$.

Julius Hamilton (Jun 08 2024 at 18:22):

First I’ll try to respond to @Tim Hosgood:

imagine you have a diagram of diagrams:

A “diagram” is nothing more than a synonym for a functor.

A diagram of diagrams is a functor on functors.

We usually think of a diagram as a “sub-part” of a category.

To say “a diagram of diagrams” makes me think we are choosing objects from a category of diagrams.

I think the question, “What diagrams exist, in category $C$?” is too vast; because for each choice of objects in $C$, we also have a choice for the indexing category.

I imagine the exponential object $B^A$ in the category of categories is the collection of all functors $F: A \to B$. But this cannot be - those functors do not form a category, since they don’t compose with one another. Or, perhaps they do, if we seek “commutative squares” as morphisms, like we do in the Arrow category of a category.

Thus, I am working up to determining if it is manageable to construct “the category of all functors into some category $C$”. My guess is yes - in $\mathbf Cat$, functors are morphisms; and somebody told me how all the morphisms into an object are a powerful way to characterize that object.

For now, regarding “a diagram of diagrams”, I’ll envision we have diagrams in a category being indexed by disparate categories. For those diagrams, we respectively index them.

let's say you had $D\colon I\times J\to\mathcal{C}$

It seems here we have a functor from a product category into a category.

you can equivalently (by "currying") describe this a either a functor $D_{IJ}\colon I\to[J,\mathcal{C}]$ or $D_{JI}\colon J\to[I,\mathcal{C}]$ (where I'm using square brackets to denote the category of functors and natural transformations).

I’ll need to think more about this. The product category $I \times J$ acts as a 2-dimensional index - a matrix of categories. $[J, \mathcal{C}]$ is the category of functors from $J$ to $\mathcal C$. I’m going to try to prove why $D: I \times J \to \mathcal{C}$ is equivalent to $D: I \to [J, \mathcal C]$.

And eventually, the goal will be to show there is a canonical morphism from $colim_J lim_I D_{JI}$ to $lim_I colim_J D_{IJ}$.

Spencer Breiner (Jun 08 2024 at 21:23):

In the functor category the functors are the objects, not the arrows, so they don't need to compose. These form a category where the arrows are natural transformations (not squares of functors).