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Stream: learning: questions

Topic: Finite completeness/cocompletenes of the cospan category

Adittya Chaudhuri (Aug 27 2025 at 13:32):

Let $\mathsf{C}$ be a category with finite limits and colimits. Let $Co(\mathsf{C})$ be the category whose objects are cospans in $\mathsf{C}$ and morphisms are morphisms of cospans in $\mathsf{C}$ (i.e a triplet of morphisms in $\mathsf{C}$ such that the necessary diagram commutes). Composition law is derived from the composition law of $\mathsf{C}$. Although I am yet to check every detail, but it seems that by diagram chasing one can show $Co(\mathsf{C})$ is finitely complete and cocomplete. (I think I checked correctly for the case of pullback and terminal objects by diagram chasing). If the result is true (i.e $Co(\mathsf{C})$ is finitely complete and cocomplete when $\mathsf{C}$ is finitely complete and cocomplete), then I am looking for some references where such result is mentioned.

John Baez (Aug 27 2025 at 13:37):

I'm hoping there's a general result that goes like this: if $\mathsf{C}$ is a category with finite limits and colimits, and $\mathsf{D}$ is a finite category, then $\mathsf{C}^{\mathsf{D}}$ has finite limits and colimits, computed pointwise. You need the case where

$\mathsf{D} = \bullet \leftarrow \bullet \rightarrow \bullet$

I'm more familiar with this result: if $\mathsf{C} = \mathsf{Set}$, and $\mathsf{D}$ is any category, then $\mathsf{C}^{\mathsf{D}}$ has all limits and colimits, computed pointwise.

Adittya Chaudhuri (Aug 27 2025 at 13:39):

Thanks very much. Is there any reference where I can find a result in this direction?

John Baez (Aug 27 2025 at 13:40):

I can give you references for the latter result, with $\mathsf{C} = \mathsf{Set}$, but I bet someone here knows where to find references for the result you really want: if $\mathsf{C}$ is a category with finite limits and colimits, and $\mathsf{D}$ is a finite category, then $\mathsf{C}^{\mathsf{D}}$ has finite limits and colimits, computed pointwise.

John Baez (Aug 27 2025 at 13:42):

(I don't know if the finiteness of $\mathsf{D}$ is relevant, but you have it in your example!)

Adittya Chaudhuri (Aug 27 2025 at 13:42):

Thanks. I see. Actually, when I proved the existence of the pushout by diagram chasing, the proof was straighforward but very complicated. I had to use a big board to really prove it in detail. That's why I thought, may be there could be some other shorter/smarter ways.

Adittya Chaudhuri (Aug 27 2025 at 13:43):

John Baez said:

(I don't know if the finiteness of $\mathsf{D}$ is relevant, but you have it in your example!)

Yes true.

John Baez (Aug 27 2025 at 13:45):

I think the finiteness of $\mathsf{D}$ is irrelevant because the limits and colimits in a functor category are computed pointwise... you don't need to think about the whole category $\mathsf{D}$ when computing a limit or colimit in $\mathsf{C}^{\mathsf{D}}$

The result you want is mentioned here, with @Kevin Carlson providing an answer, but no reference.

Adittya Chaudhuri (Aug 27 2025 at 13:46):

John Baez said:

I think the finiteness of $\mathsf{D}$ is irrelevant because the limits and colimits in a functor category are computed pointwise... you don't need to think about the whole category $\mathsf{D}$ when computing a limit or colimit in $\mathsf{C}^{\mathsf{D}}$

Thanks. Yes, I agree.

Adittya Chaudhuri (Aug 27 2025 at 13:46):

John Baez said:

The result you want is mentioned here, with Kevin Carlson providing an answer, but no reference.

Thanks very much!!

John Baez (Aug 27 2025 at 13:50):

Maybe a good category theory book that discusses functor categories would have this result, if what you want is a reference.

Adittya Chaudhuri (Aug 27 2025 at 13:50):

Thanks. I will check some standard category theory books for such results.

Adittya Chaudhuri (Aug 27 2025 at 14:14):

Although I am yet to find a text book reference, but I just checked that it is mentioned in the nLab page on functor category

Ralph Sarkis (Aug 27 2025 at 14:33):

One such reference is Sections 8.5 and 8.6 in Awodey's Category Theory 2nd edition. Wait... Only the colimit statement is with an arbitrary codomain category (the limit statement uses Set).

Adittya Chaudhuri (Aug 27 2025 at 14:56):

Thanks very much.

James Deikun (Aug 27 2025 at 15:06):

The limit statement with an arbitrary codomain category follows from the colimit statement by duality.

Adittya Chaudhuri (Aug 27 2025 at 16:22):

Yes. Thanks.

Adittya Chaudhuri (Aug 29 2025 at 12:57):

Let $\mathsf{A}$ and $X$ be finitely cocomplete categories and $F \colon \mathsf{A} \to \mathsf{X}$ be a finite colimit preserving functor. Let $\mathbb{O}pen_{F}(\mathsf{X})$ be the associated symmetric monoidal double category whose horizontal 1-morphisms are $F$-structured cospans. Now, I considered the category $\mathbb{O}pen_{F}(\mathsf{X})_{1}$, whose objects are $F$-structured cospans and morphisms are morphisms of $F$-structured cospans.

I just finished verifying the following:

I (hopefully correctly) showed that $\mathbb{O}pen_{F}(\mathsf{X})_{1}$ has finite colimits by showing the existence of an initial object and the exisitence of all pushouts. However, I find my proof of the existence of pushout - a very tedious diagram chasing method (with some not so difficult manipulation of universal properties). So, definitely, I would like to see a more clever/shorter proof of the fact that $\mathbb{O}pen_{F}(\mathsf{X})_{1}$ has finite colimits . I would be more happier if this result follows from some more general fact like the way @John Baez suggested for the case of usual cospan categories here #learning: questions > Finite completeness/cocompletenes of the cospan category @ 💬 .

John Baez (Aug 29 2025 at 14:26):

What is your ultimate goal here? In my paper with Kenny on structured cospans, we show that when $F: \mathsf{A} \to \mathsf{X}$ is a functor preserving finite colimits between categories that have such colimits, we get a weak category $\mathbb{O}\mathbf{pen}(\mathsf{X})$ internal to $\mathbf{Rex}$. Later we show this gives a symmetric monoidal double category, the usual double category of structured cospans also called $\mathbb{O}\mathbf{pen}(\mathsf{X})$.

This means that we have categories with finite colimits called $\mathbb{O}\mathbf{pen}(\mathsf{X})_0$ and $\mathbb{O}\mathbf{pen}(\mathsf{X})_1$, and source and target maps

$s, t : \mathbb{O}\mathbf{pen}(\mathsf{X})_1 \to \mathbb{O}\mathbf{pen}(\mathsf{X})_0$

that preserve finite colimits, and then a composition map that's associative up to isomorphism, where the isomorphism obeys the pentagon identity, and so on. See the material from Definition 3.1 to Theorem 3.7.

So perhaps you're trying to do something that was already done? Or perhaps I'm misunderstanding you.

Adittya Chaudhuri (Aug 29 2025 at 18:31):

John Baez said:

What is your ultimate goal here?

Thanks!!

My initial goal is to define a notion of $\mathbb{O}pen_{F}(\mathsf{X})_1$ valued co-sheaf (covariant sheaf) on the interval site (Definition 2.2 of this paper) using the definition of co-sheaf as defined in the Proposition 2.7 of the same paper. In order to apply this definition, I need $\mathbb{O}pen_{F}(\mathsf{X})_1$ to have pushouts. This cosheaf on interval site will capture the idea of an underlying static object that is accumulated over time (as explained in the same paper).

My next goal is to combine "cosheaf" structure with "the double category of structured cospan structure" by defining a suitable double category whose horizontal 1-morphisms are $\mathbb{O}pen_{F}(\mathsf{X})_1$ valued co-sheafs. (I have thought about a candidate for the same, but I need to work a bit more before I can tell it in a precise way). In a way, my final goal is to construct a double category which will give us compositional framework for time-varying objects. In a special case, I would like to study the compositional theory of time varying graphs with polarities. Probably, I want to explore how our homology monoids (defined in our paper) behave in this new framework.

John Baez (Aug 29 2025 at 18:40):

Okay. I don't understand that, and I especially don't understand why you'd be talking about $\mathbb{O}pen_{F}(\mathsf{X})_1$ (the category of loose morphisms) without also talking about $\mathbb{O}pen_{F}(\mathsf{X})_0$ (the category of objects). But anyway, as I mentioned, they both have finite colimits given your asssumptions:

Let $\mathsf{A}$ and $X$ be finitely cocomplete categories and $F \colon \mathsf{A} \to \mathsf{X}$ be a finite colimit preserving functor.

Adittya Chaudhuri (Aug 29 2025 at 18:43):

I am talking about $\mathbb{O}pen_{F}(\mathsf{X})_1$ valued cosheaf because I want to combine two compatible coaheafs (representing time varying open systems) into another cosheaf. By compatibility, I mean source-target. Here, source and target are copresheafs, but valued in $\mathsf{A}$.

Adittya Chaudhuri (Aug 29 2025 at 18:47):

In the double category that I want to construct, I want my two morphisms to be triplets of natural transformations. (source, horizontal 1-morphism, target) with some compatibility conditions.

Adittya Chaudhuri (Aug 29 2025 at 18:49):

I am implicitly considering the elements of $\mathbb{O}pen_{F}(\mathsf{X})_0$ by taking $\mathsf{A}$-valued source and target copresheafs.

Adittya Chaudhuri (Aug 29 2025 at 18:52):

May be what I am saying is too vague at the moment. I hope in a few days, I can make the above ideas more concrete.