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Stream: theory: category theory

Topic: Chimera categories

Keith Elliott Peterson (Mar 01 2024 at 07:45):

We can define categories internally, and we can define enriched categories.

Apart from enriched categories, I have not seen anyone define a generalized notion of category where the object of objects and the object of morphisms come from different categories.

Let $O$ and $M$ be two categories with enough pullbacks, and $F : O \rightarrow M$ and $G:M \rightarrow O$ two functors between $O$ and $M$ .

Let $o,p \in O$ and $m,n \in M$ be objects in $O$ (resp. $M$ ) such that $o$ and $p$ (resp. $m$ and $n$ ) are the the object and morphism objects in $O$ (resp. $M$ ) and gives rise to an internal category.

Keith Elliott Peterson (Mar 01 2024 at 07:46):

Further, let $F(o) \cong m, F(p) \cong n,$ and $G(m) \cong o, G(n) \cong p,$ .

We will then consider the category $\text{bicograph}_{F,G},$ with $\text{Obj}(\text{bicograph}_{F,G}) := \text{Obj}(O)\coprod\text{Obj}(M),$ and with

$Hom_{\text{bicograph}_{F,G}}(x,y) := \left\{\begin{matrix} Hom_O(x,y) & if\; x,y \in O\\ Hom_M(x,y) & if\; x,y \in M\\ Hom_M(F(x),y) & if\; x \in O ,y \in M\\ Hom_O(G(x),y) & if\; x \in M ,y \in O \end{matrix}\right.$

(I think I got that right)

Then a chimera category $C$ is to be defined as a category internal to $\text{bicograph}_{F,G},$ , with the object of objects being $o$ and the object of morphisms being $n$ , with a lot of the nitty-gritty machinery of source, target, composition, and relevant commuting diagrams already in place. They are just being extended via $F$ and $G$ . In other words, a chimera category is a category with objects in one category and morphisms in another. A truly strange beast.

Keith Elliott Peterson (Mar 01 2024 at 07:46):

In fact, I suspect that $p\in O$ is superfluous, but I'll leave that as an exercise to be proven or disproven.

Exercise 1:
Prove or disprove the above statement.

Trivially then, every category internal to a category is also a chimera category.

Exercise 2:
Prove or disprove that every enriched category is a chimera.

Exercise 3:
Prove or disprove that an $n$ -category is a chimera.

Can you think of any other chimeras? What strange beasts have been hiding and lurking around the mathematical corner?

(Also, for audiences that use Chinese characters, I recommend '龙范畴,' or if you're ambitious, '龘范畴'.)

fosco (Mar 01 2024 at 10:07):

https://en.wikipedia.org/wiki/Amphisbaena

Mike Shulman (Mar 01 2024 at 16:09):

I haven't read carefully, but it's possible this could be an instance of an enriched indexed category, which is a general notion of "category whose objects and morphisms come from different categories".

John Onstead (Mar 01 2024 at 20:04):

This is a very interesting construction!
I don't have enough skill to formally attempt the exercises but did want to put my thoughts out there to see if I'm understanding things correctly (probably not). For 1, every internal category is of course a chimera since you can take category O and M to be the same via the identity functor. Exercise 3 follows from 2 since an n-category is a category enriched in nCat. As for exercise 2, my initial thought was to construct a chimera where the object of objects is from Set (and so you have a set of objects like in an enriched category) and the object of morphisms is in some base category you want to enrich in. However upon further reflection I'm not sure how this leads to enriched categories after all for two reasons. First, you would somehow need to take the monoidal structure of the base category into consideration (since enrichment happens in monoidal categories), which is extra structure that is not considered by the chimera construction. Second, the "enrichment" would be global to the entire chimera category rather than localized between each pair of objects, since one is considering a single object of all morphisms rather than multiple hom objects of morphisms, one between each pair. In any case, my answer to #2 is "no" because even if you could get some construction like the one described to work, there's an issue with the sizes of the involved categories. For instance, a category internal to Set is a small category while a category enriched in Set is a locally small category. Thus the chimera construction would only be able to give "small" enriched categories rather than "locally small" enriched categories.
Edit: actually on second thought maybe you don't need to take monoidal structure (which is needed to define enriched composition) into account, if this is accounted for in the definition of chimera category like how the axioms of composition in a small category are accounted for by the axioms of an internal category in Set. But then wouldn't this allow you to "enrich" in non-monoidal categories? I'm still confused!

Keith Elliott Peterson (Mar 02 2024 at 05:36):

Mike Shulman said:

I haven't read carefully, but it's possible this could be an instance of an enriched indexed category, which is a general notion of "category whose objects and morphisms come from different categories".

I'm not sure if the above construction gives indexed categories. I could be wrong and missing something though.

John Onstead (Mar 02 2024 at 06:00):

@Keith Elliott Peterson You've piqued my interest when it comes to the relationship between enriched and chimera categories, and now I am very curious about what it is. Mind dropping a few hints?

Mike Shulman (Mar 02 2024 at 06:34):

The name "enriched indexed category" is chosen because they simultaneously generalize enriched categories and indexed categories, but they also generalize ordinary internal categories, so they might include yours as well.

Keith Elliott Peterson (Mar 02 2024 at 06:46):

John Onstead said:

Keith Elliott Peterson You've piqued my interest when it comes to the relationship between enriched and chimera categories, and now I am very curious about what it is. Mind dropping a few hints?

If it helps, the diagram I have in mind is this:
image.png

Keith Elliott Peterson (Mar 03 2024 at 09:09):

(deleted)

Nathanael Arkor (Mar 03 2024 at 10:23):

@Keith Elliott Peterson: did you mean to post the same message again?

Keith Elliott Peterson (Mar 03 2024 at 10:29):

How strange