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Stream: learning: reading & references

Topic: Cartesian closed objects?

Vincent Moreau (Mar 05 2024 at 14:01):

Dear category theorists,

There is a notion of cartesian object in a 2-category with finite 2-products, which appears for example in the article A 2-categorical approach to change of base and geometric morphisms I by Carboni, Kelly and Wood. A cartesian object in $\mathbf{Cat}$ is a cartesian category.

Does there exist an analogous notion of "cartesian closed object" in a sufficiently good 2-category? Ideally, $\mathbf{Cat}$ would be good enough and cartesian closed objects there would give back cartesian closed categories.

I'm also interested in hypothetical "monoidal objects" and "monoidal closed objects", but I would already be happy to learn about the more restrictive, cartesian setting :)

Kenji Maillard (Mar 05 2024 at 15:48):

Would pseudomonoids be the "monoidal objects" you are looking for ?

Mike Shulman (Mar 05 2024 at 15:50):

You can do this in a [[2-category equipped with proarrows]], or a similar thing such as a [[Yoneda structure]], which has cartesian products (in the arrow direction) yielding a monoidal structure that is compact closed in the proarrow direction. In that case, the product functor $A\times A \to A$ (or any more general monoidal structure $m :A\otimes A \to A$ induces a proarrow "$\hom_A(m(x,y),z)$" from $A$ to $A\otimes A$, which by the compact closed structure can be transferred into a proarrow from $A^o \otimes A$ to $A$, and you then ask for this proarrow to be representable by an arrow, the internal-hom.

Mike Shulman (Mar 05 2024 at 15:51):

I think the original reference for this idea is Day and Street, "Monoidal bicategories and Hopf algebroids".

Vincent Moreau (Mar 05 2024 at 16:19):

Kenji Maillard said:

Would pseudomonoids be the "monoidal objects" you are looking for ?

Yes, indeed!

Vincent Moreau (Mar 05 2024 at 16:22):

Mike Shulman said:

You can do this in a [[2-category equipped with proarrows]], or a similar thing such as a [[Yoneda structure]], which has cartesian products (in the arrow direction) yielding a monoidal structure that is compact closed in the proarrow direction. In that case, the product functor $A\times A \to A$ (or any more general monoidal structure $m :A\otimes A \to A$ induces a proarrow "$\hom_A(m(x,y),z)$" from $A$ to $A\otimes A$, which by the compact closed structure can be transferred into a proarrow from $A^o \otimes A$ to $A$, and you then ask for this proarrow to be representable by an arrow, the internal-hom.

I see, thank you!

Notification Bot (Mar 05 2024 at 16:23):

Vincent Moreau has marked this topic as resolved.

Notification Bot (Mar 05 2024 at 17:22):

John Baez has marked this topic as unresolved.

John Baez (Mar 05 2024 at 17:25):

By the way, monoidal objects make sense in any monoidal 2-category - just as monoid objects make sense in any monoidal category! This pattern, which is expected to continue, is an example of the "microcosm principle": objects with some structure tend to make sense in a context with the same sort of structure.

John Baez (Mar 05 2024 at 17:26):

Here's another example of the microcosm principle: cartesian objects make sense in any cartesian 2-category - that is, category with finite products. Trying to get cartesian objects to make sense in even greater generality is being a bit "pushy", in my opinion: breaking out of the obvious pattern suggested by the microcosm principle, and perhaps reaching toward some new pattern.

John Baez (Mar 05 2024 at 17:30):

And here's another important example: symmetric monoidal objects make sense in any symmetric monoidal 2-category.

(Just as monoidal objects are usually called "pseudomonoids" or "monoidales", symmetric monoidal objects are called "symmetric pseudomonoids".)

John Baez (Mar 05 2024 at 17:30):

Also, wherever I said "2-category" you could and arguably should generalize a bit and say "bicategory".

Vincent Moreau (Mar 06 2024 at 07:52):

I wonder, does the notion of 2-category equipped with proarrows which has cartesian products in the arrow direction, as suggested by Mike to be a good framework for cartesian closed objects, fit with the microcosm principle? The connection seems less direct than what you describe with monoidal objects and monoidal 2-categories, or cartesian objects and cartesian 2-categories.

John Baez (Mar 06 2024 at 16:39):

I suspect the microcosm principle, while somewhat mysterious already for 2-categories, is even less well understood for double categories.