Talk:Compact closed category

trace

The definition here is not the usual definition of trace. Trace of f:A --> A should be a morphism I to I, where I is the unit for the monoidal structure. Think in Vect, where Vect(k,k) \cong k. Then tr(f) = e_A c (f # 1) n_A, where n and e are the coevaluation and evaluation maps, c is the braiding or symmetry, and "#" is the tensor product. This gives the usual notion of trace.

For traces in categories that are not compact, you need a pivotal structure.

Also, adjoint are of course only defined for morphisms. In order to say that an object is a left adjoint, you need to mention that you are looking at a monoidal category as a one-object bicategory. There is obviously no need for that, because you can define a left dual as you do.

Should say, adjoints are defined for functors, so if you want to say A^* is the left adjoint for A then it is better to say A^* \otimes - is the left adjoint of A \otimes -. Or indeed, explain the one object bicategory version. —Preceding unsigned comment added by 60.241.132.115 (talk) 11:12, 6 December 2007 (UTC)

Compact closed vs. Rigid

As someone pointed out on the rigid category page, these two articles may be referring to the same thing. It seems to me that they are, and they should be merged.

I see two small differences:

• the condition that the monoidal category be symmetric in the compact closed article.
• the definition of a dual is different: for the rigid article, a dual is merely the internal hom [X, 1], whereas in the compact closed article, a dual also includes the morphisms to the tensor product.

A closer look at references should help. Perhaps there are two conventions current for the definition of a dual, in which case they both need to be acknowledged.

Unique to the rigid article:

• an alternative definition of a dual
• Citation of the source of the definition of rigidity
• Note that internal hom's exist in a rigid category

Unique to the compact category article:

• Citation of original(?) source of definition of compact closed
• Motivation for definition
• Examples

There are also a few unique comments in both. The rest needs to be merged. Expz (talk) 13:35, 15 December 2009 (UTC)

The rigid article now has a section stating this: I quote:

Alternative Terminology A monoidal category where every object has a left (resp. right) dual is also sometimes called a left (resp. right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous category. An autonomous category that is also symmetric is called a compact closed category.

So, no merge. linas (talk) 02:15, 25 August 2012 (UTC)

I left a {{Duplication}} tag on the article. Defining "non-symmetric compact closed categories" seems to duplicate the article on a Rigid category. The original reference the article was based on defines ccompact closed categories to be symmetric, and so does the nLab. 1234qwer1234qwer4 14:16, 16 September 2026 (UTC)

The non-symmetric definition apparently includes being bi-closed, so is apparently more general than rigidity. However, I'm struggling to find a source for this that isn't just a scrape of this article. At the very least, this does not appear to be a well-recognised definition. If it's kept at all, the sections should be renamed so that the definition for the symmetric case is clearly the main one. Edit: The definition for the non-symmetric case is clearly just rigidity, apologies. As for the claim about it being both rigid and (bi)closed; it appears (see https://arxiv.org/pdf/2301.03545v2) that the former implies the latter. The most appropriate thing here is probably just to note that some sources (especially comp linguists) use "non-symmetric compact closed" to mean "rigid" and link to the article on rigid categories. Adkbeckett (talk) 11:25, 18 September 2026 (UTC)

I've found and added a source for the non-symmetric definition. This appears to be mainly in use in computational linguistics. I also renamed the sections as I suggested. Adkbeckett (talk) 12:54, 18 September 2026 (UTC)

I'm now slightly skeptical of this claim that non-symmetric compact closed, as defined here, is stronger than rigidity. The claim that compact closed = rigity + biclosed is not found in the source for the definition. It appears that the linguists' interest in the non-symmetric case is allowing for left- and right-duals/adjoints which are not necessarily isomorphic - is this allowed in rigid categories? Adkbeckett (talk) 13:01, 18 September 2026 (UTC)

@FaviFake, the duplication banner was only referring to parts of the article. Your edit does not look helpful to me; I would suggest leaving this to someone who knows how to organise the content between the articles. 1234qwer1234qwer4 00:09, 26 September 2026 (UTC)

 Done, hopefully you'll find a solution. I did not notice this discussion after seeing the article pop up on the backlog. FaviFake (talk) 00:13, 26 September 2026 (UTC)

Unit introductions seems reversed

"where ρ, λ are the introduction of the unit on the left and right, respectively, and is the associator"

This seems backwards, its the inverses of ρ, λ that to the introductions, in other sources, and to cause the equations to make sense. Not being a mathematician myself, I'll leave this as a comment rather than edit the article now, better if an actual mathematician does that. AveryDAndrews (talk) 22:22, 27 August 2024 (UTC)