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• CommentRowNumber1.
• CommentAuthorJohn Baez
• CommentTimeMar 31st 2011
• (edited Mar 31st 2011)

I made a little addition to opposite category, pointing out some amusing nuances regarding the opposite of a $V$-enriched category when $V$ is merely braided. This remark could surely be clarified, but I think you’ll get the idea.

(In case you’re wondering why I did this, it’s because I needed a reference for “opposite category” in a blog entry I’m writing.)

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeMar 31st 2011

Are there actually more than two ways? I seem to remember thinking about this but I don’t remember the answer. I mean, obviously you could try twisting the two objects around each other arbitrarily many times, but would the resulting composition still be associative?

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 8th 2019
• (edited Aug 8th 2019)

This article does not mention opposite functors (F^op: C^op→D^op if F:C→D) or opposite natural transformations (t^op: G^op→F^op if t:F→G).

These are often denoted by the same letter as the original functor and natural transformations, without the superscript. But it may be desirable to distinguish them for the same reason that we distinguish C and C^op. For instance, a natural transformation F^op→G^op and a natural transformation F→G are very different things.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeAug 8th 2019

I typically write $F^{op}$ myself. I can’t remember the last time I had occasion to write $t^{op}$, but that also makes sense to me.

My own inclination would be to add material to opposite category, but I wouldn’t object to creating other articles.

• CommentRowNumber5.
• CommentAuthorMike Shulman
• CommentTimeAug 8th 2019

Writing $F^{op}$ and $t^{op}$ makes extra sense if you regard $(-)^{op}$ as the name of the entire “oppositization” 2-functor $Cat^{co} \to Cat$. I’d also probably suggest adding a section here for now.

• CommentRowNumber6.
• CommentAuthorDmitri Pavlov
• CommentTimeAug 9th 2019

Opposite functors, opposite natural transformations.