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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

Thanks! I added some remarks about dual objects in Prof.

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.

Shall we add a section about this? Or perhaps create separate articles opposite functor and opposite natural transformation?

• 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.

• CommentRowNumber7.
• CommentAuthorvarkor
• CommentTimeSep 28th 2022

Add redirect for “op”.

• CommentRowNumber8.
• CommentAuthorvarkor
• CommentTimeSep 28th 2022

Mention the relationship between opposite monoidal categories and opposite 2-categories.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeMay 20th 2023

• CommentRowNumber10.
• CommentAuthornonemenon
• CommentTimeSep 6th 2023

Is there a palatable description of the opposite category of the category of models of a general Lawvere theory? I am specifically interested in the dual category of Monoids, details of which seem completely absent in the literature. I would assume there are at least some folklore concerning of this natural question, in case anyone can give me some direction. Thank you.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeSep 6th 2023

Opposite categories of commutative monoids, at least, have been discussed as categories of generalized affine schemes. I have compiled some references here, but there are more.

• CommentRowNumber12.
• CommentAuthornonemenon
• CommentTimeSep 7th 2023

Thank you.

• CommentRowNumber13.
• CommentAuthorJohn Baez
• CommentTimeJun 1st 2024

I added a result saying that there are essentially just two autoequivalences of the 1-category Cat: the identity and “op”.

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeJun 1st 2024
• (edited Jun 1st 2024)

I have adjusted wording and hyperlinking (here)

and added the original references Toën (2005), Thm. 6.3; Barwick & Schommer-Pries (2011,21), Rem. 13.16; Ara, Groth & Gutiérrez (2013, 15).

(The MO-discussion initiated by Campion is really about $Ho(Cat)$, instead.)