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.)

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

]]>Thank you.

]]>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.

]]>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.

]]>added pointer to:

- Saunders MacLane, §II.2 of:
*Categories for the Working Mathematician*, Graduate Texts in Mathematics**5**Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

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

]]>Add redirect for “op”.

]]>Opposite functors, opposite natural transformations.

]]>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.

]]>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.

]]>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?

]]>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?

]]>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.)

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