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When reading through Loregian and Riehl - Categorical notions of fibration, on p. 3 there is a remark that confuses me about notation. Given a $2$-category $\mathcal C$ one usually defines $\mathcal C^\text{op}$ to be the category $\mathcal C$ but with inverted $1$-cells and $\mathcal C^\text{co}$ to be the category $\mathcal C$ but with inverted $2$-cells.

Many constructions like for example limits and colimits seem to be named according to the co-notation but are mathematically op-dual. Similarly in the example above.

Hence the question is, whether there are any interesting constructions that are named “correctly” and also whether I am even correct about this feeling about things being switched up here.

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The problem is that for a long time there were only 1-categories and hence only one kind of duality, and sometimes people called it "op" and sometimes "co". Colimits and cofibrations were therefore the only possible notion of dual for limits and fibrations, and take place in the opposite category, which was also the only notion of dual for a category, and denoted "op" as being short for "opposite". Sometimes both "op" and "co" terminologies are in use for the same thing, e.g. "oplax" and "colax".

One should not attempt to remedy this by departing from the standard definitions of $C^{\rm op}$ and $C^{\rm co}$ for 2-categories. Among other reasons, they align with standard notions of duality for enriched categories: for a general $V$ there is only one notion of opposite $V$-category, denoted $C^{\rm op}$, and when a 2-category is regarded as a $\rm Cat$-enriched category this produces the 1-cell dual.

I think the most satisfying perspective on this (though not fully satisfying) is that the 2-cell dual of a 2-category, $C^{\rm co}$, corresponds to classical dual constructions (often denoted by "co") on its objects, regarded as generalized categories. For instance, while a comonad on a category $A$ is equivalently a monad on the category $A^{\rm op}$, a comonad in a 2-category $C$ (on an object $A\in C$) is equivalently a monad in $C^{\rm co}$. (A monad in $C^{\rm op}$ is just the same as a monad in $C$.)

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  • $\begingroup$ The perspective in last paragraph seems a little bit like a coincidence that wasn‘t really planned. Since I assume for $3$-categories and so on it just continues, always inverting the highest cells if I guess correctly. $\endgroup$
    – HDB
    Dec 21, 2021 at 17:49
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    $\begingroup$ @Bixxli I don't know anything about who originally chose the notation $C^{\rm co}$ for the 2-cell dual (Benabou perhaps?) and why. It seems at least possible to me that whoever it was had this perspective in mind. It was certainly chosen long before anybody was thinking very much about $n$-categories for $n>2$. $\endgroup$ Dec 22, 2021 at 4:22

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