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This is a boring question about $2$-categorical terminology.

When I began to learn about $2$-categories, I read papers using the words "lax functors", "oplax functors", "lax transformation" and "oplax natural transformation".

The first time I got the feeling something was wrong with that terminology was when I realized that the meanings of "lax" in "lax functor" and in "lax transformation" do not seem to be the same. For instance, if $u$ and $v$ are lax functors, say from $\mathcal{A}$ to $\mathcal{B}$, any "lax transformation" from $u$ to $v$ gives rise to a lax functor $h : \Delta[1] \times \mathcal{A} \to \mathcal{B}$ making the "obvious" diagram commute, i.e. there is some kind of "elementary (lax) homotopy" from $u$ to $v$. But the same conclusion holds if we replace "lax transformation" with "oplax transformation"! Therefore, an "oplax transformation" gives rise to a "lax homotopy". That seems pretty awkward to me. Is there a reason why one should use this terminology? Why does one not merely say "transformation" and "optransformation"?

In fact, there is worse. According to the prevalent terminology, the classical "dual notion" to lax functors are "oplax functors". But most people seem happy with the convention according to which ${?}^{op}$ denotes $?$ after inversion of the $1$-cells and ${?}^{co}$ denotes $?$ after inversion of the $2$-cells. Since "oplax functors" are nothing else than lax functors from $\mathcal{B}^{co}$ to $\mathcal{A}^{co}$, why should they be called "oplax"? "Colax" seems quite a better word from this viewpoint. I have just noticed that the nLab makes this choice. Perhaps some people here could tell me the history of both words and what is the current view towards them, at least in the anglo-saxon world.

Note that "optransformation" (or "oplax transformation", for that matter) are consistent with this convention, since this is nothing but a ("lax") transformation from $v^{op}$ to $u^{op}$. However, in ordinary category theory (i.e. $1$-category theory), the duality (which is taken with respect to $1$-cells, since there is no $2$-cells) is denoted by the prefix "co-", as in "comonads", "colimits", &c. This is not consistent with the $2$-categorical convention that the process of reversing $1$-cells should be denoted by "op-".

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Perhaps you need to invent the boring-question tag. :) – Will Sawin Nov 10 '12 at 20:27

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