I would like to know the standard usage of "lax colimit" and "oplax colimit" in the 2-categorical literature. The nLab does not give an explicit definition of "lax colimit", as far as I can see, and I don't know what the most reliable source is. I think I have seen at least one paper using each convention, but I have not encountered the notion often enough to have a good sense of whether this one of those places where the terminology is not really standardized, or there is general agreement with a few exceptions.


Concretely, given a diagram X : I → C in a 2-category C (for my purposes indexed by a 1-category I), suppose I have a cone (Y, {gi}i∈Ob I, {αf}f∈Mor I), with gi : Xi → Y, and for f : i → j in I, αf : gjf → gi (such that various diagrams commute). Is this a lax colimit cone or a oplax colimit cone?



Terminology: lax vs. oplax colimits
I agree with Finn that the way to derive the correct choice of lax vs oplax is to connect it back to natural transformations. Of course, as Finn pointed out, there is controversy over the choice for natural transformations, but my views on that are clear at the [nlab page](http://ncatlab.org/nlab/show/lax+natural+transformation) so I'll just write using that terminology.


However, in contrast to Finn, I think what you've got there is actually a *lax* cocone, since it is given by a lax natural transformation $\* \to C(X-,Y)$, where $\*:I^{op}\to Cat$ is the functor constant at the point. It's true that, as Finn says, it is also an *oplax* natural transformation from $X$ to $\Delta\_Y$, where $\Delta\_Y$ is the functor $I\to C$ constant at $Y$. But I think it's better to think of a cone as a transformation $\* \to C(X-,Y)$, since this is the version that generalizes to *weighted* limits: for any weight $J:I^{op}\to Cat$, a $J$-weighted cocone is a transformation of the appropriate sort $J\to C(X-,Y)$.


The weighted-limit perspective on lax (co)limits is especially valuable because of the existence of lax morphism classifiers. Namely, for any weight $J$ there is another weight $J^\dagger$ such that lax transformations out of $J$ are the same as strict (or pseudo) transformations out of $J^\dagger$. Thus, lax $J$-weighted limits are the same as ordinary $J^\dagger$-weighted limits, so that lax $J$-weighted cones and limits are the same as ordinary $J^\dagger$-weighted cones and limits. Thus a "lax limit" is really just a particular type of *weighted* limit, whose weight happens to be of the form $J^\dagger$. Similarly, there is an oplax morphism classifier $J^\diamond$.


I think the choice I'm proposing is fairly widespread in Australia. For instance, it's the one used [here](http://arxiv.org/abs/math/0702535) and [here](http://www.maths.usyd.edu.au/u/stevel/papers/talgl.pdf) and [here](http://www.tac.mta.ca/tac/reprints/articles/4/tr4.pdf). Actually, I'm not sure offhand whether I've even ever seen the other choice in print.


