Terminology: lax vs. oplax colimits

Question

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, {g_i}_{i∈Ob I}, {α_f}_{f∈Mor I}), with g_i : X_i → Y, and for f : i → j in I, α_f : g_jf → g_i (such that various diagrams commute). Is this a lax colimit cone or a oplax colimit cone?

Answer 1

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 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 and here and here. Actually, I'm not sure offhand whether I've even ever seen the other choice in print.

Answer 2

Well, the smarmy answer is that it's neither, because you haven't given it a universal property. What you have is what I would call an oplax (co)cone, an 'oplax' transformation $X \Rightarrow \Delta_Y$. But there is no agreement in the literature over whether this sort of transformation (with the 2-cells in the naturality squares going 'upwards') should be called 'lax' or 'oplax'. See the discussion here, under '"Lax" versus "oplax"'.

Whichever you choose, the lax colimit of X will satisfy the 'usual' universal property, which I think is a natural equivalence $C(\operatorname{lcolim} X, A) \simeq \mathrm{Lax}(X, \Delta_A)$.