Let $f : C \rightarrow D$ denote an arrow of a 2-category. Does "cone to $f$" have a commonly accepted meaning?

Question

Let $f : C \rightarrow D$ denote an arrow in a $2$-category. Is there a commonly accepted meaning for the term "cone to $f$," generalizing the case where $f$ is a functor and $C$ and $D$ are categories? I'd like the cones to an arrow $f : C \rightarrow D$ to form a category in a natural way, so that its terminal object, if it exists, could reasonably be called the "limit" of $f$.

Answer 1

Following Riehl and Verity, it would appear that the correct generalisation of "limit" is the notion of (absolute right) lift. First, the definition:

An absolute right lift of $y : T \to Y$ along $p : X \to Y$ is a morphism $x : T \to X$ together with a 2-cell $\alpha : p \circ x \Rightarrow y$, such that for any $t : T' \to T$, $x' : T' \to X$ and $\alpha' : p \circ x \Rightarrow y \circ t$, there is a unique 2-cell $\beta : x' \Rightarrow x \circ t$ such that $\alpha' = (\alpha t) \bullet \beta$.

Unfolding this a little, we see that $(x, \alpha)$ is a terminal object in a certain category of "solutions" to the lifting problem, and that the same is true for $(x \circ t, \alpha t)$. The word "absolute" refers to the latter, and we need it for parametricity reasons.

To apply this to the theory of limits, we must assume that our 2-category is cartesian closed. Let $y : 1 \to [C, D]$ be the exponential transpose of $f : C \to D$ and let $p : D \to [C, D]$ be the morphism induced by the unique morphism $C \to 1$. In the case where the 2-category is $\mathfrak{Cat}$, the category of "solutions" to the lifting problem is precisely the category of cones to $f$. Thus, a right lift of $y : 1 \to [C, D]$ along $p : D \to [C, D]$ is precisely a limit(ing cone) for $f$; we get absoluteness for free in this case. More generally, replacing $1$ with a general category $T$, we get $T$-parametrised limits for $T$-parametrised diagrams.

One could probably make some modified definitions in the case where our 2-category is not cartesian closed (or even monoidal closed), but I do not know how well that works.