Why is this operator called a lift in a PDE context? Let $L$ be some differential operator and consider the PDE $Lu = 0$ with some boundary conditions. Let $W$ denote the space that solutions live in and $B$ the space that boundary conditions live in.
Suppose I've constructed a bounded operator $J:B\to W$ such that for $g\in B$, $Jg$ solves the PDE. Suppose also that I have a bounded trace operator $T:W\to B$.
My question is as follows: I've seen $J$ being referred to as a lift operator, particularly on page 8 here. But I'm struggling to reconcile this with the definition given in Wikipedia (I know nearly nothing of category theory). The only commutative diagram I can construct that makes $J$ a lift is somewhat trivial:

Here $I$ is just the identity. Is this the right way to think about $J$ as a lifting?
 A: Yes, this is the right way to think about this from the completely abstract perspective of lifts in the sense of category theory.  From the perspective of PDE and function spaces, "lift" is often the name we give to an operator that serves as a right inverse to the trace operator, i.e. a $J$ such that $T \circ J = I$.  This is exactly the commutative diagram you've written above.  In this context it's also reasonable to use the word "extension" in place of "lift" as we can think about the operator as extending the boundary data into the bulk.
When we construct such lifting operators, it is frequently the case that we begin in the canonical model problem where we want to take a function $f$ defined on $\mathbb{R}^{n-1}$ and construct a function $F$ on the upper half-space $\mathbb{R}^n_+ = \{x \in \mathbb{R}^n : x_n >0\}$ in such a way that $F$ is nice enough for us to use our trace operator and $TF = f$.  Here there is a strong natural language motivation for describing the operator as a lifting because in some sense we lift the data "up" from the set $\{x_n =0\}$ into the set above, $\{x_n >0\}$.  Personally, this is what I have in mind when I call such operators a lifting.
There is obviously no uniqueness in building such a lifting, and this is where we get nice connections to PDE and related areas.  We can try to single out a particular way of lifting such that the resulting lifted / extended function satisfies some auxiliary conditions.  In particular, we can ask that it satisfies some PDE.  I can't help but mentioning one of the most striking examples in which we want to construct a right inverse to the trace operator $T : H^1(\mathbb{R}^n_+) \to H^{1/2}(\mathbb{R}^{n-1})$.  In this case it's natural to look for a lifting $u \in H^1(\mathbb{R}^n_+)$ of $f \in  H^{1/2}(\mathbb{R}^{n-1})$ that minimizes the $H^1$ norm of $u$.  In other words, we want to find $u$ such that
$$
\int_{\mathbb{R}^n_+} |\nabla u|^2 + |u|^2 = \min\left\{ \int_{\mathbb{R}^n_+} |\nabla v|^2 + |v|^2 : v \in H^1(\mathbb{R}^{n}_+) \text{ and } Tv =f \right\}.
$$
It turns out that we can do this with the calculus of variations (and some more Sobolev theory), and the resulting $u$ then satisfies the elliptic Dirichlet-type problem
$$
\begin{cases}
-\Delta u + u =0 &\text{in } \mathbb{R}^n_+ \\
u = f & \text{on } \mathbb{R}^{n-1}.
\end{cases}
$$
The map $J$ defined by $f \mapsto Jf = u$ is then exactly as in your post when the linear operator is $L= -\Delta + I$.
