I watched a bunch of Tim Logvinenko's video, talking about generalized braid category and its representations. The origin of the story is trying to understand Caustic-Kamnitzer's construction, namely, what acts on the ambient space's coh category rather than the slice's.

I didn't understand CK's construction, so I probably should go back and read.

I am reading their old paper, almost 20 years old, https://arxiv.org/pdf/math/0701194

First they give a quick review of the Reshtikhin-Turaev theory, that assigns to tangle $T$ a linear map $\psi_T: V^{\otimes n} \to V^{\otimes m}$.

Then they say what a weak categorification is, which is a graded dg category, that assigns to a tangle $T$ some functor $\Psi_T: D_n \to D_m$, (why they say this only upto isomorphism?)

Bernstein-I.Frenkel-Khovanov conjectured a weak categorification, and proved by Stroppel. Here $D_n$ is some direct sum of categories associated to category $O$ for $gl_n$. Khovanov's $D_n$ is as graded module over graded algebra, combinatorial approach.

In this paper, CK uses $D_n = D(Y_n)$. They also construct, from a tangle $T$, a functor $\Psi(T)$, by composing the elementary functors: merging $F_n^i$, splitting $G_n^i$, and braiding $T$. Here merge and split between $Y_n$ and $Y_{n-2}$ are realized by a correspondence $X_n^i$

What is the space $Y_n$? First, we fix an $(N, N)$ nilpotent element $z \in End(\C^{2N})$.

Question: Why we don't care about how large $N$ is?