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--- blog:2025-07-02 [2025/07/03 06:58] – created pzhou
+++ blog:2025-07-02 [2025/07/03 09:00] (current) – pzhou
@@ Line 35: / Line 35: @@
 What is the so called inflation-deflation, on pg 13? it is push-forward from the origin fiber, or pullback to the origin fiber
-{{:blog:pasted:20250703-063453.png}}
+{{:blog:pasted:20250703-063453.png?600}}
 let me try to understand (3.1.4). We start life with a cube worth of categories, and functors lining the edges. We assume there is a biCartesian fibration, which means there exists (up to contractible choices) a unique lift of an edge if we fix the starting point, or the ending point.
+We look at a square in the cube, and project the cube to the square. Say a 3-dim cube, so we are left with 1-dim interval in the fiber. We start with the fiber in the upper right corner. we start with the initial node in that fiber, run co-cartesian extension to the full fiber, then for each fiber position, we have a natural transformation. what's wrong with that? why we need to do deflation? ok, whatever. we still get [1]^{n-1}, in which one direction is in the diagonal base direction, and (n-2) is the fiber direction. And this cube is in the category of functors from this tip to that tip.
+the total fiber of (3.1.5) as the BC defect. Question, if we permute the ordering of the index, is it still the same thing? why the first two indices?
+If we want to use 'right-adjoint' for the merging, then we are not using $NH_2 \otimes_{1,1} M$ to do extension, rather we are doing $Hom_{1,1}(NH_2, M)$
+OK, maybe I want to say, there are two ways of doing restriction, yes. there is a 'left-adjoint restriction', which is, you Reeb flow to the stop, cut, then un-flow a bit. ok, i will do that.