i want to translate, wedrich-dyckerhoff, into our own language.

First, what is Beck-Chevalley stuff? I am reading page 10 of DW. There are two ways to go from $(2,1)$ to $(1,2)$ partition on the line, one is break-then-merge, the other is merge-then-break. The two ways are different, obviously. And even better, there is a relation between the two ways, how? We have adjunctions, in our setting, splitting is the right adjoint (splitting is restriction) of merge. (we can have various versions of adjoints, don't worry, or limit oneself).

We have these two adjunctions

• [ (2) -> (1,1) -> (2) ] ==> [(2) -id-> (2) ]
• [(1, 1) -id-> (1,1)] ==> [ (1, 1) -> (2) -> (1,1) ]

What's the condition of an $A_1$-schober? cofiber of the 2nd guy is an auto-equivalence. Fiber of the first guy is also an equivalence.

If we want to match with the diagram DW consider, we need to identify the left-adjoint with merge, and right-adjoint with split

We are going to use these adjunctions to get the sweeping move

• (2,1) -> (1,1,1) -> (1,2)
• (2,1) -> (1,1,1) -> (1,2) -> (3) -> (1,2)
• (2,1) -> (1,1,1) -> (2,1) -> (3) -> (1,2)
• (2,1) -> (3) -> (1,2)

what are we trying to say now? we can say

• (a+b,c) -> (a,b,c) -> (a,b+c)
• (a+b,c) -> (a,b,c) -> (a,b+c) -> (a+b+c) -> (a, b+c)
• (a+b,c) -> (a,b,c) -> (a+b,c) -> (a+b+c) -> (a, b+c)
• (a+b,c) -> (a+b+c) -> (a,b+c)

There is a natural transformation from the top row to the bottom row, but

Given this commutative square (1,1,1) -> (2,1) or (1,2) -> (3), it is neither a pullback square or a pushforward square.

let me read the construction 3.5. OK, you have something X, living over the $k$-cube. You don't want a section, no. you want a 'fibered map'

you first map a thickened cube, to the cube, where the initial time slice maps to the origin, and the final slice map to identity. Then you want to map the thickend cube to X, so that fiber goes to fiber. what's going on here? You want these arrows, along the time direction, to be 'cocartesian', meaning having the initial-factor-through property, as GPT teaches me.

What is the so called inflation-deflation, on pg 13? it is push-forward from the origin fiber, or pullback to the origin fiber

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.