Differences

This shows you the differences between two versions of the page.

--- blog:2023-03-10 [2023/03/11 05:45] – pzhou
+++ blog:2023-03-10 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 57: / Line 57: @@
 $$ v \mapsto (T_1 v, T_2 v) \mapsto (T_1 T_1 v, T_1 T_2 v; T_2 T_1 v, T_2^2 v) $$
 hence, we need to have $T_i^2 = T_i, T_1 T_2=0$. The only chance here is that $(T_1, T_2)= (1,0)$ or $(T_1, T_2) = (0, 1)$.
+That's only for rank-1 comodule. How about
 You know what is the easiest way to get comodule? Just take the image of $L$. Suppose we have $(V, W)$ upstairs, and we get $V \oplus W$ downstairs. Then, we need to know how does $L (V,W) \mapsto LRL(V,W)$
-$$ V \oplus W \to (V_1 \oplus W_1) \oplus (V_2 \oplus W_2). $$
+$$ V \oplus W \to (V_1 \oplus W_1) \oplus (V_2 \oplus W_2), (v,w) \mapsto ( (v,0), (0, w)) $$
 so there is only one reasonable thing to do.
+Summary, what is the comonad structure? It is equip $\Om V$ with the trail of paths (here two paths) as it moves up and down.
+What is the comodule structure? It is dividing $v$ into $v=v_1 + v_2$. Some sort of idempotent projection.
+==== Another example ====
+Consider the skeleton of a cross, and consider the localization to the vertical line and horizontal line.
+Danny says, these two localization functors preserves limit (left exact), hence satisfies Barr-Beck.
+when you kill a linking disk (cocore to some non-compact component of a skeleton), everybody will do negative Reeb flow to snap on the skeleton again. How do I know if that motion preserves limit?
+The word 'preserve limit' is too abstract. (why do I want to preserve limit? and what kind of limit do I care? finite limit? infinite limit? homotopy limit?)
+So, don't worry about the words if you don't know what it means. Let's just see, if you can understand the comodule.
+So, we understand $\dcal$, which is just $Vect \times Vect$. And the comodules $\Omega$ is doing what? It is
+$$ \Omega(V_1 ,  V_2) = (V_1 + V_2[1], V_2 + V_1) $$
+so it is clear, what is the co-unit map. And what is $\Om (V_1 ,  V_2) \to \Om \Om (V_1 ,  V_2)$? First, what is $