• Comonad
• todo list

$\gdef\ccal{\mathcal{C}}$ $\gdef\dcal{\mathcal{D}}$ $\gdef\lra{\leftrightarrow}$ $\gdef\xto{\xrightarrow}$ $\gdef\Om{\Omega}$

What a comonad? Given two categories $$ L: \ccal \lra \dcal: R$$ We can form the comonad $\Om = LR \in End(\dcal)$, which have $$ \epsilon: \Om \to 1_\dcal, \quad L \eta R: \Om \to \Om \circ \Om $$

What is a comodule over $\Om$? We need an object $X$ in $\dcal$, with the structure map $$ a: X \to \Om X $$ that satisfies conditions like $$ X \xto{a} \Om X \xto{\epsilon} X == id_X $$ and one more condition.... $$ X \xto{a} \Om X \rightrightarrows \Om \Om X $$ where the second step is like, either stretch out $\Om$, or pull out from $X$.

That's a lot of conditions, you see. It is not easy to be a comodule.

Consider sheaves on two points map to one point, we have functors $L = p_!, R = p^!, \Om = LR$ $$ \Om(V) = V \oplus V. $$ What's the structure maps on $\Om$?

出了趟门，得瑟了。我对$\Om(V)$说。$V$说，不是我的错，$\Om$对谁都这样。

$\Om$ automatically comes with structure maps $$ \Om(V) \to V, \quad LR V \to V, \quad id_{RV}: RV \to RV $$ now, why we have adjoint? I think it must be the summation $ V \oplus V \to V$, there is nothing better than summation. It generalizes to many points. and we have another one $$ \Om(V) \to \Om \Om V, \quad V\oplus V \to V\oplus V \oplus V\oplus V. $$ How does this work? We know that $LR(V) \to L (RL) R(V)$, so if we have $(W_1, W_2) \in \ccal$, what does $W \to RL(W)$ do? $RL(W) = (W_1 \oplus W_2,W_1 \oplus W_2)$, so it must be the.

Let's do a bit marking. $\Om(V) = V e_1 \oplus V e_2$, so we keep track of which vector space is sitting at which point. So, we have $$ R(V) \to RLR(V), \quad (V_1, V_2) \to (V_1 \oplus V_2, V_1 \oplus V_2) $$ so when they get pushed down, we have $$ V_1 \oplus V_2 \to V_{11} \oplus V_{12} \oplus V_{21} \oplus V_{22} $$ and this map is $$ V_1 \to V_{11}, \quad V_2 \to V_{22} $$ the subscript is the vector space's travel passport, where she keeps with her, and collect a stamp whenever she vist upstairs. (each time, we get $p^!$, she would split up, the universe bifurcate).

Let's try to build some $\Om$ comodule, basically, we need to decide how to map $$ V \to \Om V = V_1 \oplus V_2 $$ suppose we have some linear transformation $v \mapsto T_1 v \oplus T_2 v$. Assum $V$ is rank $1$. Then $T_i$ are scalar, and we have $T_1 + T_2 = 1$.

Then, we check the $V \to \Om V \to \Om \Om V$, if we use $\Om$ to split, then we have $$ v \mapsto (T_1 v, T_2 v) \mapsto (T_1 v, 0; 0, T_2 v) $$ if we use co-action of $V$, then we get $$ v \mapsto (T_1 v, T_2 v) \mapsto (T_1 T_1 v, T_1 T_2 v; T_2 T_1 v, T_2^2 v) $$ 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 $L (V,W) \mapsto LRL(V,W)$ $$ V \oplus W \to (V_1 \oplus W_1) \oplus (V_2 \oplus W_2), (v,w) \mapsto ( (v,0), (0, w)) $$ 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.

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 $