• 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