• learn what is Barr-Beck (don't worry about Lurie, I am not $\infty$ yet).

$\gdef\colim{\text{colim}}$

Let me follow Branter's note . And, Akhil Matthew's Serre Criterion for affiness.

Category warm-up.

• What is a left-adjoint functor? $\otimes$, $i^*$ (restriction to open set)
• What is a colimit? (taking cokernel of $A \to B$, taking coequalizer)
• Who preserves colimit? $Hom( colim_i A_i, B) = \lim_i Hom(A_i, B)$ by definition. Thus, if $L$ is a left-adjoint, we have $$Hom(L (colim A_i), B) = Hom( (colim A_i), R B) = lim Hom(A_i, RB) = lim Hom(LA_i, B) = Hom(colim L A_i, B)$$
• That was too abstract. What is an example? No, remember it, like 'colimit commute with left-adjoint', say it 100 times.
• If $P$ is a compact projective, then $Hom(P, -)$ commute with all colimit? Why?
  • $P$ is compact, means that $Hom(P, -)$ preserves filtered colimit (can there be an example, where $P$ does not

What is a filtered colimit? It is a colimit over a filtered category, where every two objects can map to a third common object, and every two morphism (within the same hom space) can be composed with a third, to coequalize. As written, it is obvious that, pushout square, direct sum of stuff (even finitely many), cokernel (a special kind of pushout) are not filtered colimit.

see this note

what is a directed set? it is a set with partial ordering, such that every two elements has a common downstream guy. So, it is less general than a filtered set, where morphism between two objects can be not just $<$ relation.

$\Q$ is a direct limit (union) of copies of $\Z$. Indeed, it is like $\Q = \cup_{n > 0} \frac{1}{n} \Z$. As a $\Z$-module. very nice.

What is a localization of a ring? where the ring is viewed as a module over itself, and we allow for bigger and bigger denominators? maybe.

what does flat module mean? tensor is good?

I have this conflicting intuition: for a filtered diagram, every two nodes eventually will meet and will stay together ever after, but there will be more and more nodes as you 'go to the right, go to infinity'. how can that happen? well, it is just like life, each individual will die, but the human society lives on. fine.

why filtered colimit preserves flatness?