• 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? If $M_i$ are flat modules, and $M = \colim_i M_i$ is a filtered colimit, we need to show that for any $A \to B$ injection, we have $M \otimes A \to M \otimes B $ is still injective. Why? suppose we have $\sum_i m_j a_j$ sent to $\sum_j m_j b_j$ which is zero. Here we have finitely many $m_j$ involved in the colimit, each of them represent a coherent string of showers of modules in the system. The point is that, all the $m_j$ have a common home somewhere downstream. We can use property of that common home.

Fact: all colimit preserves right-exatness. namely, if we have a surjection $M_i \to N_i$, then the direct sum of them still is a surjection. Why arbitrary colimit does not perserve left-exactness? Here is an example: take $0 \into \Z$, and $\Z \into \Z$, and $\Z \into \Z$, and consider the pushout diagram from the first to the 2nd,3rd. then the termwise pushout gives me $\Z^2 \to \Z$, but it doesn't preserved left exactness.

You see, colimit is a useful notion, without which we cannot talk about union, cokernel, many things. filtered colimit is another useful notion, with good properties, but too strict. They are different.

Let me do this exercises:

• If $P$ is compact, then $\Hom(P, -)$ commute with filtered colimit. yes, definition. And only filtred colimit. For example, consider $P = \Z/2\Z$ as $\Z$-mod, and consider the cokernel of $\Z \xto{2} \Z$, we get $\Z/2$ as the 'colimit of the pushout'. However, if we apply the $Hom(P, -)$, termwise, we get $0 \xto{2} 0$, hence the pushout is $0$. You can say, the failure is because we didn't use derived hom.
• If $P$ is projective, then, there is no higher hom. $Hom(P, -)$ is right exact.
• finite limit and filtered colimit commute in SET.

Warm-up: what is limit of a category in set? it is like taking global section of a sheaf, but much much more derived. it involves choosing for each object $i \in I$, a corresponding value $x_i \in X(i)$, such that they are compatible along maps. sometimes, it is not even possible, like, you have two arrows $a,b: i \to j$, but there is no $c_i \in X(i)$ such that $a(c_i) = b(c_i)$. bad.

Warm-up: what is the filtered colimit in Set? It is a disjoint union of the valued set, quotient out by some realtion.

Warm-up: what is an arbitrary colimit in Set? For example, how do you take the 'pushout' in SET? Well, the best thing that I