Topic: Derived Geometry and Stacks

# Motivation for Stacks
![](attachments/Pasted%20image%2020220220034547.png)

![](attachments/Pasted%20image%2020220220013200.png)
![](attachments/Pasted%20image%2020220220013228.png)
![](attachments/Pasted%20image%2020220220013305.png)
![](attachments/Pasted%20image%2020220220013345.png)
![](attachments/Pasted%20image%2020220220020647.png)
- ![](attachments/Pasted%20image%2020220220021141.png)
What we would really like is a quasi-projective variety $\Mg$ with a universal family $U_{g} \rightarrow \Mg$ such that the fiber of a point $[C] \in \Mg$ is precisely that curve. This is where the difficulty lies-automorphisms of curves obstruct the existence of such a family-and this is the main reason we want to expand our notion of a geometric space from schemes to algebraic stacks. Algebraic stacks provide a nice approach ensuring the existence of a universal family but it is by no means the only approach.

![](attachments/Pasted%20image%2020220220021521.png)
![](attachments/Pasted%20image%2020220220022548.png)
![](attachments/Pasted%20image%2020220220022804.png)
![](attachments/Pasted%20image%2020220220023234.png)
![](attachments/Pasted%20image%2020220220024255.png)



![](attachments/Pasted%20image%2020220220024421.png)
![](attachments/Pasted%20image%2020220220024458.png)


# Problems with representability

![](attachments/Pasted%20image%2020220220024721.png)

![](attachments/Pasted%20image%2020220220024707.png)

# Enlarging the category of schemes

![](attachments/Pasted%20image%2020220220033400.png)
![](attachments/Pasted%20image%2020220220033455.png)

# Misc

![](attachments/Pasted%20image%2020220220033933.png)
![](attachments/Pasted%20image%2020220220033955.png)

# Prestacks

![](attachments/Pasted%20image%2020220220034206.png)
![](attachments/Pasted%20image%2020220220034222.png)
![](attachments/Pasted%20image%2020220220034322.png)
![](attachments/Pasted%20image%2020220220034507.png)

![](attachments/Pasted%20image%2020220220034634.png)

![](attachments/Pasted%20image%2020220220034714.png)
![](attachments/Pasted%20image%2020220220034802.png)

# Misc
![](attachments/Pasted%20image%2020220220035038.png)
![](attachments/Pasted%20image%2020220220035230.png)
![](attachments/Pasted%20image%2020220220035346.png)
![](attachments/Pasted%20image%2020220220035415.png)

![](attachments/Pasted%20image%2020220220035446.png)
![](attachments/Pasted%20image%2020220220035535.png)

![](attachments/Pasted%20image%2020220220140146.png)

![](attachments/Pasted%20image%2020220220141323.png)
![](attachments/Pasted%20image%2020220220141523.png)


# End