---
created: 2023-06-08T18:58
updated: 2023-06-08T19:01
---

---
date: 2022-03-23 22:28
modification date: Wednesday 23rd March 2022 22:28:47
title: a stack is a category fibered in groupoids
aliases: [a stack is a category fibered in groupoids, fibred category, fibered category]
---


---

- Tags 
	- #todo/untagged
- Refs:
	- <https://arxiv.org/pdf/0806.4160.pdf#page=18>
- Links:
	- #todo/create-links 

---

![](2023-06-08.png)

# a stack is a category fibered in groupoids

Fix a category $\cat T$, eg $\cat T = \Top$, then consider $\cat X\in \Cat\slice {\cat T}$ a category over $T$ defined by $\cat{X} \mapsvia \pi \cat{T}$. Then $\cat X$ is **fibered in groupoids** iff

- Diagrams in $\cat{T}$ lift for every $x \mapsvia{f} y$ in the base $\cat{T}$ and every $X\in \cat{X}$ with $\pi(X) = x$, there is a $Y\in \cat{X}$ and a morphism $X \mapsvia{\tilde f} Y$ such that $\pi(\tilde f) = f$:

\begin{tikzcd}
	{\cat{X}} && X && \textcolor{rgb,255:red,92;green,92;blue,214}{\exist Y} \\
	\\
	{\cat{T}} && x && y
	\arrow["f", from=3-3, to=3-5]
	\arrow["\pi", from=1-3, to=3-3]
	\arrow["\pi", from=1-1, to=3-1]
	\arrow["\pi"', from=1-5, to=3-5]
	\arrow["{\exists \tilde f}"', color={rgb,255:red,92;green,92;blue,214}, dashed, from=1-3, to=1-5]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMiwyLCJ4Il0sWzQsMiwieSJdLFswLDIsIlxcY2F0e1R9Il0sWzAsMCwiXFxjYXR7WH0iXSxbMiwwLCJYIl0sWzQsMCwiXFxleGlzdCBZIixbMjQwLDYwLDYwLDFdXSxbMCwxLCJmIl0sWzQsMCwiXFxwaSJdLFszLDIsIlxccGkiXSxbNSwxLCJcXHBpIiwyXSxbNCw1LCJcXGV4aXN0cyBcXHRpbGRlIGYiLDIseyJjb2xvdXIiOlsyNDAsNjAsNjBdLCJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19LFsyNDAsNjAsNjAsMV1dXQ==)

- Partial triangles lift uniquely:

\begin{tikzcd}
	&&&&& Z \\
	{\cat{X}} && X && Y \\
	&&&&& z \\
	{\cat{T}} && x && y
	\arrow[from=4-3, to=4-5]
	\arrow["\pi", from=2-3, to=4-3]
	\arrow["\pi", from=2-1, to=4-1]
	\arrow["\pi"', from=2-5, to=4-5]
	\arrow[from=2-3, to=2-5]
	\arrow[from=3-6, to=4-5]
	\arrow[from=4-3, to=3-6]
	\arrow[dashed, from=1-6, to=2-5]
	\arrow[dashed, from=1-6, to=3-6]
	\arrow["{\exists !}", dashed, from=2-3, to=1-6]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMiwzLCJ4Il0sWzQsMywieSJdLFswLDMsIlxcY2F0e1R9Il0sWzAsMSwiXFxjYXR7WH0iXSxbMiwxLCJYIl0sWzQsMSwiWSJdLFs1LDIsInoiXSxbNSwwLCJaIl0sWzAsMV0sWzQsMCwiXFxwaSJdLFszLDIsIlxccGkiXSxbNSwxLCJcXHBpIiwyXSxbNCw1XSxbNiwxXSxbMCw2XSxbNyw1LCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNyw2LCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNCw3LCJcXGV4aXN0cyAhIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d)

Definition: for a fixed $x\in \cat{T}$, define the **fiber** as $\cat{X}_x\leq \cat X$ the subcategory of objects $X$ with $\pi(X) = x$. One can check that $\cat{X}_x\in\Grpd$.

If $\cat{X}$ satisfies [descent](Unsorted/descent.md), it is a stack:
![](attachments/Pasted%20image%2020220323224541.png)
![](attachments/Pasted%20image%2020220413092555.png)

![](attachments/Pasted%20image%2020220504000131.png)
![](attachments/Pasted%20image%2020220504000247.png)

# As a presheaf of groupoids

![](attachments/Pasted%20image%2020220504000547.png)
![](attachments/Pasted%20image%2020220504000611.png)

# To prestacks and stacks

![](attachments/Pasted%20image%2020220504000729.png)

See [descent data](Unsorted/descent.md):
![](attachments/Pasted%20image%2020220504000832.png)
![](attachments/Pasted%20image%2020220504000840.png)

## Descent condition

![](attachments/Pasted%20image%2020220504001128.png)
![](attachments/Pasted%20image%2020220504001138.png)


![](attachments/Pasted%20image%2020220504001200.png)
![](attachments/Pasted%20image%2020220504001217.png)
# Examples

![](attachments/Pasted%20image%2020220504000302.png)
![](attachments/Pasted%20image%2020220504000447.png)
![](attachments/Pasted%20image%2020220504000523.png)

![](attachments/Pasted%20image%2020220413092624.png)
![](attachments/Pasted%20image%2020220413092640.png)

- $\BG\to \Top$ where $\BG$ is the groupoid of principal $G\dash$bundles $P\to X$ and the functor is $(P\to X)\mapsto X$. A fiber $\BG_X$ is the groupoid of $G\dash$bundles over the space $X$.
- For a fixed space $X$, $\mcf_X \to \Top$ where $\mcf_X(T) = \Top(T, X)$ is the category of continuous maps $T\to X$ with the functor $(T\to X)\mapsto T$.  The functor $T\mapsto \mcf_X(T)$ is a sheaf of sets on $\Top$.
![](attachments/Pasted%20image%2020220504000905.png)