---
date: 2022-01-15 21:49
modification date: Monday 7th February 2022 22:21:47
title: bar construction
aliases: ["cobar construction", "cyclic bar construction", "bar construction", "monadic bar construction"]

---

Tags: 
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Refs:
[simplicial sets](Unsorted/simplicial%20set.md)

# bar construction

> The cosimplicial cobar construction keeps track, in a very explicit sense, of all the ways that a given object $F(i)$ of the diagram is mapped into. It remembers every finite sequence of arrows whose terminus is $F(i)$. The cobar construction then looks at ways of coherently mapping a "fat point" into this enormous arrangement.

![](attachments/Pasted%20image%2020220207222001.png)

![](attachments/Pasted%20image%2020220326221417.png)
![](attachments/Pasted%20image%2020220326221356.png)
![](attachments/Pasted%20image%2020220326221342.png)

# Cyclic Bar Construction
See [geometric realization](Unsorted/geometric%20realization.md), [trace](Unsorted/trace%20(monoidal%20categories).md), [factorization homology](Unsorted/factorization%20homology.md):
![Pasted image 20210630211527](attachments/Pasted%20image%2020210630211527.png)

# Simplicial bar construction

![](attachments/Pasted%20image%2020220316194934.png)
![](attachments/Pasted%20image%2020220316195018.png)

See [classifying space](Unsorted/classifying%20space.md)

# Reduced bar construction

![](attachments/Pasted%20image%2020220317200754.png)

# Comonads

![](attachments/2023-03-05cmaand.png)
![](attachments/2023-03-05cmd1.png)