---
date: 2022-02-23 18:45
modification date: Wednesday 16th March 2022 20:41:10
title: coend
aliases: [coend, end]

---


---

- Tags 
	- #homotopy/stable-homotopy #higher-algebra 
- Refs:
	- ncatlab intro: <https://golem.ph.utexas.edu/category/2014/01/ends.html>
- Links:
	- [Kan extension](Unsorted/Kan%20extension.md)
	- [tensored](tensored.md)

---

# coend

![](attachments/Pasted%20image%2020220427230809.png)

- Analogy: limits are right adjoints to diagonals, and ends are right adjoints to homs.


![](attachments/Pasted%20image%2020210511005841.png)

## Ends and Coends


[Intuition for coends](https://mathoverflow.net/questions/78471/intuition-for-coends)

:::{.definition title="Ends and Coends"}
Definitions:

- End of a functor $F: \cat{C}\op\times \cat{C} \to X$: an equalizer
\[
\int_{\cat C} F \da \int_{x} F(x, x) \rightarrow \prod_{x \in \cat C} F(x, x) \rightrightarrows \prod_{\cat C(x, y) } F\left(x, y\right)
.\]

- Coend: a coequalizer
\[
\int^{\cat C} F \da \int^{y} F(y, y) \leftarrow \coprod_{y \in \cat C} F(y, y) \leftleftarrows \coprod_{\cat C(x, y) } F\left(y, x \right)
.\]

:::

![](attachments/Pasted%20image%2020220316204049.png)
![](attachments/Pasted%20image%2020220316204109.png)

:::{.example title="?"}
Examples of (co)ends

- Can realize global sections:
\[
\globsec{X; \mcf} = \int_{U \in \Open(X)\op} \mcf(U)
.\]

- Can realize natural transformations as ends:
\[
\Mor_{\Fun}(F, G) = \int_c \cat{C}(F(c), G(c))
,\]
realizing them as a coherent family of morphisms.

:::

- Idea: given the singular set functor $S(\wait): \Top\to\sset$ where $S(\wait)([n]) = \Top(\Delta^n, \wait)$, construct a left adjoint $L$ (geometric realization).
This should give a bijection
\[
\Top(LX, Y) \iso \sset(X, S(Y))
,\]
where homs on the right-hand side are natural transformations.

- Do this by bending natural transformations:
\[
X([n]) \to \Top(\Delta^n, Y) \leadsto \Top(X[n] \times \Delta^n, R(X))
,\]
where for every map on the right-hand side there is a map $R(X)\to Y$ making a diagram commute.
The solution: a coend
\[
R(X) \da \int^n X([n]) \times \Delta^n
.\]

- Think of functors like modules and coends like tensor products.

- Think of ends as generalizations of limits to profunctors.

  - Need to replace cones of functors with *wedges* of profunctors.

- Alternative (co)end characterization:


![2021-10-03_19-44-13](attachments/2021-10-03_19-44-13.png)


![](attachments/Pasted%20image%2020220320035837.png)




# Grothendieck construction

![](attachments/Pasted%20image%2020220320040149.png)


# Misc

![](attachments/Pasted%20image%2020220325164046.png)