---
date: 2022-01-15 21:49
modification date: Wednesday 16th February 2022 21:39:26
title: limit
aliases: [colimit, colim, filtered category]

---

Last modified date: <%+ tp.file.last_modified_date() %>

---

- Tags: 
	- #higher-algebra/category-theory 
- Refs:
	- #todo/add-references
- Links:
	- #todo/create-links 

---

# Limits (Categorical)

Modern POV: realize in terms of a [Kan extension](Unsorted/Kan%20extension.md):
![](attachments/Pasted%20image%2020220320032421.png)

![](attachments/Pasted%20image%2020220531025324.png)

## Mnemonics to distinguish lims and colims

| Inverse/projective limit                                                          | Direct/colimit                                                                   |
| --------------------------------------------------------------------------------- | -------------------------------------------------------------------------------- |
| $\lim$                                                                            | $\colim$                                                                         |
| Above: ![](attachments/Pasted%20image%2020220603121253.png)                       | Below: ![](attachments/Pasted%20image%2020220603121316.png)                      |
| terminal cones                                                                    | initial cocones                                                                  |
| subobjects of $\displaystyle\prod$                                                | quotient objects of $\Disjoint$ or $\bigoplus$                                   |
| terminal objects                                                                  | initial objects                                                                  |
| pullbacks                                                                         | pushouts                                                                         |
| products                                                                          | coproducts                                                                       |
| kernels                                                                           | cokernels                                                                        |
| equalizers                                                                        | coequalizers                                                                     |
| commutes with contravariant hom: $\cat{C}(A, \lim B_i) = \lim \cat{C}(A, B_i)$    | commutes with covariant hom: $\cat{C}(\lim A_i, B) \cong \colim \cat{C}(A_i, B)$ |
| continuous                                                                        | cocontinuous                                                                     |
| "impose equations"                                                                | "glue objects"                                                                   |
| $\ZZpadic = \lim(\cdots \mapsvia{\mod p^2} \ZZ/p^2\ZZ \mapsvia{\mod p} \ZZ/p\ZZ)$ | $\ZZ(p^\infty) = \colim(\ZZ/p\ZZ \mapsvia{\times p} \ZZ/p^2\ZZ \mapsvia{\times p} \cdots)$                     |

Idea: the limit should be the *closest* object to all of the $F(X_i)$.

# The Calculus of Limits

- Commuting lims:
	- Self-commuting: lims commute with lims, colims commute with colims.
	- There is a morphism $\colim \lim A_{i, j} \to \lim \colim A_{i, j}$ which need not be an isomorphism, i.e. limits need not commute with colimits in general.
	- **Filtered** colimits commute with **finite** limits in $\Set$.
- Preservation of properties:
	- Limits respect finite group actions, i.e. $(\lim A_i)/G \iso \lim (A_i/G)$.
	- Finite colims of compact objects are compact
- The acronyms for adjoints:
	- RAPL: right adjoints preserve limits
	- LAPC: left adjoints preserve colimits
	- LARE: Left adjoints are right-exact 
	- RALE: right adjoints are left-exact
    
# Definitions

![](attachments/Pasted%20image%2020220316203247.png)

## Limits

![](attachments/Pasted%20image%2020220124120148.png)
![](attachments/Pasted%20image%2020220124120203.png)
![](attachments/Pasted%20image%2020220124120224.png)
![](attachments/Pasted%20image%2020220203235717.png)


## Colimits


For $F: \cat I \to \cat C$, define $\chi_X: \cat I\to \cat C$ for $X\in \cat C$ regarded as a set (groupoid with only identity morphisms) to be the **constant functor** $Y\mapsto X$ for all $Y$ and $f\mapsto \id_X$ for all $f$.
Then 
$$
\lim_I F(\wait) \da {\Cat}(\chi_{(\wait)}, F): \cat C \to \Set
$$
i.e. the limit is the functor sending $X$ to natural transformations between $\chi_X$ and $F$.
If this functor is representable, it can be identified with an object in $\cat C$.


Definition in terms of cones and cocones:

![Pasted image 20211003193342](attachments/Pasted%20image%2020211003193342.png)
![Pasted image 20211003193427](attachments/Pasted%20image%2020211003193427.png)

# Examples

- $\lim(\bullet \rrarrows \bullet)$ is an equalizer
- For $I$ an index set regarded as a groupoid with only identities, $\lim(\bullet, \bullet, \cdots) = \prod_{i\in I} \bullet_i$ . Use that $\lim_I F(X) = \ts{X\to X_i}$ if $F(i) \da X_i$.
- $\lim(\bullet \to \bullet \from \bullet) = \bullet\fiberprod{\bullet}\bullet$ is a fiber product.

![](attachments/Pasted%20image%2020220316204004.png)



# Filtered categories and colimits

![](attachments/Pasted%20image%2020220316203344.png)


![](attachments/Pasted%20image%2020220317232012.png)
![](attachments/Pasted%20image%2020220317232039.png)
![](attachments/Pasted%20image%2020220317232055.png)
![](attachments/Pasted%20image%2020220318002919.png)