---
date: 2022-02-23 18:45
modification date: Friday 18th March 2022 00:23:10
title: infinity categories
aliases: [infinity categories, infinity category, infty-category, infty-categories]

---


---

- Tags 
	- #higher-algebra/derived #higher-algebra/infty-cats #resources #todo/learning 
- Refs:
	- [Unsorted/Introduction to infinity categories](Unsorted/Introduction%20to%20infinity%20categories.md) 
	- [A crash course on infty cats](https://people.math.harvard.edu/~yifei/indcoh/HigherCats.pdf) #resources/videos 
	- Short notes: <https://people.maths.ox.ac.uk/brantner/L4.pdf>, <https://www.imo.universite-paris-saclay.fr/~brantner/Spring22L2.pdf> #resources/notes 
	- [Seminar notes](attachments/highercats.pdf) #resources/notes 
	- <https://pi.math.cornell.edu/~dmehrle/notes/cornell/18fa/homotopy> #resources/notes 
	- A short course on infty categories: <https://people.math.rochester.edu/faculty/doug/otherpapers/groth_scinfinity.pdf#page=1> #projects/lecture-notes 
	- Recs from Glaeser
		- For treatments that cover quasi-categories in detail, there's  Cisinski's book (https://cisinski.app.uni-regensburg.de/CatLR.pdf), and Rezk's notes <https://faculty.math.illinois.edu/~rezk/quasicats.pdf>.
		- For "big-picture" treatments (which gloss over technical details in order to get quickly to some interesting applications) there's 
		- Mazel-Gee's notes (https://etale.site/teaching/w21/math-128-lecture-notes.pdf) which @Reuben Stern (they/them) already mentioned, as well as 
		- Groth's short course on infinity-categories (https://arxiv.org/pdf/1007.2925.pdf). 

- Links:
	- [Kan extension](Unsorted/Kan%20extension.md)
	- [simplicial set](Unsorted/simplicial%20set.md)
	- [Kan complex](Unsorted/Kan%20complex.md)
	- [cartesian fibration](cartesian%20fibration.md)
	- [stable infinity category](stable%20infinity%20category)
	- [pregeometry](pregeometry.md)
	- [infty topos](infty%20topos)
	- [presentable category](presentable%20category)
	- [localization (category theory)](Unsorted/localization%20(category%20theory).md)
	- [derived ring](derived%20ring.md)
	- [nerve](Unsorted/nerve.md)
	- [modules over a category](Unsorted/modules%20over%20a%20category.md)
	- [simplicial category](Unsorted/simplicial%20category.md)
	- [higher category](higher%20category.md)
	- [Kan complex](Kan%20complex.md)
	- [Kan extension](Kan%20extension.md)
	- [simplicial set](simplicial%20set.md)
	- [stable infinity category](stable%20infinity%20category.md)
	- [infinity groupoids](infinity%20groupoids.md)
	- [classifying space](classifying%20space.md)
	- [homotopy type](homotopy%20type.md)
	- [Kan fibration](Kan%20fibration)
	- Models:
		- [quasicategory](quasicategory.md)
		- Complete [Segal spaces](Segal%20spaces.md)
		- [Gamma space](Gamma%20space)?
	- [skeleta](skeleta.md)
	- [hypercovering](Unsorted/hypercovering.md)
	- [hyper-descent](hyper-descent.md)
	- [Waldhausen S construction](Waldhausen%20S%20construction.md) for infinity categories
	- [[tangent category]]
	- [factorization homology](Unsorted/factorization%20homology.md)

---

# infinity categories


# Misc

![](attachments/Pasted%20image%2020220425095015.png)

What is an infinity category?

![](attachments/Pasted%20image%2020220318002310.png)
![](attachments/Pasted%20image%2020220318002449.png)

How to build an infty category:
![](attachments/Pasted%20image%2020220429235334.png)

:::{.definition title="$\infty\dash$Category"}
An $\infty\dash$category $\mathcal{C}$ is a (large) [simplicial set](simplicial%20set.md)] $\mathcal{C}$ such that any diagram of the form

\begin{tikzcd}
	{\Lambda_i^n} && {\mathcal{C}} \\
	\\
	{\Delta_n}
	\arrow[from=1-1, to=3-1]
	\arrow[from=1-1, to=1-3]
	\arrow["{\exists}"', from=3-1, to=1-3, dashed]
\end{tikzcd}

![Pasted image 20210515015420](attachments/Pasted%20image%2020210515015420.png)

admits the indicated lift, where $\Lambda_i^n$ is an $i\dash$horn (a simplex missing the $i$th face) for $0 < i < n$.
:::

- All inner horns are fillable, i.e. [simplicial set](simplicial%20set.md) are *inner* Kan complexes.
- Different to Kan complexes, which include all $i$.

# Notes

- ∞-categories form a (large) ∞-category. 
- The [Segal condition](Segal%20condition), essentially characterizes $\infty\dash$categories among simplicial [infinity groupoids](infinity%20groupoids.md)

- Given two ∞-categories $\cat D, \cat C$, there is a **functor ∞-category** $\Fun(\cat D, \cat C)$. 
	- In terms of [quasicategory](quasicategory.md), the hom here is internal hom in [simplicial set](simplicial%20set.md). 
	- Example: for a given ∞-category $\cat I$ we have the ∞-category of [presheaves](presheaves) $\Fun(\cat I\op , \inftyGrpd)$

-In practice, ∞-categories are constructed from existing ones by constructions that automatically guarantee that the result is again an ∞-category, 
	- The construction typically uses universal properties in such a way that the resulting ∞-category is only defined up to equivalence
- Can take a [homotopy category](homotopy%20category.md)

- For each $n \geq 0$ there is a cat $\Delta[n] = \nerve{\ts{0 \leq  1 \leq \cdots \leq n}}$.
- Commutative triangles in $\cat C$: objects in the functor category $\Fun(\Delta[2], \cat C)$
- $\inftycat \leq \Kan$: infinity categories are a subcategory of Kan complexes.


## Adjunctions

![Pasted image 20210603191341](attachments/Pasted%20image%2020210603191341.png)

![Pasted image 20210603191352](attachments/Pasted%20image%2020210603191352.png)

![Pasted image 20210731191442](attachments/Pasted%20image%2020210731191442.png)



## Examples

![](attachments/Pasted%20image%2020220318002719.png)

# Endomorphism categories

![](attachments/Pasted%20image%2020220318003654.png)

# Algebras

![](attachments/Pasted%20image%2020220318003729.png)

# Misc

![](attachments/Pasted%20image%2020220319214242.png)
![](attachments/Pasted%20image%2020220319214304.png)
![](attachments/Pasted%20image%2020220319214319.png)

![](attachments/Pasted%20image%2020220320040545.png)