---
date: 2022-02-23 18:45
modification date: Wednesday 16th March 2022 20:08:59
title: simplicial set
aliases: [simplicial set, simplicial sets, simplicial, sSets, mapping space]

---


---

- Tags 
	- #homotopy/stable-homotopy #higher-algebra/infty-cats 
- Refs:
	- <https://nilesjohnson.net/drafts/Johnson_Yau_ring_categories.pdf#page=1057>
	- [attachments/SimplicialHomotopyLectures.pdf](attachments/SimplicialHomotopyLectures.pdf) 
	- Chapter 3 <https://mandal.ku.edu/M996QuillenKTheory/0TheKBooK.pdf>
	- <http://pi.math.cornell.edu/~web7400/7400-notes_05_10_15.pdf>
	- [attachments/Gyan_Singh_sSet.pdf](attachments/Gyan_Singh_sSet.pdf)
- Links:
	- [Model structure on sSet](Unsorted/Model%20category%20theory#Model%20structure%20on%20sSet)
	- [nerve](Unsorted/nerve.md)

---

# simplicial set

![](attachments/Pasted%20image%2020220426231231.png)

# Notes

- $\sSet = \Fun(\opcat{\Delta}, \Set) = \Presh(\Delta, \Set)$
- Functors $\Delta\op\to\Set$ where $\Delta \leq \Finset$ are totally ordered finite sets $[n] = \ts{0, 1, 2, \cdots, n}$ for all $n\geq 0$, with order-preserving set-maps.
	- I.e. presheaf on $\Delta$.
	- Morphisms are natural transformations.
- Used to define [quasicategory](Unsorted/quasicategory.md).

- The "dual" of geometric realization is totalization? #todo/questions 

![Pasted image 20211108235121.png](Pasted%20image%2020211108235121.png)



![](attachments/Pasted%20image%2020220318001815.png)

## Mapping objects

![](attachments/Pasted%20image%2020220318001851.png)
![](attachments/Pasted%20image%2020220318001938.png)
![](attachments/Pasted%20image%2020220318002011.png)
![](attachments/Pasted%20image%2020220318002624.png)

## Joins

![](attachments/Pasted%20image%2020220318002748.png)

# Op

![](attachments/Pasted%20image%2020220318003457.png)

# Constant objects

![](attachments/Pasted%20image%2020220329175243.png)

# Model structure

![](attachments/Pasted%20image%2020210516140133.png)