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date: 2021-10-21 18:42
modification date: Saturday 23rd October 2021 22:23:13
title: homotopy type
aliases: ["homotopy hypothesis", "spaces", "anima", "infty groupoids", "homotopy types"]

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

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- Refs:
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# homotopy type

![](attachments/Pasted%20image%2020220528010836.png)

If the the homotopy hypothesis holds, TFAE:
	- The [infinity category](Unsorted/infinity%20categories.md) of spaces (modeled by [simplicial sets](Unsorted/simplicial%20set.md)): take $\sset$, take the [Dweyer-Kan localization](Dweyer-Kan%20localization) along weak homotopy equivalences, then take the [homotopy coherent nerve](homotopy%20coherent%20nerve).
	- Take the subcategory of [Kan complexes](Unsorted/Kan%20complex.md) that is enriched over $\Kan$ and use a standard [mapping complex](mapping%20complex) construction.
	- [infinity groupoids](infinity%20groupoids.md), 
	- anima

![](attachments/Pasted%20image%2020220408154904.png)
![](attachments/Pasted%20image%2020220408154837.png)

# Anima

![](attachments/Pasted%20image%2020220420095634.png)