<p>Two big applications of homotopy theory in theoretical computer science are</p>

<ol>
<li><p><a href="https://homotopytypetheory.org/" rel="nofollow">Homotopy Type Theory</a> revealed a completely unexpected connection between the theory of the typed lambda calculus and homotopy theory. As a quick intuition, think of it as either a (vast) generalization of the connection between intuitionistic logic and topological spaces, or a language for doing "synthetic homotopy theory".</p></li>
<li><p>The <a href="https://ncatlab.org/nlab/show/directed+homotopy+theory" rel="nofollow">directed version of algebraic topology and homotopy theory</a> (ie, where paths are not reversible) has been developed precisely with applications to computer science in mind. The intuition is that the possible evaluations of a concurrent program correspond to a space, program executions correspond to paths in that space, and synchronization primitives correspond to obstructions. By considering the geometric properties of these spaces/programs, it becomes possible to develop tools for reasoning about their behaviour. </p></li>
</ol>

CC BY-SA 3.0
2016-07-20T15:36:29.570Z
2016-07-20T15:36:29.570Z
Neel Krishnaswami
<p><em>My answer to a related <a href="https://cstheory.stackexchange.com/a/31577/12739">post: Applications for set theory, ordinal theory, infinite combinatorics and general topology in computer science?</a>:</em></p>
<p><a href="http://www.math.utu.fi/icalp04/godel2004.html" rel="nofollow noreferrer">The 2004 Gödel Prize</a> was shared by the following two papers:</p>
<ul>
<li><em>The Topological Structure of Asynchronous Computation</em>.<br />
By Maurice Herlihy and Nir Shavit, Journal of the ACM, Vol. 46 (1999), 858-923</li>
<li><em>Wait-Free k-Set Agreement Is Impossible: The Topology of Public Knowledge</em>.<br />
By Michael Saks and Fotios Zaharoglou, SIAM J. on Computing, Vol. 29 (2000), 1449-1483.</li>
</ul>
<p>Quotes from <a href="http://www.math.utu.fi/icalp04/godel2004.html" rel="nofollow noreferrer">the 2004 Gödel Prize:</a></p>
<blockquote>
<p>The two papers offer one of the most important breakthroughs in the theory of distributed computing.</p>
<p>The discovery of the <strong>topological nature of distributed computing</strong> provides a new perspective on the area and represents one of the most striking examples, possibly in all of applied mathematics, of the use of topological structures to quantify natural computational phenomena.</p>
</blockquote>
<hr />
<p><em>Added:</em></p>
<p>A book on this topic:</p>
<p><a href="http://store.elsevier.com/Distributed-Computing-Through-Combinatorial-Topology/Maurice-Herlihy/isbn-9780124045781/" rel="nofollow noreferrer">Distributed Computing Through Combinatorial Topology, 1st Edition, 2013</a></p>

CC BY-SA 3.0
2016-07-22T16:14:20.170Z
While I am a big fan of these results, it's not clear to me if they count as *homotopical* algebra so much as just *homological* algebra...
Joshua Grochow
CC BY-SA 3.0
2016-07-23T01:46:13.820Z
@JoshuaGrochow Honestly, I know little about these results. I am interested in distributed computing and know of these results. However, I have little mathematical background to understand them well. Please feel free to modify/delete my answer. Thanks.
hengxin
CC BY-SA 3.0
2016-07-22T01:50:36.463Z
2016-07-22T01:50:36.463Z
2020-06-17T09:38:08.363Z
hengxin
<p>I am an homotopy theorist, interested in computer science.</p>

<p>I would like to ask what are some interesting applications of homotopical algebra (model categories, infinity categories, simplicial categories, etc) in theoretical computer science?</p>

CC BY-SA 3.0
2016-07-22T01:45:15.297Z
Related: [here](http://cstheory.stackexchange.com/q/2898/12739) and [here](http://cstheory.stackexchange.com/q/31325/12739).
hengxin
<big-list><algebraic-topology>
What are some interesting applications of homotopical algebra in theoretical computer science?
user40853
# What are some interesting applications of homotopical algebra in theoretical computer science?

(Asked by: username_0 on 2016-07-20)

I am an homotopy theorist, interested in computer science.

I would like to ask what are some interesting applications of homotopical algebra (model categories, infinity categories, simplicial categories, etc) in theoretical computer science?

**Tags:** big-list, algebraic-topology
> **Comments:**
> **username_1**: Related: [here](http://cstheory.stackexchange.com/q/2898/12739) and [here](http://cstheory.stackexchange.com/q/31325/12739).  

---

## 2 answers

### Answer by username_2 on 2016-07-20. Score: 3

Two big applications of homotopy theory in theoretical computer science are

1. [Homotopy Type Theory](https://homotopytypetheory.org/) revealed a completely unexpected connection between the theory of the typed lambda calculus and homotopy theory. As a quick intuition, think of it as either a (vast) generalization of the connection between intuitionistic logic and topological spaces, or a language for doing "synthetic homotopy theory".
2. The [directed version of algebraic topology and homotopy theory](https://ncatlab.org/nlab/show/directed+homotopy+theory) (ie, where paths are not reversible) has been developed precisely with applications to computer science in mind. The intuition is that the possible evaluations of a concurrent program correspond to a space, program executions correspond to paths in that space, and synchronization primitives correspond to obstructions. By considering the geometric properties of these spaces/programs, it becomes possible to develop tools for reasoning about their behaviour.


---

### Answer by username_1 on 2016-07-22. Score: 2

*My answer to a related [post: Applications for set theory, ordinal theory, infinite combinatorics and general topology in computer science?](https://cstheory.stackexchange.com/a/31577/12739):*

[The 2004 Gödel Prize](http://www.math.utu.fi/icalp04/godel2004.html) was shared by the following two papers:

* *The Topological Structure of Asynchronous Computation*.  
  By Maurice Herlihy and Nir Shavit, Journal of the ACM, Vol. 46 (1999), 858-923
* *Wait-Free k-Set Agreement Is Impossible: The Topology of Public Knowledge*.  
  By Michael Saks and Fotios Zaharoglou, SIAM J. on Computing, Vol. 29 (2000), 1449-1483.

Quotes from [the 2004 Gödel Prize:](http://www.math.utu.fi/icalp04/godel2004.html)

> The two papers offer one of the most important breakthroughs in the theory of distributed computing.
>
> The discovery of the **topological nature of distributed computing** provides a new perspective on the area and represents one of the most striking examples, possibly in all of applied mathematics, of the use of topological structures to quantify natural computational phenomena.

---

*Added:*

A book on this topic:

[Distributed Computing Through Combinatorial Topology, 1st Edition, 2013](http://store.elsevier.com/Distributed-Computing-Through-Combinatorial-Topology/Maurice-Herlihy/isbn-9780124045781/)

> **Comments:**
> **username_3**: While I am a big fan of these results, it's not clear to me if they count as *homotopical* algebra so much as just *homological* algebra...  
> **username_1**: @JoshuaGrochow Honestly, I know little about these results. I am interested in distributed computing and know of these results. However, I have little mathematical background to understand them well. Please feel free to modify/delete my answer. Thanks.  

---

