4 ms·

Gives three examples of bits of mathematics with really strong concrete uses. Then says that category theory "does not itself solve hard problems in topology o

by twic 8y ago

Gives three examples of bits of mathematics with really strong concrete uses. Then says that category theory "does not itself solve hard problems in topology or algebra. It clears away tangled multitudes of individually trivial problems. It puts the hard problems in clear relief and makes their solution possible.". And doesn't give any examples. Still sounds useless to me.

ironSkillet 8y ago
I agree, the article does not actually present anything concrete. For me, the usefulness of category theory is in providing a simplifying and high level set of abstractions that can serve to illuminate the structural essence of a problem domain. At first, I thought it was just abstract nonsense (and this is coming from someone who has a PhD in mathematics), but after playing around with it a bit, I've started to appreciate the power of the concepts. The most interesting "application" of category theory that I've seen (as it relates to software engineering) is the connection between databases and category theory, e.g at https://johncarlosbaez.wordpress.com/2018/06/06/applied-category-theory-course-databases/ https://johncarlosbaez.wordpress.com/2018/06/06/applied-cate.... In short, relational schemas are categories, and common data migration operations can very naturally can be described in the language of category theory (functors, pullbacks etc).
- twic 8y ago
  > The most interesting "application" of category theory that I've seen (as it relates to software engineering) is the connection between databases and category theory So is this application productive? It's nice to be able to describe something that already exists in terms of category theory, and even nicer if those descriptions are concise and satisfying. But does that description enable anything new? Did we get any new migration operations, normal forms, etc?
  - wisnesky 8y ago
    Yes, for example the dual of join (called 'sigma'), which only exists for databases in categorical normal form. There are many academic papers and an open-source project ('categorical query language') here: http://www.categoricaldata.net http://www.categoricaldata.net.
    - twic 8y ago
      Sounds interesting! I had a look, but couldn't find out what sigma does. This is not quite the same level as, say, the PostgreSQL documentation: http://www.categoricaldata.net/help/instancesigma.html http://www.categoricaldata.net/help/instancesigma.html Is there somewhere i can go for a worked example of using sigma?
      - wisnesky 8y ago
        Sure, here is an example that uses sigma to compute an equivalence relation: http://cql.conexus.ai/quotient.php http://cql.conexus.ai/quotient.php
deleted 8y ago
[deleted]
johndcook 8y ago
Here's an example. It's not mentioned in the OP, but in a link in the OP. Microsoft's LINQ was designed using principles from category theory. They didn't use category terminology in the final product, and rightfully so.
evrydayhustling 8y ago
Yup, no applications provided. Only here would this title be clickbait, but here we are. I would suggest "Category theory may inspire results in other fields", but specific example of that is also not present.
tomjakubowski 8y ago
> Gives three examples of bits of mathematics with really strong concrete uses. The point, I think, is not that those fields of mathematics have obvious concrete applications to computers, but that they support each other in how they are applied to computer problems. Like, one reason numerical analysis is useful is that it helps you better solve differential equations, a common way to model a linear system. So linear algebra, of course, has these great applications in computers, but they're still pretty indirect because it's not like we program computers by feeding them linear maps (or whatever).
sl8r 8y ago
FWIW, I think the best "applications" (if they can be called that) come from domains like algebraic topology and algebraic geometry, where some concepts come up so frequently that it's useful to formalize them in category theory. The homology proof of Brouwer's fixed point theorem [1] is one good example. Things like Eilenberg-Maclane spaces are another (really is most natural to think of them the thing that represents some functor). I wouldn't say that you can't do this stuff without category theory, but it does make it easier / clearer. (Similarly, you can do a lot of geometry without coordinates, but coordinates definitely make some stuff easier / clearer.) [1] https://www.wikiwand.com/en/Brouwer_fixed-point_theorem#/A_proof_using_homology https://www.wikiwand.com/en/Brouwer_fixed-point_theorem#/A_p... [2] https://www.wikiwand.com/en/Eilenberg%E2%80%93MacLane_space https://www.wikiwand.com/en/Eilenberg%E2%80%93MacLane_space

by twic 8y ago

ironSkillet 8y ago

I agree, the article does not actually present anything concrete. For me, the usefulness of category theory is in providing a simplifying and high level set of abstractions that can serve to illuminate the structural essence of a problem domain. At first, I thought it was just abstract nonsense (and this is coming from someone who has a PhD in mathematics), but after playing around with it a bit, I've started to appreciate the power of the concepts. The most interesting "application" of category theory that I've seen (as it relates to software engineering) is the connection between databases and category theory, e.g at https://johncarlosbaez.wordpress.com/2018/06/06/applied-category-theory-course-databases/ https://johncarlosbaez.wordpress.com/2018/06/06/applied-cate.... In short, relational schemas are categories, and common data migration operations can very naturally can be described in the language of category theory (functors, pullbacks etc).

twic 8y ago
> The most interesting "application" of category theory that I've seen (as it relates to software engineering) is the connection between databases and category theory So is this application productive? It's nice to be able to describe something that already exists in terms of category theory, and even nicer if those descriptions are concise and satisfying. But does that description enable anything new? Did we get any new migration operations, normal forms, etc?
- wisnesky 8y ago
  Yes, for example the dual of join (called 'sigma'), which only exists for databases in categorical normal form. There are many academic papers and an open-source project ('categorical query language') here: http://www.categoricaldata.net http://www.categoricaldata.net.
  - twic 8y ago
    Sounds interesting! I had a look, but couldn't find out what sigma does. This is not quite the same level as, say, the PostgreSQL documentation: http://www.categoricaldata.net/help/instancesigma.html http://www.categoricaldata.net/help/instancesigma.html Is there somewhere i can go for a worked example of using sigma?
    - wisnesky 8y ago
      Sure, here is an example that uses sigma to compute an equivalence relation: http://cql.conexus.ai/quotient.php http://cql.conexus.ai/quotient.php

deleted 8y ago

[deleted]

johndcook 8y ago

Here's an example. It's not mentioned in the OP, but in a link in the OP. Microsoft's LINQ was designed using principles from category theory. They didn't use category terminology in the final product, and rightfully so.

evrydayhustling 8y ago

Yup, no applications provided. Only here would this title be clickbait, but here we are. I would suggest "Category theory may inspire results in other fields", but specific example of that is also not present.

tomjakubowski 8y ago

> Gives three examples of bits of mathematics with really strong concrete uses. The point, I think, is not that those fields of mathematics have obvious concrete applications to computers, but that they support each other in how they are applied to computer problems. Like, one reason numerical analysis is useful is that it helps you better solve differential equations, a common way to model a linear system. So linear algebra, of course, has these great applications in computers, but they're still pretty indirect because it's not like we program computers by feeding them linear maps (or whatever).

sl8r 8y ago

FWIW, I think the best "applications" (if they can be called that) come from domains like algebraic topology and algebraic geometry, where some concepts come up so frequently that it's useful to formalize them in category theory. The homology proof of Brouwer's fixed point theorem [1] is one good example. Things like Eilenberg-Maclane spaces are another (really is most natural to think of them the thing that represents some functor). I wouldn't say that you can't do this stuff without category theory, but it does make it easier / clearer. (Similarly, you can do a lot of geometry without coordinates, but coordinates definitely make some stuff easier / clearer.) [1] https://www.wikiwand.com/en/Brouwer_fixed-point_theorem#/A_proof_using_homology https://www.wikiwand.com/en/Brouwer_fixed-point_theorem#/A_p... [2] https://www.wikiwand.com/en/Eilenberg%E2%80%93MacLane_space https://www.wikiwand.com/en/Eilenberg%E2%80%93MacLane_space