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The paper argues that category theory is useful for computer science. Okay, which areas of computer science? I've studied quite a lot of algorithms, computation

The paper argues that category theory is useful for computer science. Okay, which areas of computer science? I've studied quite a lot of algorithms, computational complexity, cryptography, computer graphics, machine learning - none of it seems to get any sizeable benefit from category theory. Most CS folks who like category theory seem to be on the functional programming side of the fence, and use it to basically shovel code around, making things parameterize and plug together in just the right ways. Which is fine, but a far cry from all of CS!

mrcactu5 12y ago
Not all Category Theory is about FP. I am trying to use Category Theory to understand the programming patterns in the book Python in Practice. The book talks about abstract notions like decorators and factories and proxies. These layers of abstraction don't really change how your program runs, but they help you be more oraganized as a programmer. That's what Category Theory is about. In fact, any case in programming where you are drawing diagrams with boxes and arrows, that can be considered category theory.
tel 12y ago
Here are a few examples. Here's operational transforms from CT [0][1]. Here's probability theory (and thus much of ML) [2][3][4]. Lambda calculus arises as a certain kind of category. Operads form a model for wiring diagrams. Database schemata are categories [5]. [0] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.7744 [1] http://bosker.wordpress.com/2012/05/10/on-editing-text/ [2] ncatlab.org/nlab/show/Giry's+monad [3] arxiv.org/pdf/1410.4432v1.pdf [4] https://golem.ph.utexas.edu/category/2014/10/where_do_probability_measures.html#more [5] http://arxiv.org/abs/1302.6946
- gohrt 12y ago
  You can cast the broadest abstractions in a categorical light, but that ignores the theoretical details that are 99% of the theory of probability/wiring/databases that is of interest to most scholars and practictioners. Yes, it's interesting, but it's not at all clear that category theory pays back the opportunity cost for someone who is interested in understanding field X beyond the generalized abstraction.
  - tel 12y ago
    It's not clear, no. CT is another language for abstraction and the same argument as always hold: languages are equally expressive so it's "just a matter of taste" as to what you use. Category theory emphasizes focusing on the relations between thinfs instead of the things themselves. This can be a natural fit for some theories. For instance, anyone studying probability theory knows that random variables are already functions instead of "objects" but they are the primary thing to study. The CT-minded expression of probability theory is obviously equivalent to the Kolgomorov one, but takes that r.v. emphasis further. You can decide whether or not that's a good thing.
jonsterling 12y ago
Category theory is a useful tool for understanding the syntax and semantics of programming languages, and not just functional ones. The common practice of using category theory to provide abstraction within functional languages is interesting, but not that interesting; unfortunately, it is what most people bring up, since it is so played-up by the Haskell community. I would say the heart of computer science lies in type theory, computability, semantics, logic, etc., and category theory is evidently useful there. (I'm not talking about these things as limited to so-called "functional" languages, but rather in their capacity as forming a complete, coherent and unified theory for computation, functional or not.) Certain other subfields of CS (the focuses of the so-called "American" combinatorial school, for instance) seem to get less use out of CT, I'll grant.
- cousin_it 12y ago
  Does category theory lead to any interesting results in computability?
  - jonsterling 12y ago
    I am not aware that it does on its own, but I am not an expert in category theory, so I might be wrong. Many (most?) applications of category theory are just that—they tend to elegantly marshall disparate theories into a more general framework. For a very round-about answer, however—since Brouwer, we have understood that a computable function corresponds to a continuous function; continuity has a nice characterization in category theory in terms of sheaves; therefore category theory provides a basis for understanding computability. Or: the syntax of (some, but not all) computable functions is the internal language of an elementary topos. I could go on and on... The pattern here is that we could have done all this without category theory (and in fact we did!): but it provides a unifying and enriching perspective as one of the three cornerstones of computer science: logic, type theory and category theory. (See Bob Harper's amusing post on the topic for more: http://existentialtype.wordpress.com/2011/03/27/the-holy-trinity/ http://existentialtype.wordpress.com/2011/03/27/the-holy-tri...)
    - 12y ago

mrcactu5 12y ago

Not all Category Theory is about FP. I am trying to use Category Theory to understand the programming patterns in the book Python in Practice. The book talks about abstract notions like decorators and factories and proxies. These layers of abstraction don't really change how your program runs, but they help you be more oraganized as a programmer. That's what Category Theory is about. In fact, any case in programming where you are drawing diagrams with boxes and arrows, that can be considered category theory.

tel 12y ago

Here are a few examples. Here's operational transforms from CT [0][1]. Here's probability theory (and thus much of ML) [2][3][4]. Lambda calculus arises as a certain kind of category. Operads form a model for wiring diagrams. Database schemata are categories [5]. [0] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.46.7744 [1] http://bosker.wordpress.com/2012/05/10/on-editing-text/ [2] ncatlab.org/nlab/show/Giry's+monad [3] arxiv.org/pdf/1410.4432v1.pdf [4] https://golem.ph.utexas.edu/category/2014/10/where_do_probability_measures.html#more [5] http://arxiv.org/abs/1302.6946

gohrt 12y ago
You can cast the broadest abstractions in a categorical light, but that ignores the theoretical details that are 99% of the theory of probability/wiring/databases that is of interest to most scholars and practictioners. Yes, it's interesting, but it's not at all clear that category theory pays back the opportunity cost for someone who is interested in understanding field X beyond the generalized abstraction.
- tel 12y ago
  It's not clear, no. CT is another language for abstraction and the same argument as always hold: languages are equally expressive so it's "just a matter of taste" as to what you use. Category theory emphasizes focusing on the relations between thinfs instead of the things themselves. This can be a natural fit for some theories. For instance, anyone studying probability theory knows that random variables are already functions instead of "objects" but they are the primary thing to study. The CT-minded expression of probability theory is obviously equivalent to the Kolgomorov one, but takes that r.v. emphasis further. You can decide whether or not that's a good thing.

jonsterling 12y ago

Category theory is a useful tool for understanding the syntax and semantics of programming languages, and not just functional ones. The common practice of using category theory to provide abstraction within functional languages is interesting, but not that interesting; unfortunately, it is what most people bring up, since it is so played-up by the Haskell community. I would say the heart of computer science lies in type theory, computability, semantics, logic, etc., and category theory is evidently useful there. (I'm not talking about these things as limited to so-called "functional" languages, but rather in their capacity as forming a complete, coherent and unified theory for computation, functional or not.) Certain other subfields of CS (the focuses of the so-called "American" combinatorial school, for instance) seem to get less use out of CT, I'll grant.

cousin_it 12y ago
Does category theory lead to any interesting results in computability?
- jonsterling 12y ago
  I am not aware that it does on its own, but I am not an expert in category theory, so I might be wrong. Many (most?) applications of category theory are just that—they tend to elegantly marshall disparate theories into a more general framework. For a very round-about answer, however—since Brouwer, we have understood that a computable function corresponds to a continuous function; continuity has a nice characterization in category theory in terms of sheaves; therefore category theory provides a basis for understanding computability. Or: the syntax of (some, but not all) computable functions is the internal language of an elementary topos. I could go on and on... The pattern here is that we could have done all this without category theory (and in fact we did!): but it provides a unifying and enriching perspective as one of the three cornerstones of computer science: logic, type theory and category theory. (See Bob Harper's amusing post on the topic for more: http://existentialtype.wordpress.com/2011/03/27/the-holy-trinity/ http://existentialtype.wordpress.com/2011/03/27/the-holy-tri...)
  - 12y ago