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When CS people use category theory, they're often looking for a framework which allows them to build analogies between the lambda calculus (via a cartesian clos

by ekidd 11y ago

When CS people use category theory, they're often looking for a framework which allows them to build analogies between the lambda calculus (via a cartesian closed category) and various categories for things like mathematical logic, posets or probability. And category theory is a very natural framework for thinking about these connections. So it would be useful to have category textbooks for CS folks which spent less time on topology, and more time on familiar categories. (Also, I've never quite figured out the motivation for adjoint functors. I can understand the definitions, but I don't understand why adjoints are useful. The motivating examples almost always involve unfamiliar categories.)

JadeNB 11y ago
> So it would be useful to have category textbooks for CS folks which spent less time on topology, and more time on familiar categories. Agreed, and I should have been clearer in my post above that I was just airing some thoughts on category theory rather than meaning to imply anything negative about the importance and usefulness of X-flavoured viewpoints on category theory for various values of X (including "computer science"). With that said, don't knock topology! See Baez and Stay's 'Rosetta stone' article http://arxiv.org/abs/0903.0340 http://arxiv.org/abs/0903.0340 for a plethora of connections between such seemingly abstruse concepts as topology, and computer science. > (Also, I've never quite figured out the motivation for adjoint functors. I can understand the definitions, but I don't understand why adjoints are useful. The motivating examples almost always involve unfamiliar categories.) It may be helpful to think first of Galois connexions (https://en.wikipedia.org/wiki/Galois_connection https://en.wikipedia.org/wiki/Galois_connection and http://ncatlab.org/nlab/show/Galois+connection http://ncatlab.org/nlab/show/Galois+connection). I can't find any relevant blog posts now, but I know I have seen them discussed in a CS context that may seem more natural than that of 'abstract' adjoint functors. Here is a non-free article: http://link.springer.com/chapter/10.1007%2F3-540-17162-2_130 http://link.springer.com/chapter/10.1007%2F3-540-17162-2_130 .
- agumonkey 11y ago
  IIRC Gershom Bazerman talk https://vimeo.com/72870861 https://vimeo.com/72870861 is about that precisely. (galois, adjunctions)
tel 11y ago
They just show up everywhere, adjoints. Often you'll be able to then use their uniqueness and limit preservation properties. Typically CS style category theory is a little bereft of categories to make maximal use of their appearance, but they're there. I gave a talk at LambdaConf this year mentioning that Free/Forgetful adjoints are a great way to understand free structures. There's also the fact that for all/exists arise as adjoints and you can use this to draw immediate conclusions like product preservation.

by ekidd 11y ago

JadeNB 11y ago

> So it would be useful to have category textbooks for CS folks which spent less time on topology, and more time on familiar categories. Agreed, and I should have been clearer in my post above that I was just airing some thoughts on category theory rather than meaning to imply anything negative about the importance and usefulness of X-flavoured viewpoints on category theory for various values of X (including "computer science"). With that said, don't knock topology! See Baez and Stay's 'Rosetta stone' article http://arxiv.org/abs/0903.0340 http://arxiv.org/abs/0903.0340 for a plethora of connections between such seemingly abstruse concepts as topology, and computer science. > (Also, I've never quite figured out the motivation for adjoint functors. I can understand the definitions, but I don't understand why adjoints are useful. The motivating examples almost always involve unfamiliar categories.) It may be helpful to think first of Galois connexions (https://en.wikipedia.org/wiki/Galois_connection https://en.wikipedia.org/wiki/Galois_connection and http://ncatlab.org/nlab/show/Galois+connection http://ncatlab.org/nlab/show/Galois+connection). I can't find any relevant blog posts now, but I know I have seen them discussed in a CS context that may seem more natural than that of 'abstract' adjoint functors. Here is a non-free article: http://link.springer.com/chapter/10.1007%2F3-540-17162-2_130 http://link.springer.com/chapter/10.1007%2F3-540-17162-2_130 .

agumonkey 11y ago
IIRC Gershom Bazerman talk https://vimeo.com/72870861 https://vimeo.com/72870861 is about that precisely. (galois, adjunctions)

tel 11y ago

They just show up everywhere, adjoints. Often you'll be able to then use their uniqueness and limit preservation properties. Typically CS style category theory is a little bereft of categories to make maximal use of their appearance, but they're there. I gave a talk at LambdaConf this year mentioning that Free/Forgetful adjoints are a great way to understand free structures. There's also the fact that for all/exists arise as adjoints and you can use this to draw immediate conclusions like product preservation.