"Category Theory: An Abstraction for Anything" by Alissa Pajer (2013)

Category theory provides a mathematically sound foundation on which we can create collections of objects and express morphisms between them. Together, along with a few simple rules, a collection of objects and morphisms forms a category to which we can apply many useful results, such as the uniqueness of an identity morphism. Furthermore, once we have a category in hand, we can formally explore the relationships it has with other categories, deducing powerful and practical abstractions.
The power of category theory lies in the relative simplicity and accessibility of its definitions. From just a handful of straightforward concepts, we can formalize many concrete ideas such as directed acyclic graphs, currying, polymorphic functions, and Haskell itself. This talk will introduce the basics of category theory, while simultaneously diving into specific programming-related examples of categories, functors, and natural transformations. In addition to exploring profound and beautiful concepts, this talk aims to provide you with the tools necessary to recognize category-theoretical patterns in your own programming projects.
Alissa Pajer
Precog
@alissapajer
Alissa Pajer works as an Engineer at Precog, programming full-time in Scala. Before her coding days she studied pure mathematics, earning a Bachelors from Carleton College and a Masters from the University of Colorado. She currently lives in Boulder where she's known to run up mountains.
Recorded at Strange Loop conference (thestrangeloop.com) in St. Louis, MO, Oct 2013.

[@TheOneAnOnlyGuy]

A cool way to even extend your Option/List example diagram is the natural transformation "First" which gives you Some(a) if the list has elements, None else. It's actually an inverse in one direction but not the other, i.e., First o Option = Id, Option o First != Id_List. Oh, and this was a way to say "First" on the video, btw :P

[@lukaszkalnik5084]

This is a great introductory talk about category theory. It explains the basic concepts using correct terminology but still everything is very easily understandable.

[@micknamens8659]

13:20 In the 'optionF' definition 'foo' should be 'f'.

[@carl8703]

35:00 Another way to consider the conclusion here is in terms of algebraic data types. The set of all possible functions with signature a→b has a size equal to bᵃ (where a and b here denote the size of their sets), and the set of all possible tuples a×b has a size equal to ab (again, with a and b denoting sizes), and it is plain to see that cᵃᵇ = (cᵇ)ᵃ = (cᵃ)ᵇ, so we are left to conclude that the sets given by a×b→c, a→(b→c), and (a→b)→c must have the same number of elements. This is enough to demonstrate bijection at least, and it even suggests how function composition can be associative, though I think other work would be needed to also demonstrate isomorphism.

[@micknamens8659]

19:00 An example for a non-commuting diagram would be if your transformation depends on the contained value, e.g. transforms a Some(0) value to an empty list. But then the parameter type would be restricted to a numerical type which is against the precondition, that the transformation rule is the same for all (type) objects.

[@micknamens8659]

14:08 I doubt that 'Option(foo)' is valid Scala code to lift function 'foo', i.e. to create a function operating on objects of type 'Option[x]'. Shouldn't it be 'map(foo)' instead?