Reading Category Theory Illustrated on stream:

• Previous reading is 20251002175833
• This chapter is going to try to show us the category theory counterparts of constructs from set theory.
• Products
  • A Cartesian product of sets is a new set of every ordered combination you can make with one element from each set.
  • A Cartesian product is also known as a tuple
  • Name probably comes from the fact that the size of the resulting set is the same as the sizes of the initial sets multiplied together.
  • A product's projections are functions which take the combination and extracts the value of one of the properties. For example, for a Cartesian product C of sets A and B, the projection C -> A and C -> B give you back the A and B sets respectively. In Haskell, these functions are fst and snd.
    • The Cartesian product is associative (it doesn't matter what order you do it in)
  • The examples under Defining products in terms of sets seem to be backwards. Either the Felix Hausdorff diagram is backwards, or the other two are backwards.
  • There's a section about defining products in terms of functions that I don't really understand. I think the author is just trying to say that if we assume there is some Cartesian product of two sets, that there might be more sets that add additional information to that Cartesian product which are "imposter" sets. This extra information can be discarded by converting the imposter set into the Cartesian product. However, it's not clear why this matters enough to point out, because we have no ability to discern which sets are imposters or not. It also doesn't distinguish between sets that are isomorphic. Unsure what I should learn from this section.
  • universal property is namedropped, but it is not explained.
  • I'm not sure what the category theory counterpart of cartesian products is, but this is where the products section ends.
• Sums
  • Sums are the combination of the elements from multiple sets.
  • They make the example of a child having two parents. This begs the question -- is it currently possible to have more than two parents or less than two parents?
    • It appears that having more than two parents naturally through polyspermy is not possible because the child dies. See livescience.com
    • Having more than two parents is possible with science and is done to treat mitochondrial disease or infertility. See: technologyreview.com
    • Having one parent naturally may be possible, but it is not known to science. The only known such examples in science of one parent (either from parthenogenic or androgenetic events) result in tumours. See: sciencedirect.com
    • Having one parent (aka cloning) seems to not be possible yet for humans, though primates were successfully cloned in 2018 using a process called somatic cell nuclear transfer (SCNT). See: science.org, cell.com
  • A sum of sets can have duplicate elements, but in order to do this we need to add extra structure.
  • There are also imposter sums, which are like imposter products in that they contain extra information.
  • The sum gives an or relationship between the elements. For example, if "my hair is brown" is true, then "my hair is brown or blonde" is also true.
  • While in products, you can write a function C -> A and C -> B for a Cartesian product C of A x B, you cannot do the same for a sum of A and B. This is because the sum C of A + B contains elements from A which cannot be mapped back to B and elements from B which cannot be mapped back to A.
  • Similarly, a Cartesian product cannot have its arrows flipped, because if you did, then an element in A would be mapped to multiple elements in C.
  • A product and a sum are the opposite of each other.
  • You can always create a new imposter sum with a trivial function that goes from A + B -> A + B + X and you can always create a new imposter product that goes from A x B x X -> A x B. You can see by looking at the formulas how the extra information X is on different sides of the function.
  • In category theory, anything that has this kind of relationship is said to be dual to one another. The concepts of product and sum are dual and it's why sums are known as converse product or coproduct.
  • They don't say this here, but I would assume that products can also be known as cosums.
• Defining the rest of set theory using functions
  • All of set theory can be defined entirely by functions, as was discovered by Francis William Lawvere.
  • Set Theory is usually defined in terms of sets and elements. To define it in terms of sets and functions, you can say that each element in a set has a function from a singleton set to that element.
  • In order to define a singleton set without the definition of an element, you can say that there is exactly one unique function for every set that goes to the singleton set. There could be more than one set that has this property, but if that is the case all of these sets are isomorphic to one another and are thus just the singleton set.
  • An empty set is defined as a set that has a function from it to any other set.
  • The definitions of a singleton set and an empty set are duals of each other.
  • The singleton set and empty set are also known as the terminal object and the initial object.
  • To select an element in a set, we can define a function from the singleton set to the element in a set that we want to select. We can then combine that with a function, if we want. This is how function application is done.
  • This definition of a set using only functions is known as Elementary Theory of the Category of Sets (ETCS)
  • There's another rigorous definition for set theory known as ZFC.
• Categories
  • There's a quote from a Dijkstra essay that I would like to read more of later: https://www.cs.utexas.edu/~EWD/transcriptions/EWD10xx/EWD1036.html It's quite long and I feel a bit sick.
  • The author claims that categories are often taught by introducing set theory first, but that you also don't need to know set theory to learn about category theory.
  • A category is a collection of objects that can be anything you want as well as morphisms which connect these objects to one another.
  • In category theory, the morphisms are more important than the objects.
  • Multiple morphisms from one object to another are not necessarily equivalent.
  • For a structure to be called a category, two morphisms can be composed into a third.
  • Law of Identity: A structure must have an identity morphism for each object in order for it to be called a category.
  • Law of Associativity: Morphisms that are composed have to be associative
  • This book contains commutative diagrams, in which any two paths between two objects are equivalent to each other.
  • In summary: A category is a collection of objects and morphisms that go from one object to another, where each object has to have the identity morphism and there is a way to compose two morphisms into a third one in an associative way.
• Chat is recommending that I read:
  • The Joy of Abstraction by Eugenia Cheng
  • Category Theory in Context by Emily Riehl
• As a side note, Category Theory Illustrated is by Jencel Panic.
• Submitted a PR with corrections: https://github.com/abuseofnotation/category-theory-illustrated/pull/81