Since the midth century mathematicians have tried to develop an alternative to set theory in which it would be more natural to do mathematics in terms of equivalence. In the mathematicians Samuel Eilenberg and Saunders Mac Lane introduced a new fundamental object that had equivalence baked right into it. They called it a category. Categories can be filled with anything you want.
Or you could make categories of mathematical objects: sets, geometric spaces or number systems. A category is a set with extra metadata: a description of all the ways that two objects are related to one another, which includes a description of all the ways two objects are equivalent. You can also think of categories as geometric objects in which each element in the category is represented by a point.
Imagine, for example, the surface of a globe. Every point on this surface could represent a different type of triangle. Paths between those points would express equivalence relationships between the objects. In the perspective of category theory, you forget about the explicit way in which any one object is described and focus instead on how an object is situated among all other objects of its type.
But in the second half of the 20th century, mathematicians increasingly began to do math in terms of weaker notions of equivalence such as homotopy. In these subtler notions of equivalence, the amount of information about how two objects are related increases dramatically. To see how the amount of information increases, first remember our sphere that represents many triangles. Two triangles are homotopy equivalent if you can stretch or otherwise deform one into the other.
You need to think about equivalences between all those paths, too. This path between paths takes the shape of a disk whose boundary is the two paths.
You can keep going from there. Those three-dimensional objects may themselves be connected by four-dimensional paths the path between two objects always has one more dimension than the objects themselves.
Ultimately, you will build an infinite tower of equivalences between equivalences. Several made substantial progress. Only one got all the way there. There, he began to sketch rules by which mathematicians could work with infinity categories.
This first paper was not universally well received. Lurie declined multiple requests to be interviewed for this story. What is clear is that after receiving the criticism, Lurie launched into a multiyear period of productivity that has become legendary. In Lurie released a draft of Higher Topos Theory on arxiv.
In this mammoth work, he created the machinery needed to replace set theory with a new mathematical foundation, one based on infinity categories. Algebra provides a beautiful set of formal rules for manipulating equations. Mathematicians use these rules all the time to prove new theorems. But algebra performs its gymnastics over the fixed bars of the equal sign. If you remove those bars and replace them with the wispier concept of equivalence, some operations become a lot harder.
When you move to subtler notions of equivalence, with their infinite towers of paths between paths, even a simple rule like the associative property turns into a thicket.
In Higher Algebra , the latest version of which runs to 1, pages, Lurie developed a version of the associative property for infinity categories — along with many other algebraic theorems that collectively established a foundation for the mathematics of equivalence. Can we write programs better if we resort to organizing things in terms of categories? Would it be easier to prove correctness of our programs or to discover a good algorithm to solve a task?
But these other soft questions should have positive answers. While in the previous two sections I gave concrete reasons why one might want to learn category theory, here the reason is very vague. Supposedly, learning category theory makes one a better programmer by forcing one to make connections between structures and computation.
Then when a new problem comes along, it becomes easy almost natural! Instead, I want to develop a fair fluency and categorical organization the first to sections of this article among my readers. Along the way, we will additionally implement the concepts of category theory in code.
This will give us a chance to play with the ideas as we learn, and hopefully will make all of the abstract nonsense much more concrete. Until then! Though on this blog the plan is to avoid the theory side before we investigate the algorithm from a high-level, once we get to the theory we will see an effective use of category theory in action.
Category theory has proven itself an extremely useful tool for doing mathematics, in addition to being a convenient way to think about mathematics. Like Like.
What I had in mind was more of the basic results: we want to prove that some object has said universal property? One can work with abstract nonsense and chase diagrams, or explicitly construct the object in the category of sets and prove does what it should say, for dealing with abliean categories where elements make sense. The former is obviously more convenient and perhaps more elegant, but it may not be necessary.
But perhaps I just am foolish enough to believe that there are always more ways to prove a theorem. Have I been lulled into a false sense of security by the fundamental theorem of algebra?
But then again I have no idea how one would even go about proving such a claim. I mean this exactly the sort of thing that the incompleteness theorem guarantees. Statements which are about number theory or any area of mathematics which are not provable within number theory.
I think I may have been referring too much to set theory, and of course set theory suffers from incompleteness. Yes, category theory still suffers from incompleteness any sufficiently strong formal logic will. Just thought it might be interesting. Any category is a model of these axioms, so rather trivially, category theory is incomplete. On the other hand, asking whether certain categorical foundations E. The thing is, this weakness buys us expressiveness— ZFC is cumbersome to work with structurally, precisely because it is so concrete, so rigid.
In set theory, equality ends up too strict. You cannot distinguish the circle group and the group of unit complex numbers through group homomorphisms. Doing so makes unnecessary distinctions. That is, they contain the same information given the structure they are endowed with. They define them merely up to isomorphism. And some foundations seek to do away with the notion of strict equality entirely. This question of equality also motivates some of the research in the subject.
In basic category theory, objects can be isomorphic, but morphisms have no such weak notion of equality — they are always strictly equal. The vast applicability and expressiveness of category theory leads to the observation that most structures in mathematics are best understood from a category theoretic or higher category theoretic viewpoint. Category theory is one of, if not the most, abstract fields of mathematics. This extreme generality of category theory means that it can say something about anything, but nothing too specific.
In other words, part of the growth of category is probably because you can use it to talk about damn near anything. See the applications below for examples. In this respect, category theory is like set theory. This generality mirrors the difference between strong and weak methods in artificial intelligence. Instead, specialized tools are necessary. In the same way, category theory is more of a tool for elucidating connections between mathematical structures than for solving problems — in contrast with something like linear algebra, or really any field of applied math.
The whole enterprise is bizarre. No, the Egyptians did arithmetic and some algebra. The oldest extant mathematical records deal with the Pythagorean theorem.
Animals have some notion of magnitude and many can even count. Set theory, rather than a natural extension of mathematical enterprise, seems more like something forced — the difference between English and Esperanto. As far as I can tell, the mathematical community agrees with me. But, beyond that, is their anything else exciting about category theory?
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