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# Category Theory

## Why is it interesting?

Category theory is a mathematical theory of mathematical theories. This means, category theory is all about the relationship between different mathematical theories.

This helps to understand the connections between different branches of mathematics and helps generalizing them. This helps especially when we try to put our current physical theories on a firmer ground and want to find better theories.

In some sense, category theory is relational mathematics, comparable to relational physics.

Physics involves the crown jewels of modern mathematics, so something deep might be going on, but, second, these insights remain piecemeal. There is a field of mathematics here, another there. One could get the idea that somehow all this wants to be put together into one coherent formal story, only that maybe the kind of maths used these days is not quite sufficient for doing so.

Category theory is an approach to understand these connections and categorise them. In this sense, category theory is a meta-theory of mathematics.

One of the most important results of category theory is the Yoneda lemma, which basically tells us that an "an object is completely determined by its relationships to other objects." This is surprisingly analogous to the basic idea of relational physics.

The importance of all this for physics is as follows. Lots of people working on quantum gravity like to stress the importance of "relationalism" - the idea that physical things only have properties by virtue of their relation with other physical things. For example, it only makes sense to say how something is moving *relative* to other things. This idea is an old one, going back at least to Leibniz, and attaining a certain prominence with Mach (who primarily applied it to position and velocity, rather than other properties).

Relationalism is appealing, at least to certain kinds of people, but it's a bit dizzying: if all properties of a thing make sense only in relation to other things, how do we get started in the job of describing anything at all? The danger of "infinite regress" has traditionally made certain other kinds of people recoil from relationalism; they urge that one posit of something "absolute" to get started.

Category theory provides a nice simple context to see relationalism in action, in a completely rigorous and precise form. In a category, objects do not have "innards" - viewed in isolation, they are all just featureless dots. It's only by virtue of their morphisms to and from other objects (and themselves) that they acquire distinct personalities. This is why an isomorphism between objects allows us to treat them as "the same": it establishes a 1-1 correspondence between their morphisms to, or from, other objects. (Moreover, this correspondence preserves the extra structure described above.)

This suggests that a truly relational theory of physics should take advantage of category theory. John Baez

Now, given a category C, we may ‘decategorify’ it by forgetting about the morphisms and pretending that isomorphic objects are equal. We are left with a set (or class) whose elements are isomorphism classes of objects of C. This process is dangerous, because it destroys useful information. It amounts to forgetting which road we took from x to y, and just remembering that we got there. Sometimes this is actually useful, but most of the time people do it unconsciously, out of mathematical naivete. We write equations, when we really should specify isomorphisms. ‘Categorification’ is the attempt to undo this mistake. Like any attempt to restore lost information, it not a completely systematic process. Its importance is that it brings to light previously hidden mathematical structures, and clarifies things that would otherwise remain mysterious. It seems strange and complicated at first, but ultimately the goal is to make things simpler.From Finite Sets to Feynman Diagrams by John C. Baez, James Dolan

For further motivation, see: Categorifying Fundamental Physics by John Baez and https://math.stackexchange.com/questions/312605/what-is-category-theory-useful-for

Important Concepts:

## Layman

Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party.

## Student

‘Categorification’ is the process of replacing equations by isomorphisms.From Finite Sets to Feynman Diagrams by John C. Baez and James Dolan

So: in contrast to a set, which consists of a static collection of "things", a category consists not only of objects or "things" but also morphisms which can viewed as "processes" transforming one thing into another. Similarly, in a 2-category, the 2-morphisms can be regarded as "processes between processes", and so on. The eventual goal of basing mathematics upon ω-categories is thus to allow us the freedom to think of any process as the sort of thing higher-level processes can go between. By the way, it should also be very interesting to consider "Z-categories" (where Z denotes the integers), having j-morphisms not only for j = 0,1,2,… but also for negative j. Then we may also think of any thing as a kind of process.http://math.ucr.edu/home/baez/week74.html

Categorification is best understood as the reverse of “decategorification”. This is a process which begins with some category, and produces a structure for which isomorphisms in the original category appear as equations between objects instead. Categorification is the reverse process, replacing equations in some mathematical setting with isomorphisms in some category in a consistent - but possibly nonunique - wayCATEGORIFIED ALGEBRA AND QUANTUM MECHANICS by JEFFREY MORTON

## Researcher

The motto in this section is: the higher the level of abstraction, the better.

Example1
Example2:

## History

This is a point of view that, more or less implicitly, has driven the life work of William Lawvere. He is famous among pure mathematicians as being the founder of categorical logic, of topos theory in formal logic, of structural foundations of mathematics. What is for some weird reason almost unknown, however, is that all this work of his has been inspired by the desire to produce a working formal foundations for physics. (See on the nLab at William Lawvere -- Motivation from foundations of physics).