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Hello All,

First off, while I have the platform I would like to sincerely thank John Baez, David Spivak and Brendan Fong for their efforts in bringing Category Theory to the non-working mathematician. Without them, I certainly wouldn't be writing a post introducing myself to this class. So, who am I and why am I here?

I am currently completing my Masters in Interdisciplinary Sciences (basically double major in Biology & Chemistry) at ETH in Zurich, Switzerland. I have specialized myself in Bioorganic Chemistry (particularly mechanistic enzymology) as well as Immunology. While I was born in the States, I grew up in Switzerland for the most part. Aside from trying to teach myself Category Theory, I also read, write, run and play the guitar among other things. That concludes the easy part. On to why I'm here...

The first time I came across Category Theory was around a year ago, while I was looking into mathematical structures that might serve as frameworks for thinking about cognition. Up until that point, the most structured mathematical object I had come across in my studies had been graphs (presented somewhat half-heartedly in Systems Biology). Obviously, I was also required to take basic courses in analysis, partial differential equations and linear algebra in my Bachelors, but these were taught in a very pragmatic (and thus boring) fashion. In any case, the little Graph Theory I'd seen led me to learn more, which was my first encounter with pure maths. A month later I discovered Categories and became convinced (somehow?) that knowing a bunch about them would be a good idea. Fortunately, this ended up proving true.

However, teaching oneself one of the most abstract branches of mathematics with close to no prerequisites proved difficult. Luckily, David Spivak's "Category Theory for the Sciences" attempts to build from the ground up. Nonetheless, I remember struggling with basic notions such as \(f: \mathbb R \times \mathbb R \rightarrow \mathbb R\) at the start. Eventually, I managed to work through most of the book, although I think I haven't managed to extract even half of what's in there. I've since worked through Lawvere and Schanuel's "Conceptual Mathematics", which helped me think about maps in general and am currently going through Steve Awodey's textbook in an attempt to cement the basics in a more rigorous fashion. There's still a ton to learn, but I now have some solid foundations.

I'm being this detailed because I hope to convince people who may have trouble initially in understanding some of the concepts to keep at it. Personally, thinking about this stuff has completely changed the way I look at the world. There's much to be said about where I think this new view might be useful - but I will leave that for another day. Regardless, it would be a waste if these cool ideas would remain reserved for mathematicians and a few computer scientists. To that end I wish everyone participating in the course the best of luck!

Sincerely, Marius

## Comments

Hello, Marius! That's an impressive story! It's too bad your classes on analysis and linear algebra didn't prepare you more by explaining concepts like \(f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}\) - functions between sets, and cartesian products of sets. I guess there are many ways to teach these subjects. But anyway, all these concepts can be learned on one's own if one works hard enough: these days, everything is on the internet! And it sounds like you're doing it.

I hope you ask questions about the lectures, and I hope you try the exercises and post comments about those too! A lot of people have registered for this course but most of them aren't commenting. Maybe they're too shy. I'd be very sad if only the most advanced students were commenting, leaving others intimidated. I want everyone to join in the fun.

Sometime I may ask you some questions about chemistry or immunology. I'm trying to learn a bit of biochemistry... only a microscopic amount, just enough to help me with my work.

If I learn a microscopic amount of biology, will I be a microbiologist?

`Hello, Marius! That's an impressive story! It's too bad your classes on analysis and linear algebra didn't prepare you more by explaining concepts like \\(f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}\\) - functions between sets, and cartesian products of sets. I guess there are many ways to teach these subjects. But anyway, all these concepts can be learned on one's own if one works hard enough: these days, everything is on the internet! And it sounds like you're doing it. I hope you ask questions about the [lectures](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Course), and I hope you try the [exercises](http://www.azimuthproject.org/azimuth/show/Applied+Category+Theory#Exercises) and post comments about those too! A lot of people have registered for this course but most of them aren't commenting. Maybe they're too shy. I'd be very sad if only the most advanced students were commenting, leaving others intimidated. I want everyone to join in the fun. Sometime I may ask you some questions about chemistry or immunology. I'm trying to learn a bit of biochemistry... only a microscopic amount, just enough to help me with my work. If I learn a microscopic amount of biology, will I be a microbiologist?`

Marius Furter wrote:

I certainly sympathize with you, Marius. I quite enjoyed how my linear algebra class was taught, but my differential equations class was exactly as you describe it -- pragmatic and boring. But when I talked with my best friend while

hewas taking this class, we found such nice patterns that were never touched on in lecture -- such as how the characteristic polynomial of an ODE comes from understanding the ODE as a polynomial in the differentiation operator \(D\) (as described here). I wish I knew of a book that covered differential equations from a similar angle -- I'm sure I would have found it much more engaging.It occurs to me that a similar feature shows up in linear algebra, under the Cayley-Hamilton theorem about characteristic polynomials of matrices. I'm fairly certain that this is because the differentiation operator is linear, but they sure didn't present things this way in my differential equations class.

`Marius Furter wrote: > these were taught in a very pragmatic (and thus boring) fashion. I certainly sympathize with you, Marius. I quite enjoyed how my linear algebra class was taught, but my differential equations class was exactly as you describe it -- pragmatic and boring. But when I talked with my best friend while _he_ was taking this class, we found such nice patterns that were never touched on in lecture -- such as how the characteristic polynomial of an ODE comes from understanding the ODE as a polynomial in the differentiation operator \\(D\\) (as described [here](https://math.mit.edu/~jorloff/suppnotes/suppnotes03/o.pdf)). I wish I knew of a book that covered differential equations from a similar angle -- I'm sure I would have found it much more engaging. It occurs to me that a similar feature shows up in linear algebra, under the [Cayley-Hamilton theorem](http://www.mcs.csueastbay.edu/~malek/Class/Characteristic.pdf) about characteristic polynomials of matrices. I'm fairly certain that this is because the differentiation operator is linear, but they sure didn't present things this way in my differential equations class.`

Thanks for the warm welcome! I certainly intend to participate.

John Baez wrote:

To be fair to the Professors teaching the classes, this stuff was over four years ago and I wasn't particularly engaged at the time, so I might have just missed/forgotten most of it. However, I'm fairly sure used to think of functions exclusively as curves in \(\mathbb{R}^2\) underlying certain restrictions. While I was obviously using the projections from the cartesian product intuitively in the form of coordinates, I certainly wasn't considering the set-theoretic basis for what was happening here. Essentially, I only had the example of \( \mathbb{R}^n\) which I interpreted as n-dimensional space and this interpretation is no longer natural for a general \( A \times B\). It's kinda amazing in retrospect that one can become a qualified scientist by only knowing \(\mathbb{R}^n\) and maps to/between it.

John Baez wrote:

Looking forward to that! I'll probably chime in with some examples from my fields when we look at symmetric monoidal preorders in chapter 2. It occurs to me that it might also be challenging for trained physicists/mathematicians to learn about natural sciences. It seems that in Biology one has a very large set of facts/implications which one needs to absorb and somehow extract generalities from (since there are few general laws). So, here formation of general heuristics constitutes understanding. In the more formalized sciences, it seems to be the other way around. You are presented with a general law or definition and you need to find out what it implies in order to understand it. Of course, the other directions are also important (i.e making predictions / finding more universal laws/definitions). Since we spend most of our education in one of these modes of thought, switching to the other then presents a serious hurdle. Just wondering if a 1000p+ biochemistry textbook is as intimidating for a math graduate as a rigorous math text is for a biochemist...

`Thanks for the warm welcome! I certainly intend to participate. John Baez wrote: > It's too bad your classes on analysis and linear algebra didn't prepare you more by explaining concepts like \\(f : \mathbb{R} \times \mathbb{R} \to \mathbb{R}\\) - functions between sets, and cartesian products of sets. I guess there are many ways to teach these subjects. To be fair to the Professors teaching the classes, this stuff was over four years ago and I wasn't particularly engaged at the time, so I might have just missed/forgotten most of it. However, I'm fairly sure used to think of functions exclusively as curves in \\(\mathbb{R}^2\\) underlying certain restrictions. While I was obviously using the projections from the cartesian product intuitively in the form of coordinates, I certainly wasn't considering the set-theoretic basis for what was happening here. Essentially, I only had the example of \\( \mathbb{R}^n\\) which I interpreted as n-dimensional space and this interpretation is no longer natural for a general \\( A \times B\\). It's kinda amazing in retrospect that one can become a qualified scientist by only knowing \\(\mathbb{R}^n\\) and maps to/between it. John Baez wrote: > Sometime I may ask you some questions about chemistry or immunology. I'm trying to learn a bit of biochemistry... only a microscopic amount, just enough to help me with my work. Looking forward to that! I'll probably chime in with some examples from my fields when we look at symmetric monoidal preorders in chapter 2. It occurs to me that it might also be challenging for trained physicists/mathematicians to learn about natural sciences. It seems that in Biology one has a very large set of facts/implications which one needs to absorb and somehow extract generalities from (since there are few general laws). So, here formation of general heuristics constitutes understanding. In the more formalized sciences, it seems to be the other way around. You are presented with a general law or definition and you need to find out what it implies in order to understand it. Of course, the other directions are also important (i.e making predictions / finding more universal laws/definitions). Since we spend most of our education in one of these modes of thought, switching to the other then presents a serious hurdle. Just wondering if a 1000p+ biochemistry textbook is as intimidating for a math graduate as a rigorous math text is for a biochemist...`

Biology is

utterlyintimidating to most mathematicans, because learning it seems to require memorizing huge piles of facts. Like this:Mathematicians don't like facts that they can't prove! They are happy to engage in long chains of deductive reasoning starting from some axioms. And they don't mind remembering definitions as long as they know nice examples of things that obey (or don't quite obey) these definitions. But learning facts that are true "just because they're true" is considered very unpleasant. We want to know exactly why everything is true.

I'm less intimidated than most mathematicians, because I enjoy looking for patterns, I feel there must be a "logical core" to biology, and I don't mind if there's a lot of complicated stuff that's not in this "logical core".

`Biology is _utterly_ intimidating to most mathematicans, because learning it seems to require memorizing huge piles of facts. Like this: <center><img src = "https://upload.wikimedia.org/wikipedia/commons/thumb/8/82/Calvin_cycle.svg/600px-Calvin_cycle.svg.png"></center> Mathematicians don't like facts that they can't prove! They are happy to engage in long chains of deductive reasoning starting from some axioms. And they don't mind remembering definitions as long as they know nice examples of things that obey (or don't quite obey) these definitions. But learning facts that are true "just because they're true" is considered very unpleasant. We want to know exactly why everything is true. I'm less intimidated than most mathematicians, because I enjoy looking for patterns, I feel there must be a "logical core" to biology, and I don't mind if there's a lot of complicated stuff that's not in this "logical core".`

Thanks for the insight!

I share this intuition strongly and really hope that it turns out to be true (and describable in some nice way that can be communicated back to mainstream biologists).

`Thanks for the insight! > I feel there must be a "logical core" to biology, and I don't mind if there's a lot of complicated stuff that's not in this "logical core". I share this intuition strongly and really hope that it turns out to be true (and describable in some nice way that can be communicated back to mainstream biologists).`

Hey Marius

Your comments and insights have been very helpful. I am interested in learning more about systems biology and the math behind metabolic networks. It'd be great to see how you apply CT!

`Hey Marius Your comments and insights have been very helpful. I am interested in learning more about systems biology and the math behind metabolic networks. It'd be great to see how you apply CT!`