Category Theory For Beginners: Graphs And Dynamical Systems

In this video I discuss how we can make categories of structured sets (the category of graphs, and the category of dynamical systems, and the category of functions), by considering functors. In particular, I show how the category of functors from the category with two parallel arrows, to the category Set corresponds to the category of graphs. I illustrate how graph homomorphisms correspond to the natural transformations between such functors. I also describe the category of dynamical systems, and the category of functions using this idea. I also show that the category of dynamical systems is isomorphic to a subcategory of the category of functions, which in turn, is isomorphic to a subcategory of the category of graphs. Here I use the term "dynamical system", in the same loose sense as Lawvere does in his great book "Conceptual Mathematics", to just mean a function from a set to itself. Note that the kind of graphs we discuss are also called `quivers'.
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This unlisted video describes how to find subobject classifiers in the kinds of functor categories I describe in this video, and may help people wishing to connect this material with my video on topos theory.
• Topos Theory: Finding ...

[@bhaaz]

50:30 The objection I'd have to sending the additive monoidal category to sets to describe dynamical systems is that sets can consist abstract and real things while all dynamical systems are necessarily real, not just deterministic and closed. Otherwise, superb!

[@christophealexandre1538]

This video and the channel in general deserves a lot more attention. It is incredibly useful, I am so grateful for the time you spent creating that content.

[@nathanielvirgo]

Thank you so much for these lectures, they really are amazing, and I'm learning a huge amount from them.

[@gardnjw]

With the pandemic raging outside I've finally found the time to study category theory. I have some experience with algorithms and discrete math but this stuff always seemed a little bewildering. Thank you for making these.

[@notlessgrossman163]

Thank you for these great videos.. I'm not a mathematician, I self teach myself on interesting topics. I'm trying to get a grasp of category theory and applications and your videos are a great help.

[@RichardSouthwell]

I am glad they are helping. I wish more people outside of mathematics were aware of category theory.

[@gucker]

Thank you for the another great lecture on the category theory!

[@jonaprieto]

Thanks Richard. It was a good way of explaining these examples.

[@zantine59]

I found this video incredibly useful. Thanks!

[@MrScottLi]

How is the category of dynamical systems related to monads?
I understand:
A monad = a monoid in the category of endofunctors
A dynamical system = an object in the category of endomaps

[@RichardSouthwell]

Excellent question. I had not thought of that connection. Any dynamical system could be thought of as an endofunctor of a discrete category. One could ask if any such endofunctors are monads. There is also the question of what happens when one iterates a monad.

[@notlessgrossman163]

Question : if category theory can 'model' dynamic systems, how does it fit into dynamical systems optimization problem of selecting the most efficient 'model' for the system we are analyzing? Eg. In practical terms.

[@RichardSouthwell]

I'm not sure. You could try reading 'A categorical approach to open and
interconnected dynamical systems'. Also William Lawvere wrote some interesting papers on how to model dynamical systems in more sophisticated ways.

[@kevon217]

sweet shades!

[@fgc_rewind]

now that was a cool result 1:14:47

[@nathanielvirgo]

It was so cool he had to wear shades for it

[@johnhare8208]

Am I weird in seeing set as being source edge target (s e t)

[@RichardSouthwell]

For me, it is that (l e t) stands for (left epsilon terminal), but that won't make much sense until the video on adjoint functors, which I will release after the next coming video on the wonderful Yoneda Lemma.

[@jeffreyhowarth7850]

dynamical systems like in applied category theory?

[@njitnom]

hhashahaa ur glasses man nice

[@XingHu-c8f]

1

[@burnytech]

[@hyperduality2838]

Points are dual to lines
"Perpendicularity in hyperbolic geometry is measured in terms of duality"
Lines or edges are dual to points or vertices
Vectors are dual to co-vectors (forms)
Inclusion is dual to exclusion
Unions are dual to intersections
Infinity is dual to zero