For a long time I felt like I didn't really understand the Yoneda Lemma. I knew some things that people said about it ('we can understand objects by the maps into them' and 'the Yoneda embedding is full and faithful') but the statement 'Hom(Hom(A, -), F) = F(A)' itself was something I could only use as a symbolic manipulation without understanding.
On the other hand, I did separately know facts like 'In the category of quivers there are objects which look like • and •→•, such that the maps out of them tell you exactly the vertices and edges in your quiver' and 'In the category of simplicial sets there are objects which are just an n-simplex; maps out of them are the n-simplices of the object you are mapping into'.
Somehow I only recently realised that these examples are precisely the Yoneda Lemma. These objects are precisely presheaves of the form Hom(A, -), and the Yoneda Lemma tells you what you get when you map out of them.
In particular I think it would be useful to give the quiver example to students when they learn the Yoneda Lemma.
Category theory friends: Is there a standard way to describe a functor?
I was using a two-case function to describe functor, where one case is what the functor does to objects and the other case is what the functor does to morphisms (see the image). However, I haven't been able to find a standard form in any of the literature I've been reading...
In his famous paper, "on proof and progress in mathematics", Thurston lists 8 (and implies 29 other) ways to think of the derivative.
I was bored waiting for a bus, so I tried listing the different ways I could think of what a category is (see below). Please feel free to help me add more!
A Category is...
The usual definition (omitted for space)
an abstract theory of functions / arrows (or as Awodey would say "archery")
a monoidoid
a poset with evidence (wording stolen from Alex Kavvos)
a set-enriched category
an object in CAT
a syntax for a programming language
a maze of twisted arrows all alike
a "path-complete" digraph (if there is a path x -> y there is an edge x -> y)
Just started writing up a few of my notes on introductory Category Theory. Not much here yet (it took me awhile to get Figure 1 to look right, and it's still not perfect).
Mathematicians sometime talk about algebra and geometry being dual to each other. One way to formalise this is by talking about opposite categories. If the objects of a category act like algebras, then in the opposite category they act like spaces.
But the category of finite dimensional vector spaces is its own opposite! This suggests that linear algebra is in some sense the place where algebra and geometry meet. Perhaps that explains why it's so tractable and efficacious.
I have a new(ish) preprint on the arXiv! You can find "Invariants of structures" at https://arxiv.org/abs/2402.18063. This is a somewhat embellished version of one half of my PhD thesis. A talk which I gave about this subject in the fall of 2022 is available at https://www.youtube.com/watch?v=5TeGZZ_mepc.
In this new version, I have finally added an explicit description of something I've been telling people for years: My main result shows that any first-order property of finite structures can be computed by counting small substructures. Perhaps surprisingly, this comes as a result of synthesizing a categorical treatment of Bourbaki's notion of mathematical structure with Hilbert's classical result on symmetric polynomials.
Happy New Year! It's the first #CategoryTheory#Caturday of the new year and we are totally chillin'! I worked through Test 2 of L&S this week but couldn't help myself from checking the answers in Session 18. I just /had/ to know. Learning can't happen without feedback. Speaking of learning things, Alphonse has discovered that if the ottoman is close enough to the couch he can stretch out and rest his head on it. If napping were a test, he's totally got an easy A! #math#catshttps://rruff82.github.io/LS-Categories/session18.html
Short #categorytheory lesson: There is a common figure of speech, that goes like "If x is like y, then z is like q", e.g. "If a school are like a corporation, then the teachers are like bosses". This figure of speech introduces a #functor: what are you saying is that there is a certain connection (or category-theory therms a "morphism") between schools and teachers, that is similar to the connection between corporations and bosses i.e. that there is some kind of structure preserving map that connects the category of school-related things, to the category of work-related things in which schools (a) are mapped to corporations (F a) and teacher (b) are mapped to bosses (F b). and the connections between schools and teachers (a -> b) are mapped to the connections between corporations and bosses (F a -> F b).
@dougmerritt A friend of mine ... Miles Gould ... did a first ascent of a mountain in Kyrgyzstan. He was then subsequently on a climb where one of his friends fell and died.
The emergency services said they did nothing wrong, they were well prepared, but they were just unlucky.
Yes, for we outsiders it can be a surreal community.
Miles also gave a fabulous @BigMathsJam talk on the parallels between doing a PhD in Pure Maths (in his case #CategoryTheory) and doing a first ascent. I think it's available on YouTube.
It's #CategoryTheory#Caturday again and I'm still stuck on L&S Session 15 Exercise 7! This week I break down my Python code from last week to "solve" Exercise 8, but am still just brainstorming possible solutions for 7. Alphonse is still helping, this time by sitting on the tan cushion which is now back in "ottoman"-form. It seems like a fitting metaphor for my hypothesis about my missing maps somehow. #math#catshttps://rruff82.github.io/LS-Categories/session15-p5.html
Happy #CategoryTheory#Caturday! I made some more progress on Session 15 of L&S this week trying to understand this whole "objectification in subjective" business. Edward wasn't much of a help in this regard, but in his subjective opinion he'd very much like the de-objectification of whatever that thing is on his head. #math#catshttps://rruff82.github.io/LS-Categories/session15-p3.html
Suppose we have a functor F:C → D. A right adjoint U of F can be thought of as a 'best approximation' to an inverse of F. A true inverse of F would have FU = id and id = UF, while with an adjoint we only have natural transformations FU → id and id → UF.
So U doesn't quite undo F, but it's as close as possible to an un-F functor. Functors T:C → D that arise as T = FU for some such F and U are called monads.
Of course it's possible that some monads themselves have right adjoints. The nLab has a nice page about these: https://ncatlab.org/nlab/show/adjoint+monad. It turns out that these un-T-ing functors automatically get the structure of a comonad.
A slightly less common situation is for a monad T to have a left adjoint. An example is the function monads https://ncatlab.org/nlab/show/function+monad (also known as the 'environment' or 'reader' monads). These are defined by T(A) = (B → A) for some fixed set B. In fact, on Set this family of monads are the only ones to have such a co-un-T-ing functor.
Lawvere's Conceptual #Mathematics. Friedman's The Little Schemer. These two books provide playful approach to their subject matter, toying with camp while stealthily introducing the reader to advanced concepts that they will only later in their journey recognize (#topos theory for CM, #compilers for TLS).
I consider them to be truly masterful & singular pedagogic demonstrations. What other works fit into this category?
It's technically still #Caturday so my #CategoryTheory for the week isn't late! Any confidence I might have potentially built with "Test 1" went out the window as I attempted to prove Brouwer's theorems this week. Seriously though, the level of difficulty jump between Sessions 9 and 10 was a little extreme. My only saving grace was that L&S acknowledge this fact in the introduction. I didn't know if I would make it through all 4 exercises this week, but I gave it my best effort. In feline news, Edward has been diagnosed with hyperthyroidism while Alphonse needed his insulin dosage upped to help manage his diabetes. Our vet made a wise crack about doing regular blood transfusions between the two of them. There's some kind of connection between that joke and the lack of retraction from E to I, but I'm too exhausted to make it. To make up for the semi-rushed #math this week, here's both #cats: Edward hiding from his medication and Alphonse mooching for it. https://rruff82.github.io/LS-Categories/session10.html
great example of #CategoryTheory being applied in procedural #CG via "double pushout production rules". I'm convinced that CT can be applied to most domains, it's just a matter of teaching people in the domains category theory rather than fetishizing it as a magical skeleton key to the universe. https://www.youtube.com/watch?v=FG3LbcOGHqw
Like, if you teach someone linear algebra it doesn't mean that they'll immediately be able to start producing amazing graphics shaders, even if shaders are 95% linear algebra. But #CategoryTheory detractors act disappointed that they didn't find any practical applications of it after reading Conceptual Mathematics in a few months.