A category theory study group in Boulder, CO
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Members (20)

## Stream

### Igor Polk

Discussion  -

Please, tell me a good name for a group of groups?﻿
1

Thank you, that is close! I am not able to decide on it, since my skill in group theory is very basic yet. I need to get more experience. I learn better by examples. Is there a good set of examples of applications of group theory to preferrably geometry, mechanics, or theory of automata?
I am working on a dance theory. I am theoretizing in this: assume I have a small subgroup. In my case it is a subgroup of movements. It acts, sorry, my physical analogy. The result of this is "tracked" and the result forms a certain pattern. If I change a set of movements, I am "acting" in another subgroup. And the pattern is different. Combination of these two subgroups and an element allowing changing the set of movements is what I am asking for. So far I am using a name "Pattern generator" for the combination. Does it all have some meanings to you, mathematicians? Am I too far?﻿

### Alissa Pajer

Discussion  -

Hi, wanted to let everyone know that won't be able to make a meeting tonight.﻿
1

### Luke Palmerowner

Discussion  -

Unfortunately, Tom is out of town again this Sunday, so he can't do a talk.  Does anybody have anything they'd like to research & present, or even have a discussion on / group research in real time?

Here are some topics of the top of my head:
* Cones & limits
* Category theory in programming languages
* Adjunctions (maybe not a complete treatment, but starting to think about it)
* Comma categories
* 2-categories, n-categories

Any takers?  Any other ideas (for topics or meeting styles)?﻿
1

CT for PL is really big and unwieldy. I'd propose cartesian closed categories (CCC) as a smaller, feasible topic, making firm connections with lambda calculus every step of the way.﻿

### Greg Pfeil

Discussion  -

I’m headed to the Scala meetup at 13:00 tomorrow. It’d be sweet if someone I know from this group was also around (I know you Precog people have some connection to the meetup), especially since we won’t be seeing each other tonight.

Also debating between going to either the Scala or Clojure meetup on Wednesday.﻿
1

### Luke Palmerowner

Discussion  -

Exercises for meeting #3, again in roughly increasing difficulty.  1-5 are recommended.

Functors Review

(1)
(a) Show that the covariant power set functor P : Set → Set, which takes sets to their power set (set of all subsets) and whose action on functions is defined by the direct image P(f)(A) = f[A] = { f(x) | x in A } obeys the functor laws: P(id_A) = id_P(A), and P(f;g) = P(f) ; P(g).
(b) Show that the contravariant power set functor P' : Set → Set, which takes sets to their power set and whose action on function is defined by the inverse image P'(f)(A) = f^-1[A] = { x | exists y in A, f(y) = x } obeys the contravariant functor laws: P(id_A) = id_P(A) and P(f;g) = P(g) ; P(f).

(2) Show that the following statements about opposite categories are equivalent:
(a) F is a contravariant functor C → D.
(b) F is a covariant functor C^op → D.
(c) F is a covariant functor C → D^op.
(d) F is a contravariant functor C^op → D^op.

(3) Show that Op : Cat → Cat, which take each category to its opposite category, is a functor (remember that Cat has categories as objects, functors as morphisms).  Is this functor covariant or contravariant?

Natural Transformations

(4)
(a) Show that there is a natural transformation η : I → P  (where I is the identity functor and P is the power set functor defined in exercise 1) which takes a set X to the singleton set {X}.
(b) Show that there is a natural transformation μ which takes a set of sets to their union.  (e.g. μ({{1,2},{2,3}}) = {1,2,3}   (I am now having you identify the functors between which this is natural)

(5) Finish the details of the Yoneda lemma: Given a category C, an object A of C, and a functor F : C → Set,  F(A) ≅ Nat(Hom(A, -), F)  (where Nat is the set of natural transformations between the two functors).  Note, this is an isomorphism in Set; i.e. a bijection.

Harder exercises for the enthusiast with copious free time

(6)
(a) Given functors F,G,H and natural transformations a : F→G and b : G→H, show that there is a natural transformation a;b : F→H.  (Hint: to avoid many tedious details, do this by stitching together commutative squares) Show that this notion of composition is associative.
(b) Given categories B,C,D; functors F,G : B→C and F',G' : C→D, and natural transformations a : F→G and b : F'→G', show that there is a natural transformation a•b : F;F' → G;G'.
(c)  Show the "interchange law": for natural transformations a,b,c,d with suitable signatures, (a;b)•(c;d) = (a•c);(b•d).   (Not so much hard as just an awful lot of information to keep track of.  Be diligent.)

(7) Recall the definition of a product category A×B: its objects are pairs of objects (a,b) where a in A and b in B, and its morphisms are pairs of morphisms (f,g) where f in A and g in B, as long as the domains and codomains line up.
(a) Given categories A, B and an object a in A, define the functor (a, –) : B → A×B which takes the object b in B to (a,b) in A×B.  Define its action on morphisms and show that it satisfies the functor laws.  Clearly the same holds for (–,b).
(b) Given categories A,B,C and functors F,G : A×B → C, show that a transformation t : F → G is natural if and only if it is natural both as a transformation between ((a, –) ; F) → ((a, –) ; G) (which are functors in B → C) for every a in A and as a transformation between ((–,b) ; F) → ((–,b) ; G) (which are functors in A → C) for every b in B.
This is what we mean by a transformation being natural in multiple variables.  It is equivalent to consider it natural in each variable individually, fixing the others, or natural in all of them at the same time (using a product category).﻿
1

okay I think I fixed it.  It was pretty wrong -- that's what I get for posting exercises that I haven't done myself rigorously.  :-P﻿

### Luke Palmerowner

Discussion  -

I know not everybody has email notifications on, and not everybody uses Google+ regularly (or at all besides this group), so I wanted to send a notification to draw attention to the resources that we put up recently.  The video of meeting #2, Alissa's lecture notes, and a set of exercises is posted to the community page.

If you want to receive notifications when something new gets posted, you can enable email notifications using the little bell icon to the lower left of the title panel on the left side of the screen when you view the community page.  ("Yes! More spam please!", said nobody).   I just wanted to make you aware of the option.  I'm not gonna be annoying and post any more of these each-member-tagged messages, so if you don't enable spam, you will get only meeting invites from now on!

﻿
1

Is there a way to "open up" the "community interface" to include more options? Like for instance, a plain vanilla mailing list? Would that increase participation and the shared experience?﻿

### Luke Palmerowner

Discussion  -

Here are some exercises for meeting #2, in increasing order of difficulty.  I encourage posting questions, attempts, solutions of any level in the comments.

(1) Given categories C and D, let CxD denote the category whose objects are pairs (c,d) where c is an object of C and d is an object of D, and whose morphisms are pairs (f,g) : (c,d) -> (c',d') where f : c -> c' and g : d -> d'.
(a) Show that CxD is indeed a category.
(b) Show that this construction is a product in Cat (the category of categories with functors as morphisms)

(2) Prove that disjoint union is a valid product in the category Rel (Sets as objects, relations as morphisms).

(3) Prove or disprove: functors preserve (a) products, (b) coproducts.  (Precisely: given F : C -> D is a functor, a,b objects in C, and a x b is the product of a and b in C, then F(a x b) is the (co)product of F(a) and F(b) in D).

(4) Find a coproduct in Rel or prove that one does not exist.  (Disclaimer: I haven't done this one yet, so I'm not sure about the difficulty)

(5) A terminal object in a category C is an object z such that for any object c in C, there exists a unique morphism from c -> z.   Given a category C and objects a,b in C, find a category P (in terms of C, a, and b) such that P has a terminal object if and only if the product a x b exists in C.   (Hint: the definition of product and terminal object share the words "for any ... there exists a unique morphism ...").﻿
1

Question:

How do we model CT in a typed language? There's this slipperiness of concepts, e.g. a category is an object in the functor category whose arrows are functors. As #5 has shown, a product is a an object with projections going out of it (plus whatever else to deal with the mediating arrow). It is also a terminal object in an appropriate category.

How are we to model things canonically?﻿

### Kim-Ee Yeoh

Discussion  -

Once upon a time, a professor taught quantum field theory to a class of physics students. None of them understood it. The following semester, the professor taught it again. Again, the students were utterly baffled. Undeterred, the professor offered it for a third time. This time, he understood it!

My immediate reaction on seeing the post is that, "That's nice. Wouldn't it be awesome if Luke could motivate everyone else to do the same, especially those who'd really like to improve the proving skills?"

Because no one is so utterly deficient that they can't teach a thing or two. As a brilliant teacher once said, "If all you know is how to play the black keys, well then teach 'em, all 5 of them."﻿
1

Yes! In a collective sort of fashion, like what do we as a group know about math proofs in general?

People know more than they think they do.

Also, math is stereotyped as this incredibly solitary activity, like swimming or piano-playing.

Really?﻿

### Luke Palmerowner

Discussion  -

Hey y'all, so I noticed some getting stuck on straightforward proofs, so I wrote this little guide on getting unstuck.  I don't know if it is too remedial or not, but I certainly found having these tactics spelled out for me helpful. http://lukepalmer.wordpress.com/2013/03/25/follow-your-nose-proofs/﻿
1

> So then we state CT-wise that if /phi is a monomorphism from H to G, then the order of H divides the order of G.

Which brings us back to the original question of how (or should we even) express the order of a group in terms of the category Grp, which has no primitive notion of such things, unlike set theory.

At the heart of the issue is, how much group theory can we bootstrap  to, given all that we have is the notion of a structure-preserving homomorphism?

If this sounds slightly perverse, so it is! The treasure we hope to find is undiscovered insight into group structure. Or something else!﻿

### Kris Nuttycombe

Discussion  -

This is the bit of nerdiness that I mentioned this evening. Great meetup all! I can tell I need to brush up on proving things.

http://measureofdoubt.com/2012/09/12/colbert-deconstructs-pop-music-finds-mathematical-nerdiness-within/﻿
1

### Luke Palmerowner

Discussion  -

I'm tentatively planning on having our first meeting on this coming Sunday, the 24th, in the evening (6pm-ish).  I'll make an event when it's official.

Until then, make sure that you are somewhat familiar with the prerequisite math.  You don't need to be a guru, but at least familiarize yourselves with these terms and their definitions.

* Basic set theory: surjection (a.k.a. "onto"), injection (a.k.a. "1 to 1"), bijection, power set, cartesian product.
* Group theory: group, semigroup, monoid, abelian group, group homomorphism, group isomorphism, direct product, free group
* Order theory: preorder, partial/total/well ordering, monotone function, order isomorphism

We won't need anything super deep about any of these theories.  If you haven't seen a term before or are unsure exactly what it means, look up the definition, come up with a couple examples of it, and prove that your examples do actually meet the criteria (proving it will solidify your understanding).  Feel free to post your work, ask questions, and discuss in this thread.﻿
1

### Luke Palmerowner

Discussion  -

I don't have anything prepared this week, no meeting.  Next week, Adjunctions!﻿
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### Luke Palmerowner

Discussion  -

Oops!   I accidentally set the date of the next meeting tonight, when I meant Sunday.  It's correct now, but I'm sending this just in case anybody took it literally.﻿
1

### Luke Palmerowner

Discussion  -

I am very busy this week and don't have time to write exercises on monads.  But here's first section of Mac Lane's chapter on monads, which has a couple exercises.﻿
1

Nice, thanks for putting up the pages!

If you ever get a chance to think up material on monads, please do. Even if it's only a question or two, it'll be much appreciated.﻿

### Luke Palmerowner

Discussion  -

Some people have expressed that Sunday nights are not the best time.  Please fill out this doodle poll so we can see if we can find a better time.

http://www.doodle.com/9dm7suwm58bpvieg#table﻿
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### Luke Palmerowner

Discussion  -

Here's the video from Meeting #3.  There were audio problems for the first 20 minutes or so, sorry.

Luke Palmer hung out with 1 person. #hangoutsonairKim-Ee Yeoh﻿
Meeting #3: Natural Transformations
Luke Palmer and 1 other participated
1

Cool! (Give hints!)﻿

### Alissa Pajer

Discussion  -

Here are notes (jpg format, G+ doesn't allow pdf :/) from the meeting on product and coproduct.﻿
2

### Luke Palmerowner

Discussion  -

In case it's hard to find, the recording of tonight's meeting is here:
Meeting 2: Products, Coproducts﻿
1

### Luke Palmerowner

Discussion  -

Instead of left and right composition/cancellation/inverse which are dependent on which order you write compositions, I suggest we use the prefixes pre and post, which are agnostic to the order we write them.

So a monic arrow is one that has post-inverses and an epic arrow is one that has pre-inverses.

(Or is that just setting us up to always think of arrows like functions, as if they had a well-defined notion of "data flow"?)﻿
1

> I was getting those two properties mixed up in my head.

You're in good company! Bourbaki thought that way too.

Epis appear to be even more curious. The corresponding wikipedia entry has the following para: "It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept. Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior. It is very difficult, for example, to classify all the epimorphisms of rings. In general, epimorphisms are their own unique concept, related to surjections but fundamentally different."﻿