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# Basic category theory reading group - at NYC category theory meetup

edited May 3

Over on the Category Theory Community Server, Oliver Shetler announced a reading group on category theory. Two, actually:

Hey guys, if anybody is interested, I host two Sunday 2pm reading groups on an alternating weekly basis. We've started meeting on Zoom now so it's accessible to everybody.

In the first one, we're working through Steve Roman's An Introduction to the Language of Category Theory. It's for people who want to fill holes in their basic Category Theory knowledge. We're just starting in on Chapter 2 so it's still possible to catch up easily.

In the second group, we're reading Coecke and Kissinger's Picturing Quantum Processes. We've been working through Kissinger's problem sets from when he taught a class based on the book. We're just starting on assignment 2 so it's still accessible to newcomers as well.

If you're interested, I post the events on http://www.meetup.com/category_theory/.

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edited April 26

Great, thanks for this note John!

This meetup looks promising. In particular, every other week there is a meeting to go through passages and exercises from:

I've just joined the meetup, and encourage other Azimuth people who want to get ramped up on basic category theory to give it try. The book appears to be very clear in its introduction to category theory.

I'll post some notes on the reading passage and exercises for the next meeting, which is a zoom on Sunday May 3.

Comment Source:Great, thanks for this note John! This [meetup](https://www.meetup.com/Category_Theory/) looks promising. In particular, every other week there is a meeting to go through passages and exercises from: * Steven Roman, [An Introduction to the Language of Category Theory](https://www.amazon.com/Introduction-Language-Category-Textbooks-Mathematics/dp/3319419161), Springer Compact Textbooks in Mathematics, 2017. I've just joined the meetup, and encourage other Azimuth people who want to get ramped up on basic category theory to give it try. The book appears to be very clear in its introduction to category theory. I'll post some notes on the reading passage and exercises for the next meeting, which is a zoom on Sunday May 3. 
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edited April 27

John, David, Thank you for alerting us to this reading group.

Steve Roman's book is rather expensive on Amazon. He sells directly, as a PDF, for $24.95, at his website, a book which seems to be an early draft from 2015. The title is slightly different, "An Introduction to Category Theory", the page numbers are different (because the pages are bigger). Both books have exercises. The Table of Contents is similar, but in the final version, at Amazon, he seems to have changed or added a few sections, and reworked the chapter on Adjunctions. So there are some substantial differences, it seems. The ebook at Springer for his book An Introduction to the Language of Category Theory is 46 euros, which is cheaper than Amazon's price of 65 dollars. His book seems very readable. He also has six video lectures which have gotten excellent reviews. Comment Source:John, David, Thank you for alerting us to this reading group. Steve Roman's book is rather expensive on Amazon. He sells directly, as a PDF, for$24.95, at <a href="http://www.sroman.com/CategoryBook/CategoryBook.php">his website</a>, a book which seems to be an early draft from 2015. The title is slightly different, "An Introduction to Category Theory", the page numbers are different (because the pages are bigger). Both books have exercises. The Table of Contents is similar, but in the final version, at Amazon, he seems to have changed or added a few sections, and reworked the chapter on Adjunctions. So there are some substantial differences, it seems. The ebook at Springer for his book <a href="https://www.springer.com/gp/book/9783319419169">An Introduction to the Language of Category Theory</a> is 46 euros, which is cheaper than Amazon's price of 65 dollars. His book seems very readable. He also has [six video lectures](https://www.youtube.com/watch?v=If6VUXZIB-4&list=PLiyVurqwtq0Y40IZhB6T1wM2fMduEVe56) which have gotten excellent reviews.
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Yes, his material is very clear.

Comment Source:Yes, his material is very clear.
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For Sunday May 3 meeting.

Reading assignment looks manageable, pp 43 - 56 of the text. Here are the topics covered in that passage:

• Morphisms of Functions: Natural Transformations
• Example: the determinant
• Compositions of natural transformations
• Natural isomorphisms
• Example: the double-dual
• Example: the Riesz map
• Example: the coordinate map
• Natural isomorphisms, fullness and faithfulness
• Functor categories
• The category of diagrams
• Natural equivalence
• Natural transformations between Hom functors

Upcoming sections: Yoneda Embedding, Yoneda Lemma

Comment Source:For Sunday May 3 meeting. Reading assignment looks manageable, pp 43 - 56 of the text. Here are the topics covered in that passage: * Morphisms of Functions: Natural Transformations * Example: the determinant * Compositions of natural transformations * Natural isomorphisms * Example: the double-dual * Example: the Riesz map * Example: the coordinate map * Natural isomorphisms, fullness and faithfulness * Functor categories * The category of diagrams * Natural equivalence * Natural transformations between Hom functors Upcoming sections: Yoneda Embedding, Yoneda Lemma
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Exercises for May 3 meeting:

• Ex 7, 10, 11, 12
• Choose one from 17, 18, 19
• Wenbo will present 17
• Spencer will present 18
• Oliver will present 19
• David will present 12
Comment Source:Exercises for May 3 meeting: * Ex 7, 10, 11, 12 * Choose one from 17, 18, 19 * Wenbo will present 17 * Spencer will present 18 * Oliver will present 19 * David will present 12
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Here is the gist of some of the exercises. (I have paraphrased and selected.)

Ex 7. Describe the arrow part of the hom functor $$Hom_C(B,\cdot)$$ and the naturality condition.

Ex 10. Definition of bifunctor. Show that a bifunctor is a functor in each of its arguments separately. Plus more technical questions relating to bifunctors.

Ex 11. For naturally isomorphic functors F,G, show:

• F is faithful iff G is faithful
• F is full iff G is full
• If F is naturally isomorphic to the identity functor, then it is fully faithful.
• Relationships between fullness and faithfulness under composition.

Ex 17. Let $$F,G: C \implies P$$ be functors from a category C to a preorder P.

• Describe necessary and sufficient conditions under which there is a natural transformation from F to G.
• Prove that if P and Q are preorders, then the functor category $$Q$$ is also a preorder.

Ex 18. Verify that the functor category $$D^C$$) is a category.

Ex 19. Prove that the functor category $$D^2$$ is essentially the category of arrows of D.

Comment Source:Here is the gist of some of the exercises. (I have paraphrased and selected.) Ex 7. Describe the arrow part of the hom functor \$$Hom_C(B,\cdot) \$$ and the naturality condition. Ex 10. Definition of bifunctor. Show that a bifunctor is a functor in each of its arguments separately. Plus more technical questions relating to bifunctors. Ex 11. For naturally isomorphic functors F,G, show: * F is faithful iff G is faithful * F is full iff G is full * If F is naturally isomorphic to the identity functor, then it is fully faithful. * Relationships between fullness and faithfulness under composition. Ex 17. Let \$$F,G: C \\implies P\$$ be functors from a category C to a preorder P. * Describe necessary and sufficient conditions under which there is a natural transformation from F to G. * Prove that if P and Q are preorders, then the functor category \$$Q\$$ is also a preorder. Ex 18. Verify that the functor category \$$D^C\$$) is a category. Ex 19. Prove that the functor category \$$D^2\$$ is essentially the category of arrows of D. 
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edited May 3

The exercise amounts to proving the definition of the left and right whiskering of a functor and a natural transformation.

• Right whiskering = natural transformation followed by functor
• Left whiskering = functor followed by natural transformation

In this imagery, the functor is the "whisker" in the diagram, and the natural transformation is the cat.

Comment Source:* [Presentation on exercise 12](https://www.azimuthproject.org/azimuth/files/roman-exercise-12.pdf), by David Tanzer, 5/3/2020 The exercise amounts to proving the definition of the left and right [whiskering](https://proofwiki.org/wiki/Definition:Whiskering) of a functor and a natural transformation. * Right whiskering = natural transformation followed by functor * Left whiskering = functor followed by natural transformation In this imagery, the functor is the "whisker" in the diagram, and the natural transformation is the cat. 
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Meeting tomorrow, 2-4 pm EDT

Zoom ID: 807-569-857

Please review pages 43-56 and continue onto page 57. Do exercises 6, 7 and 23.

This week, we will be covering the prerequisites for understanding the Yoneda Lemma. To that end, please read Tai-Danae Bradley's first blog post in her series on the Yoneda Lemma: https://www.math3ma.com/blog/the-yoneda-perspective

If you have time, take a peek at the second post: https://www.math3ma.com/blog/the-yoneda-embedding

Comment Source:Meeting tomorrow, 2-4 pm EDT Zoom ID: 807-569-857 Please review pages 43-56 and continue onto page 57. Do exercises 6, 7 and 23. This week, we will be covering the prerequisites for understanding the Yoneda Lemma. To that end, please read Tai-Danae Bradley's first blog post in her series on the Yoneda Lemma: https://www.math3ma.com/blog/the-yoneda-perspective If you have time, take a peek at the second post: https://www.math3ma.com/blog/the-yoneda-embedding
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edited May 24

Ex 6. Show that the hom functor $$A^{\rightarrow} = hom_X(A,.): X \rightarrow Set$$ preserves monics, that is, if $$\alpha: C \rightarrow D$$ is monic in X, then $$A^{\rightarrow}(\alpha): hom_X(A,C) \rightarrow hom_X(A,D)$$ is also monic.

Ex 7. Describe the arrow part of the hom functor $$A^{\rightarrow}$$ and the naturalness condition.

Comment Source:Ex 6. Show that the hom functor \$$A^{\rightarrow} = hom_X(A,.): X \rightarrow Set\$$ preserves monics, that is, if \$$\alpha: C \rightarrow D\$$ is monic in X, then \$$A^{\rightarrow}(\alpha): hom_X(A,C) \rightarrow hom_X(A,D)\$$ is also monic. Ex 7. Describe the arrow part of the hom functor \$$A^{\rightarrow}\$$ and the naturalness condition. 
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The homework for the next meeting is to take notes on the chapters from the text on the Yoneda Embedding and the Yoneda Lemma.

We had a good and interesting meeting. At the end of it, @AndriusKulikauskas presented the graph on the Yoneda Lemma which he had posted a while back. Hashing it out there, my conjecture is that what he was getting at is an extension to the Yoneda Lemma, in which the bijection gets enriched to a natural isomorphism. This is stated for example in the language MacLane uses, "The Yoneda map is natural in K and r."

So our strategy will be to first study the Yoneda Lemma as it is stated, as such, in terms of a bijection. Then we will cover the extension to the lemma, and revisit Andrius' diagram.

Comment Source:The homework for the next meeting is to take notes on the chapters from the text on the Yoneda Embedding and the Yoneda Lemma. We had a good and interesting meeting. At the end of it, @AndriusKulikauskas presented the graph on the Yoneda Lemma which he had posted a while back. Hashing it out there, my conjecture is that what he was getting at is an extension to the Yoneda Lemma, in which the bijection gets enriched to a natural isomorphism. This is stated for example in the language MacLane uses, "The Yoneda map is natural in K and r." So our strategy will be to first study the Yoneda Lemma as it is stated, as such, in terms of a bijection. Then we will cover the extension to the lemma, and revisit Andrius' diagram.