Category Theory Summer Study Group 2015

A suggested syllabus for a summer study group on category theory.

Syllabus

Initial details for putting  the group together can be found at http://boffosocko.com/2015/05/21/category-theory-anyone/.

Below is a handful of suggestions and thoughts relating to the study group in terms of platforms to assist us in communicating as well as a general outline for the summer.  I’m only “leading” this in the sense that I put my foot forward first, but I expect and sincerely hope that others will be active leaders and participants as well, so please take the following only as a suggestion, and feel free to add additional thoughts and commentary you feel might help the group.

Primary resources:

General Communication

Since many within the group are already members of the Google Group “Advanced Physics & Math – Los Angeles.” I suggest we use the email list here as a base of communication. I believe the group is still “private” but am happy to invite the handful of participants who aren’t already members. Those actively participating are encouraged to change their settings so that they receive emails from the group either as they’re posted, or in batches once a day.  Those subscribed only once a week or less frequently are likely to miss out on questions, comments, and other matters.

Alternately we might also use the GoodReads.com discussion group within the “Mathematics Students” group. I believe only about three of us so far may already be goodreads members, so this may require more effort for others to join.

If anyone has an alternate platform suggestion for communicating and maintaining resources, I’m happy to entertain it.

I wouldn’t be opposed to setting up a multi-user WordPress site that we could all access and post/cross-post to. Doing this could also allow for use of \LaTeX as well, which may be useful down the line. This would also have the benefit of being open to the public and potentially assisting future students. It also has built-in functionality of notifying everyone of individual posts and updates as they’re entered.

Meetings

I’ll propose a general weekly meeting online via Google Hangouts on a day and time to be determined.  It looks like the majority of respondents are in the Pacific timezone, so perhaps we could shoot for something around 7pm for an hour or so if we do something during weekdays so that East coasters can join without us running too late. If we decide to do something during the weekend, we obviously have a good bit more flexibility.

If we could have everyone start by indicating which days/times absolutely won’t work for them and follow up with their three to four preferred days/times, then we might be able to build a consensus for getting together.

Alternate videoconference options could include Skype, ooVoo, or others, in some part because I know that most participants are already part of the Google ecosystem and know that one or more potential participants is using Google Chromebooks and thus may not be able to use other platforms.  Is anyone not able to use Google Hangouts? If we opt for something else, we want something that is ubiquitous for platform, allows screen sharing, and preferably the ability to record the sessions for those who aren’t present.

Ideally the videoconference meetings will be geared toward an inverted classroom style of work in which it would be supposed that everyone has read the week’s material and made an attempt at a number of problems. We can then bring forward any general or specific conceptual problems people may be having and then work as a group toward solving any problems that anyone in the group may be having difficulty with.

I’ll also suggest that even if we can’t all make a specific date and time, that we might get together in smaller groups to help each other out.  Perhaps everyone could post one or two regular hours during the week as open “office hours” so that smaller groups can discuss problems and help each other out so that we can continue to all make progress as a group.

Primary Textbook

Spivak, David I. Category Theory for the Sciences. (The MIT Press, 2014)

Given the diversity of people in the group and their backgrounds, I’ll suggest Spivak’s text which has a gentle beginning and is geared more toward scientists and non-professional mathematicians, though it seems to come up to speed fairly quickly without requiring a large number of prerequisites.  It also has the benefit of being free as noted below.

The textbook can be purchased directly through most book retailers.  Those looking for cheaper alternatives might find these two versions useful. The HTML version should be exactly in line with the printed one, while the “old version” may not be exactly the same.

Following this, I might suggest we use something like Awody’s text or Leinster’s which are slightly more technical, but still fairly introductory. Those who’d like a more advanced text can certainly supplement by reading portions of those texts as we work our way through the material in Spivak. If all of the group wants a more advanced text, we can certainly do it, but I’d prefer not to scare away any who don’t have a more sophisticated background.

Additional References

Proposed Schedule

The following schedule takes us from now through the end of the summer and covers the entirety of the book.  Hopefully everyone will be able to participate through the end, though some may have additional pressures as the beginning of the Fall  sees the start of other courses. Without much prior experience in the field myself, I’ve generally broken things up to cover about 35 pages a week, though some have slightly more or less.  Many, like me, may feel like the text really doesn’t begin until week three or four as the early chapters provide an introduction and cover basic concepts like sets and functions which I have a feeling most have at least some experience with.  I’ve read through chapter two fairly quickly already myself.  This first easy two week stretch will also give everyone the ability to settle in as well as allow others to continue to join the group before we make significant headway.

If anyone has more experience in the subject and wishes to comment on which sections we may all have more conceptual issues with, please let us know so we can adjust the schedule as necessary.  I suppose we may modify the schedule as needed going forward, though like many of you, I’d like to try to cover as much as we can before the end of the summer.

Week One: May 24 (24 pages)

Administrative tasks

  • Purchase Textbook
  • Decide on best day/time for meeting
  • Decide on platform for meetings: Google Hangouts /Skype /ooVoo /Other
  • 1 A brief history of category theory
  • 1.2 Intention of this book
  • 1.3 What is requested from the student
  • 1.4 Category theory references
  • 2 The Category of Sets 9
  • 2.1 Sets and functions
  • 2.2 Commutative diagrams

Week Two: May 31  (50 pages)

  • 2.3 Ologs
  • 3 Fundamental Considerations in Set 41
  • 3.1 Products and coproducts
  • 3.2 Finite limits in Set

Week Three: June 7 (40 pages)

  • 3.3 Finite colimits in Set
  • 3.4 Other notions in Set

Week Four: June 14 (31 pages)

  • 4 Categories and Functors, Without Admitting It 115
  • 4.1 Monoids
  • 4.2 Groups

Week Five: June 21 (38 pages)

  • 4.3 Graphs
  • 4.4 Orders

Week Six: June 28 (19 pages)

  • 4.5 Databases: schemas and instances

Week Seven: July 5 (36 pages)

  • 5 Basic Category Theory 203
  • 5.1 Categories and functors

Week Eight: July 12 (28 pages)

  • 5.2 Common categories and functors from pure math

Week Nine: July 19 (48 pages)

  • 5.3 Natural transformations
  • 5.4 Categories and schemas are equivalent, Cat » Sch

Week Ten: July 26 (45 pages)

  • 6 Fundamental Considerations of Categories
  • 6.1 Limits and colimits

Week Eleven: August 2 (15 pages)

  • 6.2 Other notions in Cat

Week Twelve: August 9 (26 pages)

  • 7 Categories at Work 375
  • 7.1 Adjoint functors

Week Thirteen: August 16 (32 pages)

  • 7.2 Categories of functors

Week Fourteen: August 23 (19 pages)

  • 7.3 Monads

Week Fifteen: August 30 (23 pages)

  • 7.4 Operads

Additional resources

Requested/Required Responses from participants:

Preferred platform(s) for communications:

Email and/or online discussions

Platform Can use Can’t use Prefer Not to Use
Google Group
WordPress Site
GoodReads Group
Other:

Videoconferences

Platform Can use Can’t use Prefer Not to Use
Google Hangouts
Skype
ooVoo
Other

 

Dates and times you absolutely CAN’T make for meetings (please include your local time zone):

Weekdays:

Weekends:

 

Dates and times you prefer (please include your local time zone):

Weekdays:

Weekends:

 

One or two time periods during the week you could generally/reliably be available for “office hours”:

 

Any other thoughts on the material above:

  • Textbooks
  • Schedule
  • Additional resources for the group
  • Other

If you’d like to join us, please fill out the contact information and details below based on the material above:

Please indicate which videoconference platforms you are able to use by placing a checkmark in the corresponding boxes below. If you’re technically unable to use one or more, please indicate which in the “general comments” box above, and provide the reason why.

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Author: Chris Aldrich

I'm a biomedical and electrical engineer with interests in information theory, complexity, evolution, genetics, signal processing, theoretical mathematics, and big history. I'm also a talent manager-producer-publisher in the entertainment industry with expertise in representation, distribution, finance, production, content delivery, and new media.

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