Category Archives: Basic Category Theory (Chapter 5)

Week Ten Meeting: §5.1 and §5.2 of David Spivak’s Category Theory for the Sciences

Week Ten

This week’s Google Hangout (RSVP here) will cover problems/questions from week ten of the (now revised) syllabus:

  • 5.1 Categories and functors
  • 5.2 Common categories and functors from pure math

Last week we made it through most of section 5.1, so we’ll finish it up and hopefully make it through 5.2, and pending time and people’s motivation this past week, possibly break into the beginning of 5.3.


We’re behind our original schedule, but still making some relatively good progress with a core peleton of people.  Given the participation, we’ve slowed down a tad in an effort to not lose anyone, particularly with a variety of schedules and time zones.

More Resources

Now that we’re into chapter 5 where the real fun has begun, people might also consider branching out to some of the alternative and more advanced texts. Some of the video resources are probably more germane to everyone now as well. In particular, I spent some time last week with Dr. Codrington’s youtube videos (1-6 of his “Lesson One”) which are generally excellent and which follow reasonably enough the presentation of Spivak that one won’t become lost.

Technical Update

Again, as a reminder for those who have had difficulties joining in the weekly live conversation, be sure to log into Google+ a few minutes before the start of the meeting.  Once it starts the moderators send out additional invitation reminders to join the conversation which should result in a pop up on your Google+ page inviting you into the live video/audio feed.

Alternately you can go to the Category Theory Study Group’s Google+ posts page where you should see a button with a camera icon on the particular week’s post that says “Join Hangout”. Clicking it should put you into the live conversation.

Another option should be to join from your events page. If you don’t see the hangout event there, make sure we have your G+ account so we can ensure you’re included in the invitation.

Morphisms and homomorphisms

In section 5.1.1 where category is defined the author says

B. for every pair x, y ∈ Ob(C), a set HomC(x,y)∈Set; it is called the hom-set from x to y; its elements are called morphisms from x to y;2

Footnote 2 says

The reason for the notation Hom and the word hom-set is that morphisms are often called homomorphisms, e.g., in group theory.

But morphisms and homomorphisms are different, right ? Morphism can be anything, homomorphisms are more restrictive. Is that correct ? For eg: there can be morphisms between 2 groups that are not homomorphisms, right ?

Graph indexing category –

In section author talks about graph-indexing category(GrIn) and symmetric graph-indexing category(SGrIn).

Ex. asks

How many functors are there of the form GrInSGrIn?

I did not understand the answer and explanation given in the textbook. Can anyone please explain ?

Question about Questionable Category (Remark Book)

In the remark author says:

A. there is a function U:Ob(Q)→Ob(C),

B. for all a,b∈Ob(Q), we have an injection U:HomQ(a,b)↪HomC(U(a),U(b)),

Why is the function and injection both called U ? Shouldn’t the injection be called something other than U ? This is more a question of naming but I just wanted to confirm.