19 February 2018:
I don't know much that was completed this week, but:
1) I sent the math department chair, Poon, an email requesting funds for 3 grad students to attend Applied Category Theory 2018. I'm estimating $1000 for air fare and $1300 fo…
9 February 2018:
This week's progress:
1) Adam Yassine was accepted to a Mathematical Sciences Research Institute workshop called "From Symplectic Geometry to Chaos". It runs from July 23rd to August 3rd this summer. This is great because he's w…
1 February 2018:
1) Today this paper by Daniel Cicala was published in TAC!
Daniel Cicala, Spans of cospans.
Abstract. We study spans of cospans in a category C and explain how to horizontally and vertically compose these. When C is a topos …
24 January 2018:
After a slow stretch and some bad news, progress proceeds:
1) Nina Otter got not just one but two postdoc positions!
She got a 3-year offer at UCLA, and also an offer for a postdoc at the Max Planck Institute in Leipzig, Germany.…
Hi, folks! I'm teaching a course on category theory at UCR, and there are some hand-written notes here:
Category theory course.
I'm not sure it's possible to follow the course from these notes; there's a lot that I say out loud, and even some …
20 January 2018:
I've got some good news and some bad news. First for the bad news:
1) My NSF proposal for getting money to send students to the Applied Category Theory conference in Leiden was rejected. So, right now, I don't know how to get…
6 January 2018:
Here is this week's progress, as far as I know:
1) Brandon will be doing a job interview for a visiting position at Colorado College! The interview will be at the Joint Mathematics Meetings in San Diego next week.
2) John Foley …
Jesus: if you want more on (slightly) higher categories, try these papers by students of mine. That's what we're mainly working on these days! I have carefully edited all these papers:
Kenny Courser, A bicategory
of decorated cospans, Theory and…
Yes, Peter, the course is over. I don't think "temporarily" is a good adverb to insert here. Sorry!
However, the book by Fong and Spivak is still there for you to read, along with Tai-Danae Bradley's excellent short book. I also have zillions of…
Thanks, folks!
I am now teaching an advanced course in category theory at UCR; I'm trying to get someone to take really good readable notes and make those public, but it hasn't happened quite yet.
I will definitely keep updating "This Week's Progr…
Keith - a \(\mathbf{Bool}\)-profunctor \(\Phi \colon \mathcal{X} \to \mathcal{X}\) is a kind of relation from a preorder \(\mathcal{X}\) to itself, namely a feasibility relation. But we can look at the set of elements of \(\mathcal{X}\) that are r…
Christopher wrote:
Okay, one thing I keep noticing is that in these "richer" "spaces" (especially in CS applications) its often much more natural to define division then subtraction.
Yes, this is why people invented fractions long before they …
Thanks, folks! I may add Daniel's "No Pain, No Gain" threads, and others, up to the "officially announced" section of the discussions to make them easier to find.
There's a lot more pain, and gain, available from the readings I just listed. Tai-…
Igor wrote:
How do \(k\)-valued matrices act on finite sets?
They don't, nor need they. In general, morphisms don't act on objects. You just need to know what a morphism from one object to another is, and how to compose them.
To a modern …
31 December 2018:
Happy New Year!
Here is this week's progress, as far as I know:
1) Today Kenny and Daniel submitted a corrected version of their paper Spans of cospans in a topos to TAC. With luck it'll be published soon!
2) I finished the pa…
20 December 2017:
Here is this week's progress, as far as I know:
1) I told you last time how Kenny and Daniel's paper was rejected from TAC based on a mistaken counterexample to their main theorem. They explained this to the editor and referee, …
9 December 2017:
Here is this week's progress:
1) Brandon will be doing another job interview, this time for a 2-year "teaching postdoc" at the University of Minnesota. Go, Brandon, go!
2) Daniel and Kenny got a referee's report on their paper S…
1 December 2017:
This week's progress, as far as I can remember:
1) Brandon Coya is going to have a skype interview for a tenure-track job at Providence College in Rhode Island! The interview is on December 11th.
He's a bit worried about being …
27 November 2017:
This week's progress:
1) A total of 77 students applied for the "adjoint school" of Applied Category Theory 2017. 16 were chosen, and I believe they include these lucky folks:
Daniel Cicala (part of the group working with …
Keith wrote:
I've never been in a Linear algebra, how does one calculate the dual of a vector space? And what is it used for?
Let me take the second question first. In a basic math class you'd call
[ f(x) = a x + b ]
a linear function, …
Michael wrote:
Is the identity morphism an isomorphism?
Yes. Let's see why! An isomorphism is a morphism \(f \colon x \to y\) that has a morphism \(g \colon y \to x\) for which
[ gf = 1_x \textrm{ and } fg = 1_y .]
We call \(g\) an…
Igor wrote in comment #5:
In the definition of tensor product in Wikipedia it is unclear what do they mean by \(e_i \otimes f_j\), where \(e_i\) and \(f_j\) are basis vectors. They define \(\otimes\) for vector spaces, while not specifying what …
Michael wrote in comment #11:
More precisely, let \(\mathbf{V}\) be a matrix in a vector space and \(\mathbf{V^\ast}\) be matrix in its dual space. Then:
[\mathbf{V^\ast} \cdot \mathbf{V} = \mathbf{I} = \mathbf{V} \cdot \mathbf{V^\ast}]
[\m…
Michael wrote in comment #23:
Is it true that \(\cup^\ast = \cap\)?
That's a nice question! The short answer is: "Literally speaking, no. But morally speaking, yes".
It looks like it should but the \(k\) in \(\cap_V \colon k \to V \otime…
Michael wrote in comment #22:
Puzzle 284. Using the cap and cup, any morphism \(f \colon x \to y \) in a compact closed category gives rise to a morphism from \(y^\ast\) to \(x^\ast\). This amounts to 'turning \(f\) around' in a certain sen…
Daniel wrote:
Another question I feel a bit ashamed to ask, but better late than never: Is it correct that
could be drawn as
Yes! Nothing to be ashamed about there; that's a very important point and I was just too lazy to c…
I've added some fun puzzles about things you can do in a compact closed category:
Puzzle 284. Using the cap and cup, any morphism \(f \colon x \to y \) in a compact closed category gives rise to a morphism from \(y^\ast\) to \(x^\ast\). This amou…