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Chris Grossack's Blog

Chris Grossack's...
Life in Johnstone's Topological Topos 3 -- Bonus Axioms In the first post of the series, we talked about what the topological topos is, and how we can...
7 months ago
62
7 months ago
In the first post of the series, we talked about what the topological topos is, and how we can think about its objects (and, importantly, how we can relate computations in the topos $\mathcal{T}$ to computations with topological spaces in “the real world”). In part two, we...
Chris Grossack's...
Talk -- What is Factorization Homology? I was recently invited to speak at the AMS Sectional in Tallahassee, Florida. In particular, at the...
11 months ago
57
11 months ago
I was recently invited to speak at the AMS Sectional in Tallahassee, Florida. In particular, at the special session on Homotopy Theory and Category Theory in Interaction. The conference was this weekend, and I’m typing this up on my plane ride home. I had a great time, and...
Chris Grossack's...
Finiteness in Sheaf Topoi The notion of “finiteness” is constructively subtle in ways that can be tricky for people new to...
6 months ago
57
6 months ago
The notion of “finiteness” is constructively subtle in ways that can be tricky for people new to the subject to understand. For a while now I’ve wanted to figure out what’s going on with the different versions of “finite” in a way that felt concrete and obvious (I mentioned...
Chris Grossack's...
$\mathsf{B}\text{Diff}(\Sigma)$ Classifies $\Sigma$-bundles I’ve been trying to learn all about topological (quantum) field theories, the cobordism hypothesis,...
2 months ago
50
2 months ago
I’ve been trying to learn all about topological (quantum) field theories, the cobordism hypothesis, and how to use $(\infty,n)$-categories. This is all in service of some stuff I’m doing with skein algebras (which are part of a “$3+1$ TQFT” often named after Crane–Yetter, but...
Chris Grossack's...
Life in Johnstone's Topological Topos 2 -- Topological Algebras In the first post, we introduced Johnstone’s topological topos $\mathcal{T}$ and talked about what...
7 months ago
50
7 months ago
In the first post, we introduced Johnstone’s topological topos $\mathcal{T}$ and talked about what its objects look like. We showed how the interpretation of type theory in $\mathcal{T}$ gives us an “intrinsic topology” on any type we construct. We also alluded to the fact...
Chris Grossack's...
Life in Johnstone's Topological Topos 1 -- Fundamentals I’ve been thinking a lot about the internal logic of topoi again, and I want to have more examples...
7 months ago
41
7 months ago
I’ve been thinking a lot about the internal logic of topoi again, and I want to have more examples of topoi that I understand well enough to externalize some statements. There’s more to life than just a localic $\mathsf{Sh}(B)$, and since I’m starting to feel like I understand...
Chris Grossack's...
Proving Another "Real Theorem" with Topos Theory Another day, another post that starts with “So I was on mse…”, lol. Somebody asked whether...
11 months ago
38
11 months ago
Another day, another post that starts with “So I was on mse…”, lol. Somebody asked whether maximizing over a compact set is a continuous thing to do. That is, given a continuous function $f : K \times X \to \mathbb{R}$ is the function $x \mapsto \max_{k \in K} f(k,x)$...
Chris Grossack's...
Internal Group Actions as Enriched Functors Earlier today this month on the Category Theory Zulip, Bernd Losert asked an extremely natural...
a year ago
32
a year ago
Earlier today this month on the Category Theory Zulip, Bernd Losert asked an extremely natural question about how we might study topological group actions via the functorial approach beloved by category theorists. The usual story is to treat a group $G$ as a one-object...
Chris Grossack's...
Where Do Those Undergraduate Divisibility Problems Come From? Oftentimes in your “intro to proofs” class or your first “discrete math” class or something...
a month ago
27
a month ago
Oftentimes in your “intro to proofs” class or your first “discrete math” class or something similar, you’ll be shown problems of the form “prove that for $n^6 + n^3 + 2n^2 + 2n$ is a multiple of $6$ for every $n$”… But where do these problems come from? And have you ever...
Chris Grossack's...
Talk - Where Are The Open Sets? I was invited to give a talk at HoTTEST 2022, and was more than happy to accept! Ever since I was...
over a year ago
20
over a year ago
I was invited to give a talk at HoTTEST 2022, and was more than happy to accept! Ever since I was first learning HoTT I was curious how we could be sure that theorems in HoTT give us corresponding theorems in “classical” homotopy theory. Earlier this summer I spent a lot of...
Chris Grossack's...
Talk -- What is Algebraic Geometry and Why Should You Care? So an embarrassing amount of time ago (Feburary 17?) I gave a talk for the undergraduate math club...
a year ago
17
a year ago
So an embarrassing amount of time ago (Feburary 17?) I gave a talk for the undergraduate math club titled “What is Algebraic Geometry, and Why Should You Care?”. I think it went quite well, and the audience seemed like they had a good time. I really wanted to have the talk...
Chris Grossack's...
Externalizing Some Simple Topos Statements Hey all! It’s been a minute. I’ve been super busy with the UC strike and honestly I haven’t done...
over a year ago
13
over a year ago
Hey all! It’s been a minute. I’ve been super busy with the UC strike and honestly I haven’t done math in any serious capacity for almost the past month. It’s been a lot of hard work trying to get fair contracts out of the UC, but I had a lot of travel plans this December to...
Chris Grossack's...
Talk -- 2-Categorical Descent and (Essentially) Algebraic Theories A few weeks ago I gave a talk at the CT Octoberfest 2023 about some work I did over the summer that...
a year ago
12
a year ago
A few weeks ago I gave a talk at the CT Octoberfest 2023 about some work I did over the summer that I’m really proud of. Unfortunately, while writing up the result I found a 1999 paper by Pedicchio and Wood that proves the same theorem (with roughly the same proof), so I...
Chris Grossack's...
Building Objects with Category Theory Typically category theory is useful for showing the uniqueness of certain objects by checking that...
over a year ago
12
over a year ago
Typically category theory is useful for showing the uniqueness of certain objects by checking that they satisfy some universal property. This makes them unique up to unique isomorphism. However, category theory can also be used in order to show that objects exist at all, usually...
Chris Grossack's...
Estimating a Difference of Products Wow, it’s been a long time! Both since my last blog post, and since my last quick analysis trick....
a year ago
12
a year ago
Wow, it’s been a long time! Both since my last blog post, and since my last quick analysis trick. But I’ve been itching to write more blog posts lately, and I thought that something quick and easy like this would be a good way to get back into it without the kind of effort...
Chris Grossack's...
A Quick Application of Model Categories Almost exactly a year ago (time flies!) I was thinking really hard about model categories in...
a year ago
12
a year ago
Almost exactly a year ago (time flies!) I was thinking really hard about model categories in preparation for the HoTTEST summer school. I learned a TON doing this, but I’ve just today seen a really nice (and somewhat concrete) reason to care about the whole endeavor! I’d love...
Chris Grossack's...
Preprint -- The RAAG Functor as a Categorical Embedding After almost a year of sitting on my hard drive, I finally had time in August to finish revising my...
a year ago
12
a year ago
After almost a year of sitting on my hard drive, I finally had time in August to finish revising my new preprint on Right Angled Artin Groups (Raags). And in September I had time to put it on the arxiv for people to see! Within 24 hours I had an email from somebody who had...
Chris Grossack's...
A truly incredible fact about the number 37 So I was on math stackexchange the other day, and I saw a cute post looking for a book which lists,...
a year ago
11
a year ago
So I was on math stackexchange the other day, and I saw a cute post looking for a book which lists, for many many integers, facts that Ramanujan could have told Hardy if he’d taken a cab other than 1729. A few days ago OP answered their own question, saying that the book in...
Chris Grossack's...
Monoidal Monoidoidoids So I was on the nlab the other day, and I saw a fantastic joke: A 2-category is “just” a monoidal...
over a year ago
10
over a year ago
So I was on the nlab the other day, and I saw a fantastic joke: A 2-category is “just” a monoidal monoidoidoid! Here’s a screenshot in case the nlab page for 2-categories changes someday: There’s a thing called the Category Theorist’s “Just”, which describes the joy that many...