Blog

The Canonical Model Structure on Cat/Gpd

When one first starts studying category theory, they learn that the correct notion of "the same" for categories is that of equivalence, rather than the stronger notion isomorphism. For those that have studied some algebraic topology, the situation is very similar to identifying spaces if they are homotopic, rather than homeomorphic. This is often given as an analogy, but the two notions can be reconciled by giving model structures where the weaker form of identification forms the subcategory of weak equivalences. Here, we define and present the so called canonical model structure on the category of categories. The proof of each step given can also be applied to the category of groupoids.

iPad Screen Sharing to Zoom on a University Network

I am currently a teaching assistant for a first year linear algebra course. We recently made the switch to a hybrid setup, meaning that I am streaming the tutorials on Zoom as well as talking to a class in-person. Instead of writing on the physical whiteboards in the classroom, I instead write on my iPad and share the screen through Zoom. Previously, when the tutorials were just based on Zoom, this setup worked pretty well - I can share my screen over Zoom using my home WiFi network. However, I can't screen share over the university network due to the way it is set up.

Higher category theory - Basic notions

Since my last post, I have moved from Scotland to Canada to start a PhD at The University of Western Ontario. The course started on September first, so I have been here just over a month. The main subject I am taking is higher category theory, taught by my supervisor Chris Kalpulkin. In an effort to cement my understanding on the subject, I will be writing occasional blog posts on what we cover and my thoughts.

Is zero a natural number?

I help tutor an introductory course on pure mathematics at Edinburgh University. One my responsibilities is to help answer questions that the students have in person and online on Piazza; a Q&A forum used by the class. The other day, a student created a poll asking whether or not zero is a natural number. I have found that this question usually always splits the audience, with a majority usually favouring "no" as the correct answer. I remember the first time I came across this debate was during my final year of high school, when our teacher described the "home of zero" as a controversial topic. At that time, for no legitimate reason, I chose to believe that it should be a natural number and this choice has more or less stuck with me till this day. Being a student of pure mathematics, it does strike me as an odd that this causes such a debate. Having the natural numbers with or without zero doesn't affect mathematics in the slightest; we can simply define a new set to be the naturals with or without zero.

A group object in the category of groups

I think most people are surprised when they first discover what a group object in the category of groups is; I know I certainly was! I won't spoil what it is for you just now, but we will discuss it and why it is what it is later on in this post, so if you want to figure it out for yourself - read no further!