Recently I gave a talk about the homotopy groups of spheres, and as usual, I try to collect my thoughts on this blog before (or after) presenting. The homotopy groups of spheres have featured several times on this blog, and we have made some effort into calculating them for some small dimensions. In the talk I wanted to showcase some methods used to calculate these groups, as well as doing some of the “calculations”....
How can we use travel-journaling to study some highly abstract and highly complicated mathematical machinery? Can we get anything out of such an analogy? In this post we do just that, study the homotopy hypothesis and infinity groupoids through the lens of a travel ledger.
This is part four in a sort of connected story about operations in mathematics that are associative up to homotopy. It will probably be beneficial to read part 1, part 2 and part 3 in advance of this, but it is not required in theory. These previous posts do however build up some intuition and motivation for the object we are looking at today. To quickly recap what we already seen in these previous posts we recall that we started out by looking at how to transfer a group structure on a topological space through an isomorphism....
In the two last posts we have been discussing operations that are associative up to homotopy, and where such operations might arise naturally in topology. One claim I made, which I later realized was maybe a bit unmotivated and in need of some clarification was how some higher arity maps actually defined (or were defined by) homotopies between combinations of the lower arity maps. We also purely looked at this in a topological setting, but in algebraic topology we often translate to algebraic structures, so I also wanted to see clearly that the same constructions hold in that setting....
Imagine we have a system of two topological spaces $f:T\longrightarrow G$. We are often interested in knowing if a certain property on the space $G$ can be transferred through f such that we have the same property on $T$. If f is a nice enough morphism an example could be a topological invariant of $G$, for example its Euler characteristic. In this post we are more interested in transferring other things than invariants, more specifically structures....
A couple weeks ago I held a talk on introductory higher category theory. Most of the talk was based upon thing we already have discussed on this blog, such as the strict $2$-category $Cat$, bicategories, and why strictness fails for the category of topological spaces. The inly thing I talked about which I haven’t yet featured on this blog is the notion of quasi-categories, so I though that I would do that today....
A litmus test is a question asked in politics to a potential candidate for high office in which the answer determines if the person gets nominated or not. If a person or a committee holds the power of nominating candidates, they can use that power to make sure that a potential candidate holds their view on a certain matter. So, what does this have to do with mathematics, or especially with homotopy theory?...
This is part 9 of a series leading up to and exploring model categories. For the other parts see the series overview. Last time we ended by giving a definition of a homotopy between maps on the collection of bifibrant objects in a model category. Today we are going to expand further upon this idea, and try to build the theory we are familiar with for topological spaces but in the general setting....
This is part 8 of a series leading up to and exploring model categories. For the other parts see the series overview. Last time we finally defined the model category, gave some examples and tried (kind of) to give a motivation to why they are interesting and how they set the stage for homotopy theory. The first time I read the definition I was a bit confused about the lack of mention of homotopy, or at least some prototype of it that I could connect with....
This is part 7 of a series leading up to and exploring model categories. For the other parts see the series overview. Finally we have made it to the destination we set, namely, more abstraction. This post is focused on the definition and intuition on model categories, which abstracts the objects we have been studying for some weeks, namely fibrations and cofibrations. The main definition is that of a model structure on a category, which together with a nice category will form the definition of a model category....