## Homotopy in model categories

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....

June 7, 2020 · 8 min · Torgeir Aambø

## Model categories

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....

June 6, 2020 · 8 min · Torgeir Aambø

## A homotopy group of a sphere

This is part 5 of a series leading up to and exploring model categories. For the other parts see the series overview. As promised in the previous part, we are going to calculate $\pi_4(S^3)$. I think we will have to use all of the machinery (plus some new) that we have been through during this series to do the calculation. What more could we possibly need you ask? Last time we developed the machinery to calculate the cohomology of the total space of a fibration, but we want to compute homotopy....

May 28, 2020 · 13 min · Torgeir Aambø

## The Serre spectral sequence

This is part 4 of a series leading up to and exploring model categories. For the other parts see the series overview. My personal favorite part about fibrations is that they come equipped with a natural way to compute the cohomology of the total space from the cohomology of the base and the cohomology of the fibers. This process is encoded in a structure called a spectral sequence, and is a complicated object in its full generality....

May 27, 2020 · 9 min · Torgeir Aambø

## Homotopy groups

This is part 3 of a series leading up to and exploring model categories. For the other parts see the series overview. For an introduction to the material, the definitions, motivation and some examples, please read part 1 and part 2 about fibrations and fiber bundles. This and the the following parts of this series will be about their usefulness, especially in computing homology and homotopy groups. This will be done through two different techniques, namely the long exact sequence of homotopy groups, and the spectral sequence associated to a fibration....

May 16, 2020 · 7 min · Torgeir Aambø

## Fiber bundles

This is part 2 of a series leading up to and exploring model categories. For the other parts see the series overview. Yesterday we discussed the standard definition of a fibration by the homotopy lifting property, and today we are continuing that discussion, but in a more visual manner. This we will do by first looking at fiber bundles, and then generalizing them. Since fibrations are generalized fiber bundles, every fiber bundle is an example of a fibration, and they have been the most important examples for me, as they help me visualize and get intuition into the fibrations without having to really use the full generality of the definition....

May 15, 2020 · 4 min · Torgeir Aambø

## Fibrations

This is part 1 of a series leading up to and exploring model categories. For the other parts see the series overview. My main mathematical interest for the last couple years has been algebraic topology. I feel it suits my needs for intuition, and graphical picturing of what happens. A concept I have been learning more rigorously recently is fibrations, and how to use them in computing homotopy groups and homology groups of different spaces....

May 14, 2020 · 4 min · Torgeir Aambø