I really want to start writing more on this blog, and now that I'm fully procrastinating on my classes I have a great time to do it! I also want to review some basic algebraic topology stuff before I take another class in it next term. The style I'm emulating is of the [Unapologetic Mathematician](https://unapologetic.wordpress.com), who is a blogger I really like. The audience I'm writing for is... hard to define, but also the people who would read about algebraic topology on a casual blog are a small enough group that I can probably write the way I want to. I'm going to take a very casual stroll through the beginnings of algebraic topology, and talk about interesting things as I see them. This means that though I have an overall plan of things to write about, I will meander through topics and present them in not-necessarily-the-best order. I think of this as the "things I wish I knew before taking algebraic topology," because really, these things don't fit into the point where algebraic topology really kicks off (i.e. Homology and Cohomology). But before I even get to that, I'll talk about some random free group stuff (which we will need later, but which is also interesting in its own right), and also use this as an excuse to talk about some basic category theory stuff. In topology class one day, one of my friends asked the professor "What's the use of the fundamental group?" In his view, the homology groups and related constructions show up in math a lot more often than the fundamental group. This might be true, but the fundamental group lets you talk in a simple way about many really interesting things, and that is exactly what this series of posts will be about. In a sense, my whole plan is to say as much about the fundamental group as I can Thanks for reading! Next: [[Free groups]] Latest: [[Not really an intro to categorical language]] Contents: 1. [[Free groups]] - [[Minimal presentations of groups and the Frattini subgroup]] 2. [[Quotients of free groups]] 3. [[The universal property of the free group]] 4. [[A brief intro to Cayley graphs]] 5. [[The free product of groups]] 6. [[Not really an intro to categorical language]]- 7. [[A pretty bad introduction to limits]] 8. [[Free products with amalgamation]] 9. [[Coproducts in the category of abelian groups]] 10. [[Consequences of adjoint functors preserving limits]] 11. [[A worked example - the abelianization functor]] 12. [[Subgroups of free groups, Part I]] 13. [[F_2 and the Banach-Tarski paradox]] 14. [[Loops, holes, and wrapping-around]]