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Higher category theory is a bit of a strange subject. Unlike many other areas of mathematics, which focus on the implications of a set of definitions, higher category theory finds its most daunting task to be the very first definition. What is a higher category? The idea has taken shape from diverse and apparently disconnected applications, not yet fully forming into a single unified abstraction. A thousand and one attempts to define them have emerged over the decades, all proven to be ‘equivalent’, depending on how relaxed your idea of ‘equivalence’ is.

I’ve spoken recently about how arbitrary the unit interval is in the definition of homotopy. I’ve also tried to point out how replacing it with different topologies might give invariants suited to different classification problems. For instance, while $[0, 1]$ is best suited to classifying CW-complexes, the affine line $\mathbb{A}^1_k$ for a field $k$ is suitable for algebraic geometry in $k$. In order for the above statements to make sense generally however, a few issues need to be ironed out:

And we shouldn’t stand for it. But I’m getting ahead of myself. At present, all I’ve provided you is a vague-enough-to-be-deep title that really says nothing new at all. Of course homotopy is all about the unit interval. Read the definition. Let’s do just that, since it will lead us right to the reason why I wrote the first sentence of this post. Consider two topological spaces $X, Y \in \textbf{Top}$ and continuous maps $f, g : X \rightarrow Y$.

First post of many, hopefully! I’ve just got this site working, so with luck I’ll have some interesting math-related content (among other things) soon enough. See you soon!