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.
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? There is a question worth asking certain objects to check if they should be allowed to be a suitable “definition” for a certain nice structure. The question, or test, which we will look more closely at soon, first started as a conjecture by Grothendieck, named later “the homotopy hypothesis”. This conjecture is still open in the way formulated by Grothendieck, but it can be turned on its head to form this test instead. The reason this is possible is because of ambiguity in a certain definition in higher category theory, and because there is seemingly many inequivalent “definitions” for the same object. Before exploring any theory at all, the conjecture states that $\infty$-groupoids are equivalent to topological spaces. The litmus test then becomes; $X$ is considered a definition of $\infty$-groupoids if and only if all $X$s’ are equivalent to topological spaces. ...