Taxotopy Theory of Posets I: van Kampen Theorems

Abstract

Given functors $F,G:\mathcal C\to\mathcal D$ between small categories, when is it possible to say that $F$ can be "continuously deformed" into $G$ in a manner that is not necessarily reversible? In an attempt to answer this question in purely category-theoretic language, we use adjunctions to define a `taxotopy' preorder $\preceq$ on the set of functors $\mathcal C\to\mathcal D$, and combine this data into a `fundamental poset' $(Λ(\mathcal C,\mathcal D),\preceq)$. The main objects of study in this paper are the fundamental posets $Λ(\mathbf 1,P)$ and $Λ(\mathbb Z,P)$ for a poset $P$, where $\mathbf 1$ is the singleton poset and $\mathbb Z$ is the ordered set of integers; they encode the data about taxotopy of points and chains of $P$ respectively. Borrowing intuition from homotopy theory, we show that a suitable cone construction produces `null-taxotopic' posets and prove two forms of van Kampen theorem for computing fundamental posets via covers of posets.