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 $⪯$ on the set of functors $\mathcal{C}\to \mathcal{D}$, and combine this data into a `fundamental poset' $(\Lambda (\mathcal{C},\mathcal{D}),⪯)$. The main objects of study in this paper are the fundamental posets $\Lambda (1,P)$ and $\Lambda (\mathbb{Z},P)$ for a poset $P$, where $1$ 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.

26 pages

References in corpus (1)

• Simplicial complexes with lattice structures