Taxotopy Theory of Posets I: van Kampen Theorems

Abstract

Given functors F, G ∶ C → 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 C → D, and combine this data into a 'fundamental poset' (Λ(C, D), ⪯).The main objects of study in this paper are the fundamental posets Λ(1, P ) and Λ(Z, P ) for a poset P , where 1 is the singleton poset and 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.