2015/10/29 by Amit Kuber, Kuber, Amit, David Wilding +1
Mathematics · #06A06 #18A40 #54F05 #55Pxx #55Q05 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.CT #msc:06A06 #msc:18A40 #msc:54F05 #msc:55Pxx #msc:55Q05
paper · pdf · doi:10.48550/arxiv.1510.08921
26 pages
arxiv created 2015/10/29 · openalex publication_date 2015/10/29 · arxiv updated 2015/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given functors F,G:\mathcal C→\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→\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.