2010/10/21 by Tom Leinster, Leinster, Tom
Mathematics · #Category Theory (math.CT) #Dynamical Systems (math.DS) #FOS: Mathematics #General Topology (math.GN) #math.CT #math.DS #math.GN
paper · pdf · doi:10.48550/arxiv.1010.4474
81 pages. Supersedes arXiv:math/0411344 and arXiv:math/0411345. To appear in Advances in Mathematics. Version 2: tiny errors corrected
arxiv created 2010/11/09 · arxiv updated 2010/11/10
A little-known and highly economical characterization of the real interval [0, 1], essentially due to Freyd, states that the interval is homeomorphic to two copies of itself glued end to end, and, in a precise sense, is universal as such. Other familiar spaces have similar universal properties; for example, the topological simplices Deltan may be defined as the universal family of spaces admitting barycentric subdivision. We develop a general theory of such universal characterizations. This can also be regarded as a categorification of the theory of simultaneous linear equations. We study systems of equations in which the variables represent spaces and each space is equated to a gluing-together of the others. One seeks the universal family of spaces satisfying the equations. We answer all the basic questions about such systems, giving an explicit condition equivalent to the existence of a universal solution, and an explicit construction of it whenever it does exist.