TL;DR
This paper explores the phenomenon of unexpected niceness in mathematical objects, focusing on examples like polynomial rings, Witt vectors, and classifying space cohomology, highlighting how solutions to universal problems often acquire additional structure.
Contribution
It provides a survey of various instances of niceness in mathematics, emphasizing the role of universal problems and the extra structures they tend to develop.
Findings
Examples of niceness in polynomial rings, Witt vectors, and classifying spaces.
Universal problems often lead to solutions with additional structure.
Highlights the prevalence of niceness phenomena in different mathematical contexts.
Abstract
Many things in mathematics seem lamost unreasonably nice. This includes objects, counterexamples, proofs. In this preprint I discuss many examples of this phenomenon with emphasis on the ring of polynomials in a countably infinite number of variables in its many incarnations such as the representing object of the Witt vectors, the direct sum of the rings of representations of the symmetric groups, the free lambda ring on one generator, the homology and cohomology of the classifying space BU, ... . In addition attention is paid to the phenomenon that solutions to universal problems (adjoint functors) tend to pick up extra structure.
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Taxonomy
TopicsLogic, Reasoning, and Knowledge